Basic Skills in Higher Mathematics Robert Glen Adviser in Mathematics Mathematics 1(H) Outcome 1

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line Straight lines

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC Index PC(a) Gradients and straight lines PC(b) Gradients and angles PC(c) Parallel and perpendicular Click on the one you want

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Index Click on the section you want 1 What is gradient? 2 The gradient of a line 3 The equation of a line given its gradient and the intercept on the y - axis 4 The equation of a line given one point on the line and the gradient 5 The equation of a line given two points on the line

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 1 1 What is gradient?

Mathematics 1(Higher) 1.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2m 3m The gradient (slope) of this roof is 2m 3m = 2 1 What is gradient? 3

Mathematics 1(Higher) 1.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m3m 3m 3m3m 3m3m = 1 The gradient (slope) of this roof is 1 What is gradient?

Mathematics 1(Higher) 1.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 7m3m 7m = The gradient (slope) of this roof is What is gradient?

Mathematics 1(Higher) 1.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 7m 1 What is gradient? 2m 3m = = 3 7 = 2 3 Gradient 1 Check this: The steeper the slope, the greater the gradient.

Mathematics 1(Higher) 1.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 4m 5 4 What is the gradient of this roof ? 5m AB DC What is gradient?

Mathematics 1(Higher) 1.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 4m 5 4 What is the gradient of this roof ? 5m AB DC Click on the letter of the correct answer 1 What is gradient?

Mathematics 1(Higher) 1.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 4m 5 4 What is the gradient of this roof ? 5m AB DC Sorry, wrong answer Have another go! Gradient = vertical horizontal 1 What is gradient?

Mathematics 1(Higher) 1.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 4m 5 4 What is the gradient of this roof ? 5m AB DC Click on the letter of the correct answer 1 What is gradient?

Mathematics 1(Higher) 1.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3m 4m 5 4 What is the gradient of this roof ? 5m AB DC Correct! 1 What is gradient? End of Section 1

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 2 2 The gradient of a line

Mathematics 1(Higher) 2.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line  Read all lines from left to right  Line AB is uphill from left to right  Line AB has a positive gradientm AB  0 A B y x

Mathematics 1(Higher) 2.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line  Read all lines from left to right  Line PQ is downhill from left to right  Line PQ has a negative gradientm PQ  0 A B P y Q x

Mathematics 1(Higher) 2.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line  Read all lines from left to right  Line PQ has a negative gradientm PQ  0  Line AB has a positive gradientm AB  0 A B y P Q x

Mathematics 1(Higher) 2.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line A B Gradient = change in y change in x (9, 6) (0, 3) m AB = = 3 9 y x

Mathematics 1(Higher) 2.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line A B Gradient = change in y change in x (9, 6) (0, 3) m AB = = Note: we could have measured the gradient like this y x

Mathematics 1(Higher) 2.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line x Gradient = change in y change in x m PQ = = Q P (0, 7) (9, 1) y

Mathematics 1(Higher) 2.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line Gradient = change in y change in x m PQ = = Note: we could have measured the gradient like this P y (0, 7) Q(9, 1) x

Mathematics 1(Higher) 2.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line A B(9, 6) (0, 3) Gradient = change in y change in x m AB = = = y x

Mathematics 1(Higher) 2.10 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line Gradient = change in y change in x m PQ = = = - y P (0, 7) Q(9, 1) x

Mathematics 1(Higher) 2.11 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y x A formula to memorise B (x 2, y 2 ) A (x 1, y 1 ) m AB = y 2 - y 1 x 2 - x 1

Mathematics 1(Higher) 2.12 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y x A formula to memorise B (x 2, y 2 ) A (x 1, y 1 ) m AB = y 2 - y 1 x 2 - x 1

Mathematics 1(Higher) 2.13 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y x 1 Calculate the gradient of line AB B (6, 5 ) A (2, 3 ) m AB = y 2 - y 1 x 2 - x 1 = = 2424 = 1212 Did you get this answer?

Mathematics 1(Higher) 2.14 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y x 2 Calculate the gradient of line CD. D (6, 2) C (2, -1) m CD = y 2 - y 1 x 2 - x 1 = 2 - (-1) = 3434 Did you get this answer?

Mathematics 1(Higher) 2.15 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y x 3 Calculate the gradient of line EF. F (5, -1) E (-3, 3) m EF = y 2 - y 1 x 2 - x 1 = (-3) = -4 8 = End of Section 2 Did you get this answer?

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 3 3 The equation of a line given its gradient and the intercept on the y - axis

Mathematics 1(Higher) 3.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y x O (0, 3)  m = ½ (x, y)  K L Find the equation of line KL which has a gradient of ½ and passes through the point (0, 3). m KL = y - 3 x - 0 = 1212  y - 3 = ½ x  y = ½ x + 3 The equation of KL is y = ½ x + 3

Mathematics 1(Higher) 3.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y x O (0, 3)  m = ½ (x, y)  K L Find the equation of line KL which has a gradient of ½ and passes through the point (0, 3). The equation of KL is y = ½ x + 3 Formula: y = m x + c

Mathematics 1(Higher) 3.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y x O (0, c)  m (x, y)  K L The equation of line with gradient m and intercept c is: y = m x + c Memorise this

1 Find the equation of line PQ which has a gradient of -2 and passes through the point (0, 5). Mathematics 1(Higher) 3.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y x O (0, 5)  m = -2  P Q The equation of PQ is y = -2 x + 5 (x, y) Use the formula

Mathematics 1(Higher) 3.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y x O (0, -3)  m = ¾  E F 2 Find the equation of line EF which has a gradient of ¾ and passes through the point (0, -3). The equation of EF is y = ¾ x - 3 (x, y) Use the formula

Mathematics 1(Higher) 3.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept You should now do Section A1 questions on page 3 of the Basic Skills booklet. End of Section 3

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 4 4 The equation of a line given one point on the line and the gradient

Mathematics 1(Higher) 4.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (4, 3)   K L (x, y) O Find the equation of the line through the point (4, 3) with gradient 3. m KL = y - 3 x - 4 = 3  y - 3 =  y - 3 = 3x - 12  y = 3x The equation of KL is y = 3x - 9 m = 3 3(x - 4) - 9

Mathematics 1(Higher) 4.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (4, 3)   K L (x, y) O Find the equation of the line through the point (4, 3) with gradient 3. The equation of KL is y = 3x - 9 m = 3 Formula: y - b = m (x - a)

Mathematics 1(Higher) 4.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (a, b)   K L (x, y) O The equation of the line through the point (a, b) with gradient m is : m y - b = m (x - a) Memorise this

Mathematics 1(Higher) 4.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (-1, 2)   P Q (x, y) O 1 Find the equation of the line through the point (-1, 2) with gradient 2. The equation of PQ is y = 2 x + 4 m = 2 Use the formula

Mathematics 1(Higher) 4.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (-1, 2)   P Q (x, y) O 1 Find the equation of the line through the point (-1, 2) with gradient 2. The equation of PQ is y = 2 x + 4 m = 2 y - b = m (x - a) y - 2 = y - 2 = 2 (x + 1) y - 2 = 2 x + 2 y = 2 x (a, b) (x - (-1)) 2 + 4

2 Find the equation of the line through the point (6, -2) with gradient ½. Mathematics 1(Higher) 4.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (6, -2)   M N (x, y) O m = ½ Use the formula The equation of MN is 2y = x - 10

Mathematics 1(Higher) 4.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (6, -2)   M N (x, y)O 2 Find the equation of the line through the point (6, -2) with gradient ½. The equation of MN is 2y = x - 10 m = ½ y - b = m (x - a) y - (-2) = y + 2 = ½ (x - 6) 2y + 4 = 2y = x (a, b) or x - 2y - 10 = 0 Multiply both sides by 2 to clear the fraction. ½ (x - 6) x

Mathematics 1(Higher) 4.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (-1, 4)   R S (x, y) O 3 Find the equation of the line through the point (-1, 4) with gradient 2/3. The equation of RS is 3y = -2x + 10 m = -2/3 Use the formula

Mathematics 1(Higher) 4.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y x (-1, 4)   R S (x, y) O 3 Find the equation of the line through the point (-1, 4) with gradient 2/3. The equation of RS is 3y = -2 x + 10 m = -2/3 y - b = m (x - a) y - 4 = 3y - 12 = 3y = (a, b) or 2 x + 2y - 10 = 0 Multiply both sides by 3 to clear the fraction. -2/3 (x - (-1)) y- 4 = -2/3 (x + 1) -2(x + 1) -2 x+ 10

Mathematics 1(Higher) 4.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient You should now do Section A1 questions on page 3 of the Basic Skills booklet. End of Section 4 Mathematics 1(Higher) 4.10 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 5 5 The equation of a line given two points on the line

Mathematics 1(Higher) 5.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line Find the equation of the line joining the points A (3, 1) and B (6, 4). Step 1Calculate the gradient m AB = y 2 - y 1 x 2 - x 1 = = 3 = 1 Step 2 Calculate the equation y - b = m (x - a) y - 1 = y - 1 = x - 3 y = x - 2 Choose A (3, 1) as the point on the line. i.e. a = 3, b = 1 (You get exactly the same answer if you choose B.) y x (6, 4) A B (3, 1) O (a, b)   m = 1 1 (x - 3)

Mathematics 1(Higher) 5.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line The equation of CD is y = 2x Use the formula 1 Find the equation of the line joining the points C (1, 2) and D (5, 10). y x D O C (5, 10) (1, 2)   Answer coming up!

Mathematics 1(Higher) 5.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line Step 1Calculate the gradient m AB = y 2 - y 1 x 2 - x 1 = = 8484 = 2 Step 2 Calculate the equation y - b = m (x - a) y - 2 = y - 2 = 2 x - 2 y = 2 x Choose C (1, 2) as the point on the line. i.e. a = 1, b = 2 (You get exactly the same answer if you choose B.) (a, b) 1 Find the equation of the line joining the points C (1, 2) and D (5, 10). y x O (5, 10) (1, 2) D C   2 (x - 1)

Mathematics 1(Higher) 5.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line 2 Find the equation of the line joining the points G (-3, 1) and H (5, -3). x (5, -3) G H (-3, 1)   The equation of GH is 2y = - x - 1 Use the formula y Answer coming up!

Mathematics 1(Higher) 5.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line Step 1Calculate the gradient m GH = y 2 - y 1 x 2 - x 1 = (-3) = -4 8 = -½ Step 2 Calculate the equation y - b = m (x - a) y - 1 = 2y - 2 = 2y = - x Choose G (-3, 1) as the point on the line. i.e. a = -3, b = 1 (You get exactly the same answer if you choose H.) (a, b) or x + 2y +1 = 0 2 Find the equation of the line joining the points G (-3, 1) and H (5, -3). x   G H (5, -3) (-3, 1) y -½ (x - (-3)) - x

Mathematics 1(Higher) 5.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line Step 2 Calculate the equation y - b = m (x - a) y - 1 = -½(x - (-3)) 2y - 2 = - x - 3 2y = - x - 1 Multiply both sides by 2 to clear the fraction. A fuller explanation y - 1 = -½(x + 3) (a, b) 2 Find the equation of the line joining the points G (-3, 1) and H (5, -3). x (5, -3) G H   y

Mathematics 1(Higher) 5.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient m AB = y 2 - y 1 x 2 - x 1 y x (x 2, y 2 ) A (x 1, y 1 )   B y x m y = m x + c  (0, c) OO y x O y x O     y - b = m (x - a) (a, b) (x, y) (x 1, y 1 ) (x 2, y 2 ) m 1Calculate m m = y 2 - y 1 x 2 - x 1 2 y - b = m (x - a) (a, b) Summary

Mathematics 1(Higher) 5.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line You should now do Sections A2 and A3 on page 3 of the Basic Skills booklet. End of Section 5

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan     Gradients and angles

Mathematics 1(Higher) 1.1 Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan  y x A B O p q m AB = pqpq    = tan  

Mathematics 1(Higher) 1.2 Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan  y x C D O m CD = 35  = 0.70 (to 2 dp) tan 35 

Mathematics 1(Higher) 1.3 Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan  y x E F O m EF = 35  = (to 2 dp) tan 145  Line EF is downhill, so its gradient is not tan 35 . 145  Always take the angle between the line and the positive direction of the x-axis.

Mathematics 1(Higher) 1.4 Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan  m GH = = 0.53 (to 2 dp) tan 28  1 What is the gradient of the line GH (to 2 dp)? x 28  G H y O

Mathematics 1(Higher) 1.5 Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan  y x K L O m KL = 48  = (to 2 dp) tan 132  132  2 What is the gradient of the line KL (to 2 dp)?

Mathematics 1(Higher) 1.6 Outcome 1 Use the properties of the straight line PC(b) Find the gradient of a straight line using m = tan  You should now do the questions on page 7 of the Basic Skills booklet. End of PC(b)

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line

Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Index Click on the section you want 1 Parallel lines 2 Perpendicular lines 3 Equations

Mathematics 1(Higher) 1.1 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Section 1 1 Parallel lines

These lines are all parallel to each other If one of the lines has a gradient m, they all have a gradient m. Mathematics 1(Higher) 1.2 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Parallel lines have equal gradients

Mathematics 1(Higher) 1.3 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line The line y = 2x + 10 has a gradient of 2.2. So any line parallel to this one has a gradient of 2. y = 2x + 10 y = 2x + 5 y = 2x y = 2x - 5 y = 2x - 10 x y The line 2x - y + 5 = 0 also belongs to this set of parallel lines. Can you see why? 2x - y + 5 = 0 2x 2x + 5 = y  y =

Mathematics 1(Higher) 1.4 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? y = 3x - 1y = -3x + 3 y = 3x 3x + y = 33x - y = 3 AC Click on the letter of a correct answer NB There could be more than one right answer. B DE

Mathematics 1(Higher) 1.5 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? y = 3x - 1 NB There could be more than one right answer. Correct! This line has a gradient of 3. Have another go! A

Mathematics 1(Higher) 1.6 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? NB There could be more than one right answer. Have another go! Wrong! This line has a gradient of -3. y = -3x + 3 B

Mathematics 1(Higher) 1.7 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? y = 3x NB There could be more than one right answer. Correct! This line has a gradient of 3. Have another go! C

Mathematics 1(Higher) 1.8 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? NB There could be more than one right answer. 3x + y = 3 Wrong! This line has a gradient of -3. Have another go! y = -3x +3 D Click here to see all the answers

Mathematics 1(Higher) 1.9 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? Correct! This line has a gradient of 3. Have another go! Click here to see all the answers y = 3x +3 3x - y = 3 E

Mathematics 1(Higher) 1.10 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 1 Which of the following lines is/ are parallel to the line y = 3x - 5? Parallel to y = 3 x - 5 Not parallel to y = 3 x - 5 Key y = -3x +3 y = 3x - 1y = -3x + 3 y = 3x 3x + y = 33x - y = 3 ACB DE y = 3x +3

Mathematics 1(Higher) 1.11 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 2 Which of the following lines is/ are parallel to the line x + y = 8? y = x + 5y = - x + 1 y = x x + y = 10x - y = 7 Click on the letter of a correct answer NB There could be more than one right answer. A D B E C

Mathematics 1(Higher) 1.12 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 2 Which of the following lines is/ are parallel to the line x + y = 8? NB There could be more than one right answer. Wrong! This line has a gradient of +1. Have another go y = x + 5 A

Mathematics 1(Higher) 1.13 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 2 Which of the following lines is/ are parallel to the line x + y = 8? y = - x + 1 Click on the letter of a correct answer NB There could be more than one right answer. Correct! This line has a gradient of -1. Have another go B

Mathematics 1(Higher) 1.14 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 2 Which of the following lines is/ are parallel to the line x + y = 8? Click on the letter of a correct answer NB There could be more than one right answer. Wrong! This line has a gradient of +1. Have another go y = x C

Mathematics 1(Higher) 1.15 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 2 Which of the following lines is/ are parallel to the line x + y = 8? NB There could be more than one right answer. Correct! This line has a gradient of -1. Have another go y = -x +10 x + y = 10 D Click here to see all the answers

Mathematics 1(Higher) 1.16 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 2 Which of the following lines is/ are parallel to the line x + y = 8? Wrong! This line has a gradient of +1. Have another go Click here to see all the answers y = x - 7 x - y = 7 E

Mathematics 1(Higher) 1.17 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Parallel to x + y = 8 Not parallel to x + y = 8 Key 2 Which of the following lines is/ are parallel to the line x + y = 8? y = -x +10 y = x + 5y = - x + 1 y = x x + y = 10x - y = 7 A D B E C y = x - 7

Mathematics 1(Higher) 1.18 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? y = 2x - 1 y = ½ x + 1 2y = x x - 2y = 4x - 2y + 7= 0 A D B E C

Mathematics 1(Higher) 1.19 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? Wrong! This line has a gradient of 2. Have another go y = 2x - 1 A

Mathematics 1(Higher) 1.20 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? Correct! This line has a gradient of ½. Have another go y = ½ x + 1 B

Mathematics 1(Higher) 1.21 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? Correct! This line has a gradient of ½. Have another go y = ½x 2y = x C

Mathematics 1(Higher) 1.22 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? Correct! This line has a gradient of ½. Have another go y = ½ x - 2 x - 2y = 4 D Click here to see all the answers

Mathematics 1(Higher) 1.23 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? Correct! This line has a gradient of ½. Click here to see all the answers Have another go y = ½ x + 3 ½ x - 2y + 7= 0 E

Mathematics 1(Higher) 1.24 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line 3 Which of the following lines is/ are parallel to the line y = ½ x - 3? Parallel to y = ½ x - 3 Not parallel to y = ½ x - 3 Key y =½x y = ½x - 2 y = 2x - 1 y = ½ x + 1 2y = x x - 2y = 4x - 2y + 7= 0 A D B E C y = ½ x + 3 ½

Mathematics 1(Higher) 1.25 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Continue with Section 2 Perpendicular lines End of Section 1

Mathematics 1(Higher) 2.1 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Section 2 2 Perpendicular lines

Mathematics 1(Higher) 2.2 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line x y A B C D m AB = 3232 CD is perpendicular to AB. m CD = m AB  m CD = 3232  = -1

Mathematics 1(Higher) 2.3 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line x y E F G H m EF = 3434 GH is perpendicular to EF. m GH = m EF  m GH = 3434  = -1

Mathematics 1(Higher) 2.4 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line x y P Q R S m PQ = 3131 RS is perpendicular to PQ. m RS = m PQ  m RS = 3131  = -1

Mathematics 1(Higher) 2.5 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line x y m1m1 m2m2 If two lines with gradients m1 m1 and m 2 are perpendicular then m 1 × m 2 = Memorise this

Mathematics 1(Higher) 2.8 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line If two lines with gradients m1 m1 and m 2 are perpendicular then m 1 × m 2 = Parallel lines have equal gradients. Summary m m m m m x y m1m1 m2m2 1 2

Mathematics 1(Higher) 2.6 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line x y 1 For each line write down the gradient of any line a parallel to the line b perpendicular to the line 1 Answers 1 ½, , 1/3 3 3/4, -4/3 4 -1/3, 3          2 3    4     Here are the answers

Mathematics 1(Higher) 2.7 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Answers 1 4, -¼ 2 ¾, -4/3 3 -5, 1/5 4 -1, 1 5 ½, /5, 5/3 1 y = 4x y = ¾ x x + 5y = 15 3 y = -5x 4 x + y = 15 5 x - 2y + 3 = 0 Here are the answers 2 For each line write down the gradient of any line a parallel to the line b perpendicular to the line

Mathematics 1(Higher) 2.9 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line You should now do Section C1 on page 11 of the Basic Skills booklet. End of Section 2

Mathematics 1(Higher) 3.1 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Section 3 3 Equations

Mathematics 1(Higher) 3.2 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line AB has equation y = 3x + 5. Find the equation of the line  parallel to AB through (1, -2)  perpendicular to AB through (1, -2) Parallel line m AB = 3 So m parallel = 3 Point on line is (1, -2) y - b = m (x - a)a) y - (-2) = 3(x - 1) y + 2 = 3x - 3 y = 3x - 5 Perpendicular line m AB = 3 So m perp = -1/3 Point on line is (1, -2) y - b = m (x - a)a) y - (-2) = -1/3 (x - 1) 3y + 6 = - x + 3 x + 3y + 3 = 0 Click here for revision of finding equations of straight lines

Mathematics 1(Higher) 3.3 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line Find the equation of the line: 1 Through (0, 3), parallel to y = 2x +1 2 Through (1, 5), perp to y = ¼ x Through (-2, 2), parallel to x + y = 10 4 Through (5, -3), perp to y = -½ x +7 5 Through (3, -1), parallel to 2x + 3y + 5 =0 Answers 1 y = 2x 2x +3 2 y = -4x y = -x-x 4 y = 2x 2x x 3x + 2y 2y -11 = 0

Mathematics 1(Higher) 3.4 Outcome 1 Use the properties of the straight line PC(c) Find the equation of a line parallel to and perpendicular to a line You should now do Sections C2 and C3 on page 11 of the Basic Skills booklet. End of PC(c)