Lesson 8
How Many 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.
Solution
For access, consult one of our IM Certified Partners.
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.
Solution
For access, consult one of our IM Certified Partners.
Problem 3
Decide whether each equation is true for all, one, or no values of \(x\).
 \(6x4=\text4+6x\)
 \(4x6=4x+3\)
 \(\text2x+4=\text3x+4\)
Solution
For access, consult one of our IM Certified Partners.
Problem 4
Solve each of these equations. Explain or show your reasoning.

\(3(x5) = 6\)

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

\(4x  5 = 2 x\)
Solution
For access, consult one of our IM Certified Partners.
(From Unit 4, Lesson 4.)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.
Solution
For access, consult one of our IM Certified Partners.
(From Unit 3, Lesson 13.)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.
Solution
For access, consult one of our IM Certified Partners.
(From Unit 1, Lesson 7.)