Lesson 16
How Many Solutions?
Let’s solve equations with different numbers of 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 \(x\).
- \(6x-4=\text-4+6x\)
- \(4x-6=4x+3\)
- \(\text-2x+4=\text-3x+4\)
Problem 4
Solve each of these equations. Explain or show your reasoning.
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\(3(x-5) = 6\)
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\(2\left(x - \frac{2}{3}\right) = 0\)
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\(4x - 5 = 2 -x\)
Problem 5
In the picture triangle \(A’B’C’\) is an image of triangle \(ABC\) after a rotation. The center of rotation is \(E\).
![Quadrilateral A prime, B prime, C prime, D prime is an image of quadrilateral A, B, C, D after rotation around another point, E. Side A prime, B prime has length 9. Angle D has measure 45 degrees.](https://staging-cms-im.s3.amazonaws.com/aCt9jHmbP7AvKxVMUGAFEp38?response-content-disposition=inline%3B%20filename%3D%228-8.1.B.PP.Image.01b.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B.PP.Image.01b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T132134Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ecdef59b0e4fda9d9c344a7428557270e5b7874cf658d3647ffce4ae5e08ef80)
- What is the length of side \(AB\)? Explain how you know.
- What is the measure of angle \(D'\)? Explain how you know.
Problem 6
Solve each of these equations. Explain or show your reasoning.
\(2(x+5)=3x+1\)
\(3y-4=6-2y\)
\(3(n+2)=9(6-n)\)