# Solving Linear Equations

## 2 Step Equations Examples

Moving on from the Basic Linear Equations page.

Examples in this section feature 2 step equations examples.

Equations where  2  operations are performed to obtain an  x  value, instead of just  1  operation.

Examples

(1.1)

Solve for x.       5x 2  =  18

Solution

Here there's &nbsp2  operations to perform to get  x  by itself and solved.

5x 2  =  18
+          2         2

5x         =   20

Now:

\boldsymbol{\frac{5x}{5}}   =   \boldsymbol{\frac{20}{5}}       =>       x  =  4

(1.2)

Solve for x.         -2x + 3  =  7

Solution

-2x + 37
3     3

-2x        =  4

Now:

\boldsymbol{\frac{-2x}{-2}}   =   \boldsymbol{\frac{-4}{-2}}       =>       x  =  -2

Although order doesn't really matter, it often makes life easier to leave any division until last, in 2 step equations examples.

## 2 Step Equations Examples, Like Terms

To solve linear equations with more than 1 like term, the first step is to look to simplify the like terms to just a single term.

For example:

If   3x &nbspand  2x   were in an equation, they can be added together, or subtracted.

3x + 2x  =  5x

3x2x  =  x

Both cases resulting in &nbsp2  like terms becoming just a single term.

Examples

(2.1)

Solve   4x + 2x  =  48

Solution

4x + 2x  =  48       =>       6x  =  48

\boldsymbol{\frac{6x}{6}}   =   \boldsymbol{\frac{48}{6}}       =>       x  =  \boldsymbol{\frac{48}{6}}       =>       x  =  8

(2.2)

Solve   3x2  =  5x

Solution

We want to get all like terms together on each side of the   =   sign.

The quickest way here is to move the  3x  over to the right hand side.

But remembering to change the sign of  3x.

For example if you have:     62  =  4

Moving the  2  across as is results in:     6  =  42    X

But moving the  2  across and changing its sign to positive, gives:

6  =  4 + 2

So care does sometimes have to be taken.

Here:

3x2  =  5x       =>       -2  =  5x3x

-2  =  2x       =>       x  =  -1

(2.3)

Solve   5x + 2  =  203x

Solution

Move  2  across:       5x  =  -2 + 203x     =>     5x  =  183x

Now move &nbsp-3x  across:       5x + 3x  =  18     =>     8x  =  18

\boldsymbol{\frac{8x}{8}}   =   \boldsymbol{\frac{18}{8}}       =>       x  =  \boldsymbol{\frac{18}{8}}       =>       x  =  \boldsymbol{\frac{9}{2}}

(2.4)

Solve   5x + 2  =  5 + 3x

Solution

Similar to the previous example, moving all like terms to each side of the  =  should eventually give:

2x  =  3,

then:

\boldsymbol{\frac{2x}{2}}   =   \boldsymbol{\frac{3}{2}}       =>       x  =  \boldsymbol{\frac{3}{2}}

## Solving Equations with Brackets

(3.1)

Solve   12  =  4(3x + 1)

Solution

First step is to multiply out the brackets.

12  =  4(3x + 1)      =>      12  =  12x + 4

12 = 12x + 4
4             4
8 = 12x

\boldsymbol{\frac{8}{12}}   =   \boldsymbol{\frac{12x}{12}}       =>       \boldsymbol{\frac{8}{12}}  =  x       =>       x  =  \boldsymbol{\frac{2}{3}}

(3.2)

Solve   2 − (6x4)  =  2x

Solution

2 − (6x4)  =  2x      =>      26x + 4  =  2x

2 + 4  =  2x + 6x      =>      6  =  8x

\boldsymbol{\frac{6}{8}}   =   \boldsymbol{\frac{8x}{8}}       =>       \boldsymbol{\frac{6}{8}}  =  x       =>       x  =  \boldsymbol{\frac{3}{4}}

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