Lesson 16

How Many Solutions?

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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.

Solution

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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.

Solution

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

Solution

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

Solution

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(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.

Solution

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(From Unit 1, Lesson 6.)

Problem 6

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

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

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

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

 

Solution

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(From Unit 4, Lesson 13.)