Two-Step Equations - Definition & Examples

(Video) Inverse Operations to Isolate a Variable

by eHowEducation

This video by eHowEducation shows you how to solve equations that include multiple operations.

$\large{\text{Summary}}$

We can isolate variables in algebra by using inverse operations. Inverse operations are when you do the opposite of an operation in the expression in order to cancel it out. So, for example, the inverse of $+5$ would be $-5$.

In the video above, Maria uses an example problem to show you how to use inverse operations to isolate a variable. $$2x-5=7$$ We want to get rid of the $-5$. Therefore, we need to apply the inverse operation, addition. Add $5$ to both sides. $$\begin{align} 2x-5&=7\\ 2x-5\,\color{#8925AE}{+\,5}&=7\,\color{#8925AE}{+\,5}\\ 2x&=12 \end{align}$$ Next, we want to get rid of the $2$ being multiplied. To do this, we use the inverse, division. Divide both sides by $5$. $$\begin{align} 2x&=12\\ \frac{2x}{\color{#8925AE}{2}}&=\frac{12}{\color{#8925AE}{2}}\\ x&=6 \end{align}$$

$$\begin{align} 3x + 9 &= 15\\ 3x + 9\, \color{#8925AE}{-\,9} &= 15 \,\color{#8925AE}{-\,9}\\ 3x &= 6\\ \frac{3x}{\color{#8925AE}{3}} &= \frac{6}{\color{#8925AE}{3}}\\ x &= 2 \end{align}$$

Two-step equations take two operations to solve

Just like one-step equations took just one step to solve, two-step equations take, well, two steps! You'll use all the same operations (addition, subtraction, multiplication, and addition), but you'll have to use two of them instead of just one.

Two step equations look like this:

$$4x + 5 = 9\\\ \\ 0 = \frac{x}{5} - 1\\\ \\ \frac{x + 3}{20} = 2\\\ \\ 19x = x + 3\\\ \\ 2x - 5 = x + 1$$

...and so on. Each of these equations will take just two steps to solve...but the trick is taking the steps in the right order.

The right order:

To solve two-step (and longer!) equations, just isolate your variable on one side by doing operations in the following order:

1. Subtracting or Adding terms.
2. Dividing or Multiplying terms.
3. Evaluating things in Parentheses.

(If you pay extra attention, you might recognize that this is the reverse of the order of operations)

Sometimes you might have to use some of these steps more than once, and sometimes you may not use some of them at all! Let's try a problem:

We can rewrite this equation as: $$(x + 3) \div 20 = 2$$

Remember that solving inside of parentheses is the last step when isolating the variable. That means the first step is to multiply both sides by $20$ so that we can work with the terms in parentheses: $$\begin{align} \frac{(x + 3)}{20} &= 2\\ \left(\frac{(x + 3)}{20}\right)(20) &= 2(20)\\ x + 3 &= 40 \end{align}$$

Good! Now we've got a regular one-step equation waiting for us. Let's finish it up!

$$\begin{align} x + 3 &= 40\\ x + 3 - 3 &= 40 - 3\\ x &= 37 \end{align}$$

Use Opposite Operations to Get the Variable Alone on One Side

Isolating a variable means getting it all alone on one side of the equation. To do this, you need to move everything except for the variable to the other side of the equation.

You can move numbers across the equal sign in an equation by performing the opposite operation (to the number you want to move) to both sides of the equation. So if you have $x + 5 = 9$, and you want to move the $5$ (which is being added), you'd subtract $5$ from both sides. In other cases:

• If the number is being subtracted, then add to both sides to isolate the variable
• If the number is being multiplied then divide both sides to isolate the variable
• If the number is being divided then multiply both sides to isolate the variable

For example, if you wanted to isolate $x$ in the equation $\frac{x}{7} - 4 = 10$, it would look like this: $$\begin{align} \frac{x}{3} - 4 &= 10\\ \frac{x}{3} - 4\, \color{#8925AE}{+\, 4}&= 10\, \color{#8925AE}{+\, 4}\\ \frac{x}{3} &= 14\\ \frac{x}{3}\, \color{#8925AE}{\times \,3} &= 14 \,\color{#8925AE}{\times\, 3}\\ x &= 42 \end{align}$$