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## Distributed Linear Equations

#### Introduction

Linear equations are equations that describe a line on a graph. A linear equation will tell you the slope of the line (how steeply it rises or falls) and the point at which the line crosses the y-axis (where x=0). The most common form of displaying a linear equation is using the template y=mx+b, where m is the slope and b is the intercept.

Distributing an equation means multiplying out the parentheses. If you have a set of parentheses in an equation, you won't be able to solve it until you have multiplied them out.

## Lesson

Distributing means multiplying out the parenthesis in an equation. For example, if you are given the equation 2(x+3) = 10, you can distribute the two by multiplying it with both of the terms inside the parenthesis to get:

$2x + 6 = 10$

Then you subtract 6 from both sides to get the like terms together:

$2x + 6 (-6) = 10 (-6)$

$2x = 4$

$x = 2$

Remember, to do this with a linear equation, we need to get the y value on its own on the left of the equation (just y, not 2y, or $y^{2}$) and the x value and the number on the right of the equation.

So, if we get the equation:

$y = 4(x - 1)$

We would need to distribute in order to solve.

Firstly, we multiply the 4 by both terms inside the parenthesis:

$y = 4(x) + 4(-1)$

$y = 4x - 4$

Since 'solving' a linear equation just means putting it in y=mx+b format, we have now solved it.

A key point to remember is that if the number outside the parenthesis is negative, we need to distribute that too.

For example:

y= -8( $rac{1}{2}$ x + 2)

We would need to multiply the -8 by both terms inside the parenthesis:

$y= -8($$rac{1}{2}$$x) + -8(2)$

$y = -4x-16$

## Examples

$9(4x + 5) = -171$
 $9(4x + 5)$ $=$ $-171$ $(9*4x)+(9*5)$ $=$ $36x + 45$ $=$ $-171$ $-45$ $=$ $-45$ $36x$ $=$ $-216$ $/36$ $=$ $/36$ $x$ $=$ $-6$