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Rearranging equations Negative numbers Fractions Solving equations Plotting graphsThis topic is relevant for:

Here we will learn about **y = mx + c**

There are also

The equation **y = mx + c**** m** is the

**linear **equation and the variables

When we input a value for

This means that ** independent variable**, and **dependent variable** as it is determined by the value of

E.g.

Let’s look at the line

This graph has a **gradient** of **2** and a y-intercept of **1**, the coordinate (0,1).

- The term
**linear**is used to describe a straight line where the variables are raised to a power no higher than**1** - If the variables are raised to a power no higher than
**2**we refer to it as a**quadratic** - If the variables are raised to a power no higher than
**3**we refer to it as a**cubic**, and so on.

The gradient of the line tells us** how steep the line is**.

We use the letter ** m** to denote the gradient.

Imagine climbing a ladder. If the ladder is really close to the wall, the gradient of the ladder is really steep (you would almost be climbing vertically). Taking the base of the ladder away from the wall means that the gradient of the slope decreases reaching a lower point on the wall.

**The bigger the gradient the steeper the line.**

E.g.

A gradient of

See the diagram below.

The gradient of a line can be many different types of number, i.e. fractions, decimals, negatives etc.

The ** y-intercept **is the point of

E.g.

Let’s look at the equation

To find the

\begin{aligned}
&y=(5 \times 0)+7\\\\
&y=7
\end{aligned}

So when

This is the

Here is a quick summary of some equations in the form

y=mx+c

y=2x+4

y=6x-3

y=-4x

y=6-x

gradient m

2

6

-4

-1

y -intercept c

4

-3

0 (the origin)

6

In order to state the gradient and

**Rearrange the equation to make**y the subject**Substitute**x = 0 into the equation to find they -intercept**State the coefficient of**x (the gradient)

Get your free

Get your free

**y=mx+c** is part of our series of lessons to support revision on **straight line graphs**. You may find it helpful to start with the main straight line graphs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

State the gradient and

**Rearrange the equation to make**y the subject

The equation

2**Substitute x = 0 into the equation to find the y-intercept**

When

\begin{aligned}
&y=(-3\times{0})+8\\\\
&y=0+8\\\\
&y=8
\end{aligned}

The

3**State the coefficient of x (the gradient)**

The coefficient of

The gradient of the line

Solution:

State the gradient and

**Rearrange the equation to make **

Here we have the two terms of

\begin{aligned}
&y=7-x\\\\
&y=-x+7
\end{aligned}

**Substitute x = 0 into the equation to find the y-intercept**

When

\begin{aligned}
&y=-0+7\\\\
&y=0+7\\\\
&y=7
\end{aligned}

The

**State the coefficient of x (the gradient)**

The coefficient of

The gradient of the line

Solution:

State the gradient and

**Rearrange the equation to make **

Here we need to make

**Substitute x = 0 into the equation to find the y-intercept**

When

\begin{aligned}
&y=0-10\\\\
&y=-10
\end{aligned}

The

**State the coefficient of x (the gradient)**

The coefficient of

The gradient of the line

State the gradient and

**Rearrange the equation to make **

**Substitute x = 0 into the equation to find the y-intercept**

When

\begin{aligned}
&y=(\frac{1}{3}\times{0})+\frac{5}{2}\\\\
&y=0+\frac{5}{2}\\\\
&y=\frac{5}{2}
\end{aligned}

The

\[\frac{5}{2}, \quad \text{or} \quad c=\frac{5}{2}.\]

**State the coefficient of x (the gradient)**

The coefficient of

\[\frac{1}{3}.\]

The gradient of the line

\[\frac{1}{3}, \quad \text{or} \quad m=\frac{1}{3}.\]

State the gradient and

**Rearrange the equation to make **

**Substitute x = 0 into the equation to find the y-intercept**

When

\begin{aligned}
&y=(\frac{3}{4}\times{0})+5\\\\
&y=0+5\\\\
&y=5
\end{aligned}

The

**State the coefficient of x (the gradient)**

The coefficient of

\[\frac{3}{4}.\]

The gradient of the line

\[\frac{3}{4}, \quad \text{or} \quad m=\frac{3}{4}.\]

State the gradient and

\[x=\frac{y+0.85}{0.2}.\]

**Rearrange the equation to make **

**Substitute x = 0 into the equation to find the y-intercept**

When

\begin{aligned}
&y=(0.2\times{0})-0.85\\\\
&y=0-0.85\\\\
&y=-0.85
\end{aligned}

The

\[x=\frac{y+0.85}{0.2}\]

is

**State the coefficient of x (the gradient)**

The coefficient of

The gradient of the line

\[x=\frac{y+0.85}{0.2}\]

is

**Incorrect inverse operation**

When rearranging equations, instead of applying the inverse operation to the value being moved, the value is simply moved to the other side of the equals sign.

E.g.

**Stating the value of**m andc

A common error is to incorrectly state the values of m and c as a result of not rearranging the equation so that it is in form of

Take example

The correct answer for the

**Mixing up the gradient and the y-intercept**

Take, for example, the equation

1. State the gradient, m , and y -intercept, c , for the equation

y=-5x+9

m=-5, \; c=9

m=9, \; c=5

m=9, \; c=-5

m=-5, \; c=4

The coefficient of x is -5 , so m=-5

When x=0, y=9, so c=9.

2. State the gradient, m , and y -intercept, c , for the equation

y=6-x

m=6, \; c=-1

m=-1, \; c=6

m=5, \; c=6

m=0, \; c=5

y=6-x

y=-x+6

The coefficient of x is -1 , so m=-1

When x=0, y=6, so c=6.

3. State the gradient, m , and y -intercept, c , for the equation

x=2y+5

m=\frac{1}{2}, \; c=-\frac{5}{2}

m=2, \; c=5

m=-2, \; c=-5

m=1, \; c=7

y=\frac{1}{2}x-\frac{5}{2}

The coefficient of x is \frac{1}{2} , so m=\frac{1}{2}

When x=0, y=-\frac{5}{2}, so c=-\frac{5}{2}.

4. State the gradient, m , and y -intercept, c , for the equation

3x=5y-6

m=\frac{3}{5}, \; c=\frac{6}{5}

m=3, \; c=-6

m=2, \; c=-1

m=2y, \; c=5

y=\frac{3}{5}x+\frac{6}{5}

The coefficient of x is \frac{3}{5} , so m=\frac{3}{5}

When x=0, y=\frac{6}{5}, so c=\frac{6}{5}.

5. State the gradient, m , and y -intercept, c , for the straight line

2x=3(3+y)

m=\frac{2}{3}, \; c=-3

m=2, \; c=0

m=6, \; c=9

m=\frac{2}{3}, \; c=3

y=\frac{2}{3}x-3

The coefficient of x is \frac{2}{3} , so m=\frac{2}{3}

When x=0, y=-3 so c=-3.

6. State the gradient, m , and y -intercept, c , for the equation of the line

0.5x+0.75y=0.25

m=-\frac{2}{3}, \; c=\frac{1}{3}

m=0.5, \; c=0.5

m=0.75,\; c=0.5

m=1.5, \; c=0.5

y=-\frac{2}{3}x+\frac{1}{3}

The coefficient of x is -\frac{2}{3} , so m=-\frac{2}{3}

When x=0, y=\frac{1}{3}, so c=\frac{1}{3}.

1. Given that the coordinate (3,4) lies on the line y=3x+c calculate the y -intercept of the straight line.

**(2 Marks)**

Show answer

Substitute x=3 and y=4 into y=3x+c to get

4=(3\times3)+c (1mark)

**(1)**

4=9+c so

c=-5 (1mark)

**(1)**

2. (a) The coordinate A=(0,2) lies on a straight line. The gradient of the line is 5 . Using this information, state the equation of the straight line.

(b) Write the equation of a line that is parallel to a) in the form y=mx+c .

**(4 Marks)**

Show answer

(a)

A is the y -intercept so c=2 or when x=0, y=2 so c=2

**(1)**

y=5x+c (1mark)

**(1)**

y=5x+2 (1mark)

**(1)**

(b)

y=5x+c
where

c ≠ 2
(1mark)

**(1)**

3. Show m=2 for the straight line 8x-4y=12.

**(3 Marks)**

Show answer

**(3)**

You have now learned how to:

- reduce a given linear equation in 2 variables to the standard form y=mx+c

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