Understanding Equations with No Solution or Infinitely Many

1.7.3 Understanding Equations with No Solution or Infinitely Many

1.7.3 • Understanding Equations with No Solution or Infinitely Many

Activity

Noah is having trouble solving two equations. In each case, he took steps that he thought were acceptable but ended up with statements that are clearly not true.

Analyze Noah’s work on each equation and the moves he made. Were they acceptable moves? Why do you think he ended up with a false equation?

Discuss your observations with your group and be prepared to share your conclusions. If you get stuck, consider solving each equation.

Example 1

Step 1 - Original equation.

$x + 6 = 4 x + 1 - 3 x$

Step 2 - Apply the Commutative Property.

$x + 6 = 4 x - 3 x + 1$

Step 3 - Combine like terms.

$x + 6 = x + 1$

Step 4 - Subtract $x$ from each side.

$6 = 1$

Example 2

Step 1 - Original equation.

$2 (5 + x) - 1 = 3 x + 9$

Step 2 - Apply the Distributive Property.

$10 + 2 x - 1 = 3 x + 9$

Step 3 - Subtract 10 from each side.

$2 x - 1 = 3 x - 1$

Step 4 - Add 1 to each side.

$2 x = 3 x$

Why did Noah have no solution for either equation?

Compare your answer:

Your answer may vary, but here is an example: All the moves that Noah made were acceptable, but because he ended up with a false statement, that means there is no value of $x$ that makes the first equation true. We can see it in the third line: $x + 6 = x + 1$ . There is no number that can make the equation true.

Why did Noah arrive at no solution for the equation solved in Example 2?

Compare your answer:

Your answer may vary, but here is a sample.

Example 1: All the moves that Noah made were acceptable, but because he ended up with a false statement, that means there is no value of $x$ that makes the first equation true. We can see it in the third line: $x + 6 = x + 1$ . There is no number that can make the equation true.

Example 2: Noah's moves seem acceptable except for the last one. If he had subtracted $2 x$ from each side of the equation, he would have $0 = x$ . Instead, he divided both sides by $x$ which would give an undefined number if $x i s 0$ .

Noah tried to solve a third equation and ended up with a true statement, without variables.

Step 1 - Original equation.

$2 (4 + x) - 2 = 2 x + 6$

Step 2 - Apply the Distributive Property.

$8 + 2 x - 2 = 2 x + 6$

Step 3 - Combine Like Terms.

$2 x + 6 = 2 x + 6$

Step 4 - Subtract 6 from both sides.

$2 x = 2 x$

Step 5 - Subtract $2 x$ from both sides.

$0 = 0$

Since there is not a variable and the statement is true, any value of $x$ would be a solution for the original equation. So, this equation has infinitely many solutions.

Are you ready for more?

Extending Your Thinking

We cannot divide the number 100 by zero because dividing by zero is undefined. Instead, try dividing 100 by 10, then 1, then 0.1, then 0.01. What do you notice happens as you divide by smaller numbers?

Compare your answer: Your answer may vary, but here is a sample.

10, 100, 1000, 10,000. For example: The quotients are getting very large.

Now try dividing the number -100 by 10, by 1, by 0.1, 0.01. What is the same and what is different when you compare to question 1?

Compare your answer: Your answer may vary, but here is a sample.

-10, -100, -1000, -10,000. For example: The absolute values of the quotients are the same, but the sign is opposite..

In middle school, you used tape diagrams to represent division. This tape diagram shows that 6÷2=3.

Draw a tape diagram that shows why $6 \div \frac{1}{2} = 12$ .

Try to draw a tape diagram that represents 6÷0. Explain why this is so difficult.

Compare your answer: Your answer may vary, but here is a sample.

The zero takes up no space, so no matter how many copies we have it could never reach 6.

Self Check

Which of the following equations has a solution?

$10x+ 10 = 7x + 3x$
$x + 3 = x - 2$
$2x = 2x+1$
$3x = x + 18$

Additional Resources

Equations Without Solutions

What happens when we solve $5 z = 5 z - 1$ ?

Step 1 - Subtract $5 z$ to get the $z$ terms together on one side.

$5 z - 5 z = 5 z - 5 z - 1$

Step 2 - Simplify and the $z$ terms are gone!.

$0 = 1$

Since 0 does not equal $- 1$ , this equation led to a false statement. There are no solutions to this equation.

Example

Solve $5 m + 3 (9 + 3 m) = 2 (7 m - 11)$ .

Step 1 - Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.

$5 m + 27 + 9 m = 14 m - 22$

$14 m + 27 = 14 m - 22$

Step 2 - Collect all the variable terms on one side of the equation.

Use the Addition or Subtraction Property of Equality.

$14 m - 14 m + 27 = 14 m - 14 m - 22$

$27 = - 22$

Step 3 - Check the solution.

The equation $27 = - 22$ is NOT a true statement. The equation 5m+3(9+3m)=2(7m−11) has no solutions.

Infinitely Many Solutions

What if an equation results in a true statement without variables?

The equations $- 4 = - 4$ , $5 = 5$ , and $0 = 0$ are all true statements because one side is equal to the other. This means that any value of the variable in the original equation would be a solution. When this happens in the final step of an equation, there are infinitely many solutions.

Example

Solve $2 y + 6 = 2 (y + 3)$

Step 1 - Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.

$2 y + 6 = 2 (y + 3)$

Step 2 - Collect all the variable terms on one side of the equation. Use the Addition or Subtraction Property of Equality.

$2 y - 2 y + 6 = 2 y - 2 y + 6$

$6 = 6$

$2 y + 6 = 2 y + 6$

Step 3 - Check the solution. We say the solution to the equation is all of the real numbers.An equation that is true for any value of the variable is called an identity.

The solution of an identity is all real numbers.

Try it

Equations with No Solutions and Infinitely Many Solutions

Solve $10 x - 4 = 2 (5 x - 2)$ and determine the number of solutions.

Compare your answer: Your answer may vary, but here is a sample.

Solve this linear equation using a general strategy:

Step 1 - Original equation.

$10 x - 4 = 2 (5 x - 2)$

Step 2 - Distribute

$10 x - 4 = 10 x - 4$

Step 3 - Collect all the variable terms on one side of the equation. Use the Addition or Subtraction Property of Equality.

$10 x - 10 x - 4 = 10 x - 10 x - 4 - 4 = - 4$

Step 4 - Check the solution

This is a true statement so there are infinitely many solutions.

Solve $3 x - 4 = 3 (x - 2)$ and determine the number of solutions.

Compare your answer: Your answer may vary, but here is a sample.

Solve this linear equation using a general strategy:

Step 1 - Original equation.

$3 x - 4 = 3 (x - 2)$

Step 2 - Distribute

$3 x - 4 = 3 x - 6$

Step 3 - Collect all the variable terms on one side of the equation. Use the Addition or Subtraction Property of Equality.

$3 x - 3 x - 4 = 3 x - 3 x - 6 - 4 = - 6$

Step 4 - Check the solution

The equation $- 4 = - 6$ is NOT a true statement. The equation $3 x - 4 = 3 (x - 2)$ has no solutions.