Illustrative Mathematics Grade 8, Unit 4, Lesson 8: How Many Solutions?

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Illustrative Math
Grade 8

Lesson 8: How Many Solutions?

Let’s solve equations with different numbers of solutions.

Illustrative Math Unit 8.4, Lesson 8 (printable worksheets)

Lesson 8 Summary

Sometimes it’s possible to look at the structure of an equation and tell if it has infinitely many solutions or no solutions. For example, look at
2(12x + 18) + 6 = 18x + 6(x + 7)
Using the distributive property on the left and right sides, we get
24x + 36 + 6 = 18x + 6x + 42
From here, collecting like terms gives us
24x + 42 = 24x + 42
Since the left and right sides of the equation are the same, we know that this equation is true for any value of x without doing any more moves!

Similarly, we can sometimes use structure to tell if an equation has no solutions. For example, look at
6(6x + 5) = 12(3x + 2) + 12
If we think about each move as we go, we can stop when we realize there is no solution:
1/6 · 6(6x + 5) = 1/6 · (12(3x + 2) + 12) Multiply each side by 1/6
6x + 5 = 2(3x + 2) + 2 Distribute 1/6 on the right side.
6x + 5 = 6x + 4 + 2 Distribute 2 on the right side.
The last move makes it clear that the constant terms on each side, 5 and 4 + 2, are not the same. Since adding 5 to an amount is always less than adding 4 + 2 to that same amount, we know there are no solutions.

Doing moves to keep an equation balanced is a powerful part of solving equations, but thinking about what the structure of an equation tells us about the solutions is just as important.

Lesson 8.1 Matching Solutions

Consider the unfinished equation 12(x + 3) + 18 = . Match the following expressions with the number of solutions the equation would have with that expression on the right hand side.

• See Video for Whole Lesson

Lesson 8.2 Thinking About Solutions Some More

Your teacher will give you some cards.

1. With your partner, solve each equation.
2. Then, sort them into categories.
3. Describe the defining characteristics of those categories and be prepared to share your reasoning with the class.

Lesson 8.3 Make Use of Structure

For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions). If an equation has one solution, solve to find the value of x that makes the statement true.

Are you ready for more?

Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98.

1. Choose any set of three consecutive numbers. Find their average. What do you notice?
2. Find the average of another set of three consecutive numbers. What do you notice?
   • Show Answer

     The average is always the second number.

3. Explain why the thing you noticed must always work, or find a counterexample
   • Show Answer

     The average is always the second number because the number to the left is one smaller and the number to the right is one bigger.

Lesson 8 Practice Problems

1. Lin was looking at the equation 2x - 32 + 4(3x - 2462) = 14x. She said, “I can tell right away there are no solutions, because on the left side, you will have 2x + 12x and a bunch of constants, but you have just 14x on the right side.” Do you agree with Lin? Explain your reasoning.
2. Han was looking at the equation 6x - 4 + 2(5x + 2) = 16x. He said, “I can tell right away there are no solutions, because on the left side, you will have 6x + 10x and a bunch of constants, but you have just 16x on the right side.” Do you agree with Han? Explain your reasoning.
3. Decide whether each equation is true for all, one, or no values of x.
4. Solve each of these equations. Explain or show your reasoning.
5. The points (-2,0) and (0,-6) are each on the graph of a linear equation. Is (2,6) also on the graph of this linear equation? Explain your reasoning.
6. In the picture triangle A’B’C' is an image of triangle ABC after a rotation. The center of rotation is E.
   a. What is the length of side AB? Explain how you know.
   b. What is the measure of angle D'? Explain how you know.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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