1. GCSE: Straight Line Equations
Dr J Frost
Last modified: 3rd September 2014

2. GCSE specification:
Understand that an equation of the form y = mx + c corresponds to a straight line graph
Plot straight line graphs from their equations
Plot and draw a graph of an equation in the form y = mx + c
Find the gradient of a straight line graph
Find the gradient of a straight line graph from its equation
Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c)
Understand how the gradient of a real life graph relates to the relationship between the two variables
Understand how the gradients of parallel lines are related
Understand how the gradients of perpendicular lines are related
Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be -1/m
Generate equations of a line parallel or perpendicular to a straight line graph

3. 𝑥=2 What is the equation of this line?y
What is the equation of this line?
And more importantly, why is it that? 4 3 2 1 -1 -2 -3 -4 x
The line represents all points which satisfies the equation.
□ “Understand that an equation corresponds to a line graph.”
𝑥=2 ?

4. Starter A D F C B E G H What is the equation of each line? y 4 3 2 1x -1 H -2 -3 -4
What is the equation of each line?

5. Equation of a line
Understand that an equation of the form 𝑦=𝑚𝑥+𝑐 corresponds to a straight line graph
The equation of a straight line is 𝑦=𝒎𝑥+𝒄
gradient
y-intercept

6. Gradient using two points!
Given two points on a line, the gradient is:
𝑚= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
1, (3, 10)
𝑚=3 ?
𝑚=−2 ?
5, (8, 1)
𝑚=− 8 3 ?
2, (−1, 10)

7. Gradient from an Equation
Find the gradient of a straight line graph from its equation.
𝑦=1−2𝑥
Putting in form 𝒚=𝒎𝒙+𝒄:
𝒚=−𝟐𝒙+𝟏
Gradient is -2
2𝑥+3𝑦=4
Putting in form 𝒚=𝒎𝒙+𝒄:
𝟑𝒚=−𝟐𝒙+𝟒 𝐲=− 𝟐 𝟑 𝒙+ 𝟒 𝟑
Gradient is − 𝟐 𝟑 ? ?

8. Test Your Understanding
Find the gradient of the line with equation 𝑥−2𝑦=1.
𝟐𝒚=𝒙−𝟏 𝒚= 𝟏 𝟐 𝒙− 𝟏 𝟐 𝒎= 𝟏 𝟐 ?

9. Exercise 1
Determine the gradient of the lines which go through the following points.
3,5 , 5, 𝒎=𝟑 −1,0 , 4, 𝒎= 𝟑 𝟓 2,6 , 5,− 𝒎=−𝟑 4,7 , 8, 𝒎= 𝟑 𝟒 1,1 , −2, 𝒎=−𝟏 3,3 , 4, 𝒎=𝟎 4,−2 , 2,− 𝒎=𝟏 −3,4 , 4, 𝒎=− 𝟏 𝟕
Determine the gradient of the lines with the following equations:
𝑦=5𝑥− 𝒎=𝟓 𝑥+𝑦= 𝒎=−𝟏 𝑦−2𝑥= 𝒎=𝟐
𝑥−3𝑦= 𝒎= 𝟏 𝟑 2𝑥+4𝑦= 𝒎=− 𝟏 𝟐 2𝑦−𝑥= 𝒎= 𝟏 𝟐 2𝑥=3𝑦− 𝒎= 𝟐 𝟑
A line 𝑙 1 goes through the points (2,3) and 4,6 . Line 𝑙 2 has the equation 4𝑦−5𝑥=1. Which has the greater gradient:
𝒎 𝟏 = 𝟑 𝟐 𝒎 𝟐 = 𝟓 𝟒
So 𝒍 𝟏 has greater gradient. ? 1 d ? e ? a ? f ? b ? g c ? ? d 3 e ? f ? g ? ? h ? 2 a ? b ? c ?

10. Drawing Straight Linesy 4 3 2 1 -1 -2 -3 -4
Sketch the line with equation: 𝑥+2𝑦=4 x
□ “Plot and draw a graph of an equation in the form y = mx + c”
Bro Tip: To sketch a line, just work out any two points on the line. Then join up. Using 𝑥=0 for one point and 𝑦=0 for the other makes things easy.

11. Test Your Understanding4 3 2 1 -1 -2 -3 -4
Sketch the line with equation: 𝑥−3𝑦=3 x
□ “Plot and draw a graph of an equation in the form y = mx + c”

12. Finding intersection with the axis𝑦
When a line crosses the 𝑦-axis:
𝒙=𝟎
When a line crosses the 𝑥-axis:
𝒚=𝟎 ? ? 𝑥
The point where the line crosses the:
Equation
𝒚-axis
𝒙-axis
𝑦=3𝑥+1
0,1
− 1 3 ,0
𝑦=4𝑥−2
0,−2
1 2 ,0
𝑦= 1 2 𝑥−1
0,−1
2,0 ? ? ? ? ? ?

13. Equation given a gradient and point
The gradient of a line is 3. It goes through the point (4, 10). What is the equation of the line? ?
𝒚=𝟑𝒙−𝟐
Start with 𝒚=𝟑𝒙+𝒄 (where 𝒄 is to be determined)
Substituting: 𝟏𝟎= 𝟑×𝟒 +𝒄
Therefore 𝒄=−𝟐
The gradient of a line is -2. It goes through the point (5, 10). What is the equation of the line? ?
𝒚=−𝟐𝒙+𝟐𝟎

14. Test Your Understanding
Determine the equation of the line which has gradient 5 and goes through the point 7,10 .
𝒚=𝟓𝒙−𝟐𝟓
Determine the equation of the line which has gradient −2 and goes through the point 3,−2 .
𝒚=−𝟐𝒙+𝟒
Find the equation of the line which is parallel to 𝑦=− 1 2 𝑥+3 and goes through the point 6,1
𝒚=− 𝟏 𝟐 𝒙+𝟒 1 ? 2 ? 3 ?

15. Equation given two points
A straight line goes through the points (3, 6) and (5, 12). Determine the full equation of the line.
Gradient: 3
Equation: 𝒚=𝟑𝒙−𝟑 ?
(5,12) ?
(3,6)
A straight line goes through the points (5, -2) and (1, 0). Determine the full equation of the line.
(5, -2)
Gradient: -0.5
Equation: 𝒚=− 𝟏 𝟐 𝒙+ 𝟏 𝟐 ?
(1,0) ?

16. Exercise 2
Determine the points where the following lines cross the 𝑥 and 𝑦 axis.
𝑦=2𝑥 ,1 , − 1 2 ,0 𝑦=3𝑥− ,−2 , 2 3 ,0
2𝑦+𝑥= , 5 2 , 5,0
Using suitable axis, draw the line with equation 2𝑥+𝑦=5.
A line has gradient 8 and goes through the point 2,10 . Determine its equation.
𝒚=𝟖𝒙−𝟔
A line has gradient −3 and goes through the point 2,10 . Determine its equation.
𝒚=−𝟑𝒙+𝟏𝟔
Determine the equation of the line parallel to 𝑦=6𝑥−3 and goes through the point 3,10 .
𝒚=𝟔𝒙−𝟖
Determine the equation of the line parallel to 𝑦=− 1 3 𝑥+1 and goes through the point −9,5 .
𝒚=− 𝟏 𝟑 𝒙+𝟐
Determine the equation of the lines which go through the following pairs of points:
3,5 , 4, 𝒚=𝟐𝒙−𝟏 4,1 , 6, 𝒚=𝟑𝒙−𝟏𝟏 −2,3 , 4,−3 𝒚=−𝒙+𝟏 0,3 , 3, 𝒚= 𝟐 𝟑 𝒙+𝟑 4,−1 , 2,4 𝒚=− 𝟓 𝟐 𝒙+𝟗 1 5 ? ? ? 6 ? ? 2 7 ? 𝑦 5 ? ? 𝑥
5 2 ? ? 3 ? ? 4 ?

17. y 4 3 2 1 -1 -2 -3 -4
m = -1/3 ?
m = 1/2 ?
m = 3 ? x
m = -2 ?
Find the gradients of each pair of perpendicular lines. What do you notice?

18. Perpendicular Lines !
If two lines are perpendicular, then the gradient of one is the negative reciprocal of the other.
𝑚 1 =− 1 𝑚 2
Or:
𝑚 1 𝑚 2 =−1
Gradient
Gradient of Perpendicular Line 2
− 1 2 −3
1 3
1 4 -4 5
− 1 5
− 2 7
7 2
7 5
− 5 7 ? ? ? ? ? ?

19. Example Problems Q1
A line is goes through the point (9,10) and is perpendicular to another line with equation 𝑦=3𝑥+2. What is the equation of the line?
𝒚−𝟏𝟎=− 𝟏 𝟑 𝒙−𝟗
A line 𝐿 1 goes through the points 𝐴 1,3 and 𝐵 3,−1 . A second line 𝐿 2 is perpendicular to 𝐿 1 and passes through point B. Where does 𝐿 2 cross the x-axis?
𝟓,𝟎
Are the following lines parallel, perpendicular, or neither?
𝑦= 1 2 𝑥
2𝑥−𝑦+4=0
Neither. Gradients are 𝟏 𝟐 and 𝟐. But 𝟏 𝟐 ×𝟐=𝟏, not -1, so not perpendicular. ? Q2 ? Q3 ?

20. 𝒎 𝟏 =− 𝟑 𝟐 𝒎 𝟐 = 𝟐 𝟑 𝒎 𝟏 𝒎 𝟐 =−𝟏 so perpendicular.
Exercise 3
A line 𝑙 1 goes through the indicated point and is perpendicular to another line 𝑙 2 . Determine the equation of 𝑙 1 in each case.
2, 𝑙 2 :𝑦=2𝑥 𝒍 𝟏 :𝒚=− 𝟏 𝟐 𝒙+𝟔 −6,3 𝑙 2 :𝑦=3𝑥 𝒍 𝟏 :𝒚=− 𝟏 𝟑 𝒙+𝟏 0, 𝑙 2 :𝑦=− 1 2 𝑥−1 𝒍 𝟏 :𝒚=𝟐𝒙+𝟔 −9,0 𝑙 2 :𝑦=− 1 3 𝑥+1 𝒍 𝟏 :𝒚=𝟑𝒙+𝟐𝟕 10,10 𝑙 2 :𝑦=−5𝑥+5 𝒍 𝟏 :𝒚= 𝟏 𝟓 𝒙+𝟖
𝐴 2,5 𝐵 4,9
Find the equation of the line which passes through B, and is perpendicular to the line passing through both A and B.
𝒚=− 𝟏 𝟐 𝒙+𝟏𝟏
Line 𝑙 1 has the equation 2𝑦+3𝑥=4. Line 𝑙 2 goes through the points (2,5) and (5,7). Are the lines parallel, perpendicular, or neither?
𝒎 𝟏 =− 𝟑 𝟐 𝒎 𝟐 = 𝟐 𝟑 𝒎 𝟏 𝒎 𝟐 =−𝟏 so perpendicular. 4 1 𝑙
𝑦=3𝑥+5 ? 𝑥 ?
Determine the equation of the line 𝑙.
𝒚=− 𝟏 𝟑 𝒙+𝟓 ? ? ? 𝑦 5 ? 2
𝑦=−2𝑥+4 𝑙 𝑥 ?
Determine the equation of the line 𝑙.
Known point on 𝒍:
𝟐,𝟎
So equation of 𝒍:
𝒚= 𝟏 𝟐 𝒙−𝟏 3 ? ?

21. GCSE specification:
Understand that an equation of the form y = mx + c corresponds to a straight line graph
Plot straight line graphs from their equations
Plot and draw a graph of an equation in the form y = mx + c
Find the gradient of a straight line graph
Find the gradient of a straight line graph from its equation
Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c)
Understand how the gradient of a real life graph relates to the relationship between the two variables
Understand how the gradients of parallel lines are related
Understand how the gradients of perpendicular lines are related
Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be -1/m
Generate equations of a line parallel or perpendicular to a straight line graph

22. Two last things… ? 5 3 ? 4 Midpoint of two points
Distance between two points
(7,9) ? 5
5,9 3
(𝟒,𝟕.𝟓) ?
(3,6)
(3,6) 4
Just find the average of 𝑥 and the average of 𝑦.
Find 𝑥 change and 𝑦 change to form right-angled triangle.
Then use Pythagoras.

23. Past Exam Questions See GCSEPastPaper_Solutions.pptx
GCSERevision_StraightLineEquations.docx