# Two-step equations

It is valid:$\forall a, b, c$

- if $a<b$, then $\forall c\in \mathbb{R}$:

$a+c<b+c$ (1)

- if $a<b$, then $\forall c\in \mathbb{R}$:

$a \cdot c<b \cdot c$ (2)

- if $a<b$ and $b<c$

$a<c$ (3)

Most equations require more than one step to find the solution. In this lesson we are going to cover two-step equations.

**Two-step equations** are equations that can be solved in two steps. These equations can be written in form of:

- $ax+b=c$
- $ax-b=c$
- $\frac{x}{a}+b=c$
- $\frac{x}{a}-b=c$

where $x$ is the unknown variable, $a$ is the coefficient and $b$ and $c$ are numbers. The coefficient can not be $0$ or $1$.

When solving this type of equation we should follow this order of operations:

1. addition and subtraction

2. multiplication

3. division

Note that we don’t have to follow this order, but it makes the calculation a lot easier.

Let’s get started with some **examples**, one of each type of the two-step equation.

$2x+5=7$

This example can be solved in following steps:

We should subtract $5$ from both sides first.

$2x+5=7 /+(-5)$

$2x+5-5=7-5$

Now we have one step equation:

$2x=2 /:2$

$x=1$

Note that in this example we could have done division first and then subtraction, but it would make the solving of the equation more difficult since we would have to deal with fractions and it is easier to deal with regular numbers.

$4x-8=16$

The example above can be solved in steps similar to the previous example:

First we need to add 8 to both sides. The resulting equation is:

$4x-8=16 /+8$

$4x-8+8=16+8$

In this step we only need to divide both sides with 4.

$4x=24 /:4$

$x=6$

The result is $\ x = 6$.

Look at this example that you need to divide to get the unknown variable $x$:

$\frac{x}{3}+2=7 /+(-2)$

$\frac{x}{3}+2-2=7 -2$

$\frac{x}{3}=5$

In this step we are going to multiply the equation by $3$ and get the result.

$\frac{x}{3}=5 /\cdot 3 $

$\frac{x}{3}\cdot 3=5 \cdot 3$

$x=15$

The result is $x=15$.

All other two step equations are solving like those two examples.