Multi-step equations

For solvin multi-step equations lets remember once again:

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)

Multi-step equations are equations that are solved in more than two steps. Multi-step equations are just a bit more complicated than one-step or two-step equations, but they can be simplified and solved without any problem. Our goal is to find the unknown number, usually given as $x$, just like in one-step and two-step equations.
There are no specific instructions that we need to follow when solving multi-step equations.

The idea for solving multi-step equations is that we should always “group” variables (unknown numbers like $x$ or $y$, etc.) on one side of the equation and “everything else” on the other side. Then we just need to do mathematical operations that are given in the equation and solve it.

The order of operations is the same as described in the lesson about two-step equations. First we need to do addition and subtraction then multiplication and division.

Examples

Now, let’s solve some examples and show how the calculation is done.

$15-(3x+1)=4(x+1)+3x$

Before starting any calculation, we need to get rid of the parentheses if they are given in the equation. The resulting equation is:

$\ 15 – 3x – 1 = 4x + 4 + 3x$

The resulting equation can be simplified by calculating the given values on each side of the equation.

$\ 14 – 3x = 7x + 4 /+(-14-7x)$

$14-3x-14-7x=7x+4-14-7x$

Now, we have:

$-10x=-10$

The result is $\ x = 1$.

We can examine next example of multi-step equation:

$\frac{1}{3}x + 1 = \frac{1}{9}x – \frac{1}{2}$

We don’t have any parentheses we are going to start with “grouping variables” on one side and “everything else on the other side” of the equation. To do that, we are using properties (1) to (3).

$\frac{1}{3}x + 1 = \frac{1}{9}x – \frac{1}{2} /+(-1-\frac{1}{9}x)$

$\frac{1}{3}x – \frac{1}{9}x+1-1 =\frac{1}{9}x-\frac{1}{9}x – \frac{1}{2} – 1$

We are going to simplify the equation by calculating the values on each side.

$\frac{(3x – 1x)}{9} = \frac{(-1 – 2)}{2}$

$\frac{2x}{9} = \frac{-3}{2}$

After the previous step the equation is simplified. In this step we are going to multiply both sides by $18$ to get rid of the fraction on the left side of the equation.

$\ (\frac{2x}{9}) \cdot 18 = \frac{-3}{2} \cdot 18$

$\ 4x = -27 /:4$

$x=\frac{-27}{4}$

 The result is a fraction, $\ x = \frac{-27}{4}$ .

Next example:

$x+2(x+3)=36$

Since we have parentheses we need to get rid of them.

$\ x + 2x + 6 = 36$

$\ 3x + 6 = 36 /+(-6)$

Now we have a two-step equation. To solve it we need to subtract number $6$ from both sides.

$\ 3x + 6 – 6 = 36 – 6$

$\ 3x = 30 /:3$

$\ x = 10$

The result is $\ x=10$.

Multi-step equations worksheets

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