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Algebra 1

1.7.3 Understanding Equations with No Solution or Infinitely Many

Algebra 11.7.3 Understanding Equations with No Solution or Infinitely Many
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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 x + 6 = 4 x + 1 − 3 x

Step 2 - Apply the Commutative Property.

x + 6 = 4 x − 3 x + 1 x + 6 = 4 x − 3 x + 1

Step 3 - Combine like terms.

x + 6 = x + 1 x + 6 = x + 1

Step 4 - Subtract x x from each side.

6 = 1 6 = 1

Example 2

Step 1 - Original equation.

2 ( 5 + x ) − 1 = 3 x + 9 2 ( 5 + x ) − 1 = 3 x + 9

Step 2 - Apply the Distributive Property.

10 + 2 x − 1 = 3 x + 9 10 + 2 x − 1 = 3 x + 9

Step 3 - Subtract 10 from each side.

2 x − 1 = 3 x − 1 2 x − 1 = 3 x − 1

Step 4 - Add 1 to each side.

2 x = 3 x 2 x = 3 x

1.

Why did Noah have no solution for either equation?

2.

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

Are you ready for more?

Extending Your Thinking

1.

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?

2.

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?

3.

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

Tape diagram, 3 parts, each marked 2.

Draw a tape diagram that shows why   6 ÷ 1 2 = 12 6 ÷ 1 2 = 12 .

4.

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

Self Check

Which of the following equations has a solution?
  1. 10 x + 10 = 7 x + 3 x
  2. x + 3 = x − 2
  3. 2 x = 2 x + 1
  4. 3 x = x + 18

Additional Resources

Equations Without Solutions

What happens when we solve 5 z = 5 z − 1 5 z = 5 z − 1 ?

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

5 z − 5 z = 5 z − 5 z − 1 5 z − 5 z = 5 z − 5 z − 1

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

0 = 1 0 = 1

Since 0 does not equal − 1 − 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 ) 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 5 m + 27 + 9 m = 14 m − 22

14 m + 27 = 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 14 m − 14 m + 27 = 14 m − 14 m − 22

27 = − 22 27 = − 22

Step 3 - Check the solution.

The equation 27 = − 22 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 − 4 = − 4 ,  5 = 5 5 = 5 , and  0 = 0 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 ) 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 ) 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 2 y − 2 y + 6 = 2 y − 2 y + 6

6 = 6 6 = 6

2 y + 6 = 2 y + 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

Try It: Equations with No Solutions and Infinitely Many Solutions 

1.

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

2.

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

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