Lesson 16

How Many Solutions?

Let’s solve equations with different numbers of solutions.

Problem 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.

Problem 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.

Problem 3

Decide whether each equation is true for all, one, or no values of \(x\).

  1. \(6x-4=\text-4+6x\)
  2. \(4x-6=4x+3\)
  3. \(\text-2x+4=\text-3x+4\)

Problem 4

Solve each of these equations. Explain or show your reasoning.

  1. \(3(x-5) = 6\)

  2. \(2\left(x - \frac{2}{3}\right) = 0\)

  3. \(4x - 5 = 2 -x\)

(From Unit 4, Lesson 13.)

Problem 5

In the picture triangle \(A’B’C’\) is an image of triangle \(ABC\) after a rotation. The center of rotation is \(E\).

Quadrilateral A prime, B prime, C prime, D prime is an image of quadrilateral A, B, C, D after rotation around another point, E. Side A prime, B prime has length 9. Angle D has measure 45 degrees.
  1. What is the length of side \(AB\)? Explain how you know.
  2. What is the measure of angle \(D'\)? Explain how you know.
(From Unit 1, Lesson 6.)

Problem 6

Solve each of these equations. Explain or show your reasoning.

\(2(x+5)=3x+1\)

\(3y-4=6-2y\)

\(3(n+2)=9(6-n)\)

 

(From Unit 4, Lesson 13.)