Lesson 8
How Many Solutions?
Let’s solve equations with different numbers of solutions.
Problem 1
Lin was looking at the equation \(2x32+4(3x2462) = 14x\). 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 \(6x4+2(5x+2)=16x\). 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 \(x\).
 \(6x4=\text4+6x\)
 \(4x6=4x+3\)
 \(\text2x+4=\text3x+4\)
Problem 4
Solve each of these equations. Explain or show your reasoning.

\(3(x5) = 6\)

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

\(4x  5 = 2 x\)
Problem 5
The points \((\text2,0)\) and \((0,\text6)\) are each on the graph of a linear equation. Is \((2,6)\) also on the graph of this linear equation? Explain your reasoning.
Problem 6
In the picture triangle \(A’B’C’\) is an image of triangle \(ABC\) after a rotation. The center of rotation is \(E\).
 What is the length of side \(AB\)? Explain how you know.
 What is the measure of angle \(D'\)? Explain how you know.