Lesson 16

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.

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.

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

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

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

1. What is the length of side \(AB\)? Explain how you know.
2. What is the measure of angle \(D'\)? Explain how you know.

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