Equations with Many Solutions or No Solution

Warm Up

How can you give examples of equations with a given number of

solutions?

Equations that simplify to the form x=a have one solution, equations that simplify to a = a have many solutions,

and equations that simplify to the form a = b, where a ≠ b, have no

solution.

Use the properties of equality to simplify each equation. Tell whether the equation has one, zero, or infinitely many solutions.

3x - 6 = 4 + 2x x = 10

3x - 8 = 3(x - 4) + 1- 8 = -11

3x - 7 = 3(x - 3) + 2-7 = -7

Does multiplying both sides of a false statement by the same number

change the fact that the statement is false?

Usually the statement remains false when both sides are multiplied by the same number. The only exception is when both sides are multiplied by 0,

which creates 0 = 0, a true statement.

The larger of two numbers is twice the smaller. Their sum is three

times their difference.Find the numbers.

Frank solved an equation and got the result x = x. Sarah solved the same equation and got 12 = 12.

Frank says that one of them is incorrect because you cannot get

different results for the same equation. What would you say to Frank? If both results are indeed

correct, explain how this happened.

Matt said 2x - 7 = 2(x - 7) has infinitely many solutions. Is he

correct? Justify Matt’s answer or show how he is incorrect.

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