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Straight line graphs

y = mx + c

y = mx + c

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y = mx + c

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Explainer Video

Tutor: Daniel

Summary

y=mx+cy=mx+cy=mx+c​

​​In a nutshell

y=mx+cy=mx+cy=mx+c is the equation for a straight line, where mmm is the gradient of the line and ccc is the yyy-axis intercept. mmm​ tells you how steep a line is by tracking the change in yyy​ over the change in xxx​. ccc​ tells you where the line intercepts the yyy-axis. This is a handy equation because it allows you to learn a lot about a line with only a pair of coordinates. Once you understand it intuitively, you can plot out lines without graph paper.



The gradient formula

A gradient tells you how steep a line is between two points (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​). It's calculated as: 


m=ΔyΔx=y2−y1x2−x1\boxed{m = \dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}}m=ΔxΔy​=x2​−x1​y2​−y1​​​


Intuitively, you can see that a greater change in yyy​ or xxx would lead to a steeper line on paper. Therefore, a higher mmm value means a greater gradient.

Maths; Straight line graphs; KS5 Year 12; y = mx + c

Three lines with the equations y=x+4y=x+4y=x+4, y=5x+4y=5x+4y=5x+4 and y=−5x+4y=-5x+4y=−5x+4.

Once you have the coordinates of a line, you can work out the gradient easily.


Procedure

1.

​​Assign coordinates to (x1,y1)(x_1,y_1)(x1​,y1​)​​  and (x2,y2)(x_2,y_2)(x2​,y2​).

2.

​​Plug those numbers into the formula for mmm.

3.

​​Solve for mmm.


Example 1

Find the gradient for a line between points (2,3)(2,3)(2,3) and (6,7)(6,7)(6,7).


Assign values for (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​).

 x1=2x_1 = 2x1​=2, y1=3y_1 = 3y1​=3, x2=6x_2= 6x2​=6 and y2=7y_2=7y2​=7


Substitute into the formula for mmm and solve.

m=y2−y1x2−x1m=7−36−2m=44\begin{aligned}m&=\dfrac{y_2-y_1}{x_2-x_1}\\\\m&= \dfrac{7-3}{6-2}\\\\m&= \dfrac{4}{4}\\\end{aligned}mmm​=x2​−x1​y2​−y1​​=6−27−3​=44​​​​

m=1‾\underline{m=1}m=1​​​



Working backwards

You can works backwards if you are given just the gradient of a line and some coordinates. This can let you find where a line terminates.


Example 2

A line has coordinates (4,8)(4,8)(4,8) and (−1,a)(-1, a)(−1,a). The gradient of the line is −2.5-2.5−2.5​. What is the value of coordinate a?


Assign values to (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​)

​(4,8)(−1,a)(x1,y1)(x2,y2)x1=4,x2=−1,y1=8,y2=am=−2.5\begin{aligned}(4,8)&(-1,a)\\(x_1,y_1)&(x_2, y_2)\\x_1 = 4, x_2 &= -1, y_1=8, y_2=a\\m&=-2.5\\\end{aligned}(4,8)(x1​,y1​)x1​=4,x2​m​(−1,a)(x2​,y2​)=−1,y1​=8,y2​=a=−2.5​​​

Substitute into the formula for mmm

​​

 m=y2−y1x2−x1−2.5=a−8−1−4\begin{aligned}m&=\dfrac{y_2-y_1}{x_2-x_1}\\\\-2.5 &=\dfrac{a-8}{-1-4}\\\end{aligned}m−2.5​=x2​−x1​y2​−y1​​=−1−4a−8​​

Solve for aaa​


a−8−5=−2.5a=12.5+8a=20.5‾\begin{aligned}\dfrac{a-8}{-5}&=-2.5\\a&= 12.5 + 8 \\&\underline{a=20.5}\\\end{aligned}−5a−8​a​=−2.5=12.5+8a=20.5​​



The y-intercept, c

​ccc is known as the yyy​-intercept. It is the point where the line has an xxx​ value of zero. Visually, this is the point where the line hits the yyy​-axis, as the yyy​-axis lies where x=0x=0x=0​. To find the intercept from a complete line equation, all you need to do is make x=0x=0x=0​, i.e, remove the mmm​ and xxx​ from the equation.

Maths; Straight line graphs; KS5 Year 12; y = mx + c

Three lines with equations y=x+4y=x+4y=x+4, y=x+2y=x+2y=x+2 and y=x−4y=x-4y=x−4.

​

The xxx-intercept is needed less often. It is simply where the line hits the xxx​-axis. To find it, you make y=0y=0y=0​ and solve accordingly.


Example 3

What is the gradient and the y-intercept of line 3y+9x−12=63y+ 9x - 12 = 63y+9x−12=6?


Rearrange the equation into the form y=mx+cy=mx + cy=mx+c and solve.

​3y+9x−12=63y=−9x+12+63y=−9x+18y=−3x+6m=−3,c=6‾\begin{aligned}3y+9x-12&=6\\3y &= -9x +12 + 6\\3y&=-9x + 18\\y&=-3x + 6\\&\underline{m= -3, c= 6}\\\end{aligned}3y+9x−123y3yy​=6=−9x+12+6=−9x+18=−3x+6m=−3,c=6​​​​


​

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Equation of a straight line: y = mx + c

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FAQs - Frequently Asked Questions

How do you find the equation y=mx+c?

y=mx+c is the equation for a straight line, where m is the gradient/slope of the line and c is the y-intercept. If you know two points on a straight line, you can them to find m, and then from there figure out what c is.

What is c in the equation y=mx+c?

y=mx + c is the equation for a straight line, where c is the y-intercept. The point where y=c is where the line intersects the y-axis. It's also the point where x=0. The coordinates for the point hitting the y-axis on that line is (0,c).

What does y=mx+c stand for?

y=mx + c is the equation for a straight line, where m is the line's slope/gradient, and c is the y-intercept.

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