Finding equations of lines How many lines have a gradient of 2? What do we know about the equation of these lines? How many of these will pass thru (1,3)?

What we already know The equation of a line with a gradient of 2 and y intercept of -3 is y = 2x + 3 The equation of a line with a gradient of m and a y-intercept of b is y = mx + b

What is the equation of the line with a gradient of 2 and passing thru (1,3) ? y = 2x + 3

Here is an algebraic approach to this question What is the equation of the line with a gradient of 2 and passing thru (1,3) ? We know that the equation is of the form y = mx + b and we know m = 2  y = 2x + b substitute (1,3) for x and y  3 = 2(1) + b, now solve for b  b = 3 – 2 = 1  The equation of the line is y = 2x + 1

Using the algebraic method... What is the equation of the line with a gradient of 3 and passing thru (-2,2)? We know the equation is of the form y = mx + b and we know m = 3  y = 3x + b substitute x=-2, y=2  2 = 3(-2) + b  b = = 8  The equation of the line is y = 3x + 8

What have we learnt so far.. Given the gradient of a line and the y intercept you can find the equation by..... Given the gradient of a line and a point other than the y-intercept you can find the equation by.... What could we do if we don’t know the gradient? Answer: Find the gradient!

How can we find the gradient of the line joining 2 points? rise run

Find the gradient of the line joining: 1. (1,2) and (4,5) 2. (1,2) and (4,4) 3. (1,2) and (1,5) 4. (1,2) and (1,-1) 5. (1,2) and (-1,-3) Write a formula for finding the gradient of a line joining any 2 points, say (x 1, y 1 ) and (x 2, y 2 )

The formula for finding the gradient is..

Now back to finding the equation of the line... Use your formula you have just developed to find the gradient of the line joining (1,4) and (3,5) And hence find the equation of the line passing thru (1,4) and (3,5)? y= ½ x + b substitute x = 1, y = 4  4 = ½ (1) + b  b = 3 ½  y = ½ x + 3 ½ or

In the previous question, Would it have mattered if we had substituted (3,5) instead of (1,4)? m = ½ y= ½ x + b substitute x = 3, y = 5 (instead of x=1 y= 4)  5 = ½ (3) + b  b = 5 – 1 ½ = 3 ½  y = ½ x + 3 ½ or No, it doesn’t matter which point we substitute!!

Find the gradient and hence equation of the line joining (4,7) and (-1,2) Gradient y = 1x + b substitute x = -1, y = 2  2 = 1 (-1) + b  b = 3  y = x + 3

Check graphically: Draw the line joining (4,7) and (-1,2) and find it’s equation You should get y = x + 3 that we found algebraically

Explain a method for finding the equation of a line Given the gradient and y-intercept 2.Given the gradient and a point other than the y-intercept 3.Given 2 points

2 special types of lines What is the gradient of the line joining any 2 points that are horizontal ? What can you conclude about the equation of the line? y = mx + b must become y = 0x + b  y = b Horizontal lines are of the form: y = c

2 special types of lines What is the gradient of a line joining 2 points that are vertical? What can you conclude about the equation of such a line? y = mx + b ? Doesn’t apply as there is no y-intercept List 3 other points on your line What do all these points have in common? Vertical lines are of the form: x = c