# Lesson 16

How Many 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. $$4x - 5 = 2 -x$$

### Solution

(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 $$E$$.

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

(From Unit 1, Lesson 6.)

### Problem 6

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

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

$$3y-4=6-2y$$

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