Lesson 7

All, Some, or No Solutions

Let’s think about how many solutions an equation can have.

Problem 1

For each equation, decide if it is always true or never true.

  1. \(x - 13 = x + 1\)

  2. \(x+\frac{1}{2} = x - \frac{1}{2}\)

  3. \(2(x + 3) = 5x + 6 - 3x\)

  4. \(x - 3 = 2x - 3 -x\)

  5. \(3(x-5) = 2(x-5) + x\)

Problem 2

Mai says that the equation \(2x + 2 = x +1\) has no solution because the left hand side is double the right hand side. Do you agree with Mai? Explain your reasoning.

Problem 3

  1. Write the other side of this equation so it's true for all values of \(x\): \(\frac12(6x-10) - x =\)

  2. Write the other side of this equation so it's true for no values of \(x\): \(\frac12(6x-10) - x = \)

Problem 4

Here is an equation that is true for all values of \(x\): \(5(x+2) = 5x+10\). Elena saw this equation and says she can tell \(20(x+2)+31=4(5x+10)+31\) is also true for any value of \(x\). How can she tell? Explain your reasoning.

Problem 5

Elena and Lin are trying to solve \(\frac12x+3=\frac72x+5\). Describe the change they each make to each side of the equation.

  1. Elena’s first step is to write \(3=\frac72x-\frac12x+5\).
  2. Lin’s first step is to write \(x+6=7x+10\).
(From Unit 4, Lesson 4.)

Problem 6

Solve each equation and check your solution.





(From Unit 4, Lesson 6.)

Problem 7

The point \((\text-3, 6)\) is on a line with a slope of 4.

  1. Find two more points on the line.
  2. Write an equation for the line.
(From Unit 3, Lesson 12.)