Lesson 8
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.
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 \(x\).
- \(6x-4=\text-4+6x\)
- \(4x-6=4x+3\)
- \(\text-2x+4=\text-3x+4\)
Solution
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Problem 4
Solve each of these equations. Explain or show your reasoning.
-
\(3(x-5) = 6\)
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\(2\left(x - \frac{2}{3}\right) = 0\)
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\(4x - 5 = 2 -x\)
Solution
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(From Unit 4, Lesson 4.)Problem 5
The points \((\text-2,0)\) and \((0,\text-6)\) are each on the graph of a linear equation. Is \((2,6)\) also on the graph of this linear equation? Explain your reasoning.
Solution
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(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
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(From Unit 1, Lesson 7.)