When we worked with one-step equations , many of those equations we could just look at and know the answer. In this lesson, let's look at equations which require more steps to solve.

A two-step equation will require two steps to solve, and generally they have a multiplication/division and a subtraction/addition. For example:

2x + 5 =11 \quad\quad \dfrac{1}{3}h - 3 =6 \quad\quad \dfrac {9-j}{2}=5

Let's first look at using a model to solve and then how to solve algebraically.

Algebraic tiles

Algebra tiles allow us to represent an equation more visually. It is important to ensure that you are keeping the two sides of the equation balanced, so what you do to one side, you must do to the other.

Exploration

This applet represents the equation 3x+ 1=7.

You can click and drag algebraic tiles from the bottom to the gray part on the scale. Click the reset button in the top right corner to go back to 3x+ 1=7.

Take away +1 tile. What happens when we take away +1 unit from the left side of the scale?
What can you do to keep the scale balanced after removing +1 unit from the left side of the scale?
Write the new equation after removing +1 tile containing only +x tiles on the left side of the equation.
Divide the tiles into 3 equal groups on both both sides. How many +1 tiles are equivalent to each +x tile?

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Instead of taking away +1 tile from the left side of the scale, what tile can we add to +1 tile so we have zero pair and can cancel it out?

If we add the same amount to each side of the equation, it will remain balanced.

If we take away the same amount from each side of the equation, it will remain balanced. If we double, triple, or even quadruple the amounts on both sides of a scale, the scale will stay balanced. In fact, we can keep it balanced by multiplying or dividing the amounts by any nonzero number - so long as it's the same on both sides.

Examples

Example 1

Consider the following algebra tiles:

A set of tiles representing the equation 2 x minus 3 equals 5. Ask your teacher for more information.

What should we add to the left side and right side of the equation to keep only the x tiles on the left side of the equation?

Worked Solution

Create a strategy

Keep the two sides balanced by adding tiles to have zero pairs and cancel the unit tiles on the left side.

Apply the idea

Add 3 positive unit tiles to both sides of the equation to cancel out the 3 negative unit tiles.

A set of tiles representing the equation 2 x minus 3 plus 3 equals 5 plus 3. Ask your teacher for more information.

Draw the final number of algebraic tiles and write the equation to solve for x.

Worked Solution

Create a strategy

Count the final number of tiles after canceling out the unit tiles.

Apply the idea

A set of tiles representing the equation 2 x equals 8. Ask your teacher for more information

Find the value of x.

Worked Solution

Create a strategy

We divide the tiles into equal groups on both sides of the equation to find the equivalent of a single x tile.

Apply the idea

Divide by 2 groups.

A set of tiles representing the equation x is equal to 4. Ask your teacher for more information.

Idea summary

Algebra tiles allow us to represent an equation visually. It is important to ensure that we are keeping the two sides of the equation balanced, so what we do to one side, we must do to the other.

Algebraic techniques

If we don't have algebra tiles available or if we have an equation involving fractions, then solving purely algebraically is also an option.

Remember from when we solved one-step equations:

We want to get the variable by itself on one side of the equals sign.
To keep everything balanced, we must do the same operations to both sides by applying the properties of equality.
We should use opposite operations such as addition and subtraction, as well as multiplication and division to solve the equation.

Examples

Example 2

Solve the following equation: 8m+9=65

Worked Solution

Create a strategy

Use opposite operations to isolate the variable.

Apply the idea

\displaystyle 8m+9-9	\displaystyle =	\displaystyle 65-9	Subtract 9 from both sides
\displaystyle 8m	\displaystyle =	\displaystyle 56	Simplify
\displaystyle \frac{8m}{8}	\displaystyle =	\displaystyle \frac{56}{8}	Divide both sides by 8
\displaystyle m	\displaystyle =	\displaystyle 7	Evaluate

Example 3

Solve the following equation: \dfrac{x}{-9} + 10 = -5

Worked Solution

Create a strategy

Use opposite operations to isolate the variable.

Apply the idea

\displaystyle \dfrac{x}{-9} +10-10	\displaystyle =	\displaystyle -5-10	Subtract 10 from both sides
\displaystyle \dfrac{x}{-9}	\displaystyle =	\displaystyle -15	Simplify
\displaystyle \frac{x}{-9} \times(-9)	\displaystyle =	\displaystyle -15 \times (-9)	Multiply both sides by (-9)
\displaystyle x	\displaystyle =	\displaystyle 135	Evaluate

Idea summary

When solving two-step equations:

We want to get the variable by itself on one side of the equals sign
To keep everything balanced, we must do the same operations to both sides by applying the properties of equality
We should use opposite operations such as addition and subtraction, as well as multiplication and division to solve the equation

Outcomes

7.EE.B.4

Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

7.EE.B.4.A

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

6.02 Two-step equations

Introduction

Ideas

Algebraic tiles

Exploration

Examples

Example 1

Algebraic techniques

Examples

Example 2

Example 3

Outcomes

7.EE.B.4

7.EE.B.4.A

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