Visualizing Equivalent Equations on a Number Line

In the last section we showed how to solve the equation 17=2x+4 by undoing on a number line. Undoing works well if there is one instance of a variable that is being operated on by addition, subtraction, multiplication, or division. Undoing does not work as well if there are variables on both sides of the equation. In this section we will show how to use the number line to solve equations with variables on both sides of the equal sign. The number line provides a visual explanation for why equations stay equivalent when adding, subtracting, multiplying, and dividing both sides of the equation.

Discuss in Pairs

Work in pairs to solve the equation 3x+7 = 5x + 1. Then discuss the following questions.

Describe different ways that students might solve this equation.
What visualizations do you have when solving equations and how do they help you make sense of the process?
Describe some difficulties that students have when they solve equations with variables on both sides.

Using the Number Line

The equation 3x+7=5x+1 is true when the value of 3x+7 is the same as 5x+1. Since x is a variable, it can take on any value, but not all values will make the equation true. Values of x that make the equations true are called solutions of the equation. For example, if x=2, then 3(2)+7=5(2)+1 is false, so x=2 is not a solution to the equation. If x is 3, then the equation 3(3)+7=5(3)+1 is true and is a solution to the equation.

Look at the number line shown in figure 1. Labeling a tick mark both 3x+7 and 5x+1 means that both expressions represent the same value since they both name the same spot. In other words, 3x+7 is equivalent to 5x+1.

A number line with a tick mark in the middle. The top of the tick mark is labeled 3x+7 and the bottom is labeled 5x+1

Figure 1: Representing 3x+7 = 5x+1 on a number line.

There are many ways to begin solving this equation and students should be encouraged to try different approaches. A common first move when solving the equation is to subtract 3x from both sides. If we assume that x is positive, then subtracting 3x from both expressions of the symbolic equation is represented by shifting the tick mark 3x units to the left. Figure 2 shows one way to solve 3x+7=5x+1 on a number line. Take a few minutes to examine this solution. What do you notice? What do you wonder? Discuss what members of your PLC noticed and wondered about in figure 2 before reading on.

Figure 2: Operating on both sides of an equation

In figure 2, notice that the arrows showing the action are not doing and undoing as in the previous section but are moving the values to the left or the right together. We use this reasoning when we tell students that they can do the same operation to both sides of the equation. This is a different view of equation solving than undoing and the direction of the arrows highlights the differences between these approaches to solving equations. The number line shows that the equation 7=2x+1 is equivalent to the original equation. Subtracting 1 from both expressions is represented by a shift to the left of 1 unit. The division by 2 from both expression is a shrink towards 0. Since we assumed that x was positive the division by 2 brings x closer to 0 from 2x.

Notice that if we substitute the solution x=3 into each expression, we get the values of each tick mark to be equal as well as having the values correctly ordered from left to right. Once we have solved this equation and found x to be equal to 3, we can then see that the relative magnitudes are not correct. For instance, a subtraction of 3x is a slide of 9 units to the left, but this slide is shown as about the same magnitude as the subtraction of 1 on this open number line. This more flexible use of a number line is appropriate when solving equations on a number line. However, if the solution had resulted in the numbers being incorrectly ordered, we would recommend making a second number line that correctly models the solution to the equation. This is because incorrect ordering is often the result of negative numbers being involved and making a second number line that correctly models the actions with these numbers provides an important opportunity to strengthen student understanding of operations with negative values. Note that this is also true when using open number lines to represent addition and subtraction of two numbers when doing number talks.

Watch a Video

Watch a video about how to set up and solve the equation 2x+13=5x-2 by doing the same operation on both sides to generate equivalent equations.

Discuss

How does the number line help you visualize adding, subtracting, and dividing both sides of an equation?
What assumptions were made about the equation when 2x was subtracted? How would the solution shown on the number line change if that assumption was determined to not be true?

Taking the First Step

There is not one correct way to solve any equation. Rather, there are many choices that students make as they solve equations by doing the same operation on both sides of the equal sign. The next problem asks you to consider the choices students have when they solve equations, and how the number line allows them to visualize the consequences of those choices.

Discuss in Pairs

Consider the equation (2/3)x+21=3x. Figures 3, 4, and 5 show three different first moves that students might do when solving this equation. Examine the three figures, then discuss the following with a partner.

Discuss how each first move helped, or did not help, the student get closer to knowing the value of x.
Choose one the three moves and finish solving the equation on the number line. Which move did you choose and why did you choose it?
It is common for students to not consider multiplying both sides by 3 as first move in solving equations like this. Discuss how the number lines shown in figures 2 and 3 might help students to: (1) make sense of what it means to multiply both sides of the equation by 3, and (2) consider multiplication (rather than just addition or subtraction) as a helpful first step in solving equations.

A number line with 2 tick marks labeled on top from left to right 2/3x then 2/3x+21. The tick marks are labeled on the bottom from left to right 3x-21 then 3x. There are arrows connecting the tick marks from right to left labeled -21 on both the bottom of the number line and the top.

Figure 3: Subtracting 21 from both sides

A number line with 3 tick marks labeled on top from left to right 0, 2/3x, then 2/3x+21. The tick marks are labeled on the bottom from left to right 0, 3x-21, then 3x. There are arrows connecting the middle tick mark to the rightmost tick mark from left to right labeled *3 on both the bottom of the number line and the top.

Figure 4: Multiplying both sides by 3

Figure 5: Subtracting 2/3x

Try to Solve Equations on a Number Line

The worksheet Solving Equations with Variables on Both Sides on a Number Line contains several equations with variables on both sides.

Work individually for about 10 minutes to solve the equations by doing the same operation on both sides of the equal sign on a number line as well as recording the corresponding equivalent equations.
Discuss your work in pairs for another 10 minutes. Compare your assumptions and notice how your number lines are different and how they are the same.
Discuss with your PLC your experience solving equations using this method. Discuss both strengths and weaknesses with this approach.
This approach to solving equations asks you to think about operations on expressions and numbers more carefully than simply solving them symbolically. Discuss the merits of this approach and if this additional level of reasoning is something that would benefit your students at this stage.
The example strategy for equation d was more complicated than the strategy you might have used. Discuss with your group why the step showing multiplication of 2 on both sides moved the expression from right to left. Discuss why the equivalent equations d=8d+14 and -2=d label the same tick mark.

Conclusion

Solving equations on a number line by adding, subtracting, multiplying and dividing and connecting these actions to the symbols requires much more time and effort than just working with the symbolic. You and your students might become frustrated with the confusion that happens as you make sense of the movements along the number line. We have found that if students cannot make sense of these actions and how the movements correspond to operations then maybe they should not be solving equations at all. Students need to be able to visualize the actions that they are doing and the number line is one way to do that. This approach directly connects algebra sense with number sense using a visual model.