Lesson 5
Solving Any Linear Equation
Problem 1
Solve each of these equations. Explain or show your reasoning.
\(2(x+5)=3x+1\)
\(3y-4=6-2y\)
\(3(n+2)=9(6-n)\)
Solution
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Problem 2
Clare was solving an equation, but when she checked her answer she saw her solution was incorrect. She knows she made a mistake, but she can’t find it. Where is Clare’s mistake and what is the solution to the equation?
\(\begin{align} 12(5+2y)&=4y-(5-9y)\\ 72+24y&=4y-5-9y\\ 72+24y&=\text-5y-5\\ 24y&=\text-5y-77\\ 29y&=\text-77\\ y&=\frac {\text{-}77}{29}\ \end{align}\)
Solution
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Problem 3
Solve each equation, and check your solution.
\(\frac19(2m-16) = \frac13(2m+4)\)
\(\text-4(r+2)=4(2-2r)\)
\(12(5+2y)=4y-(6-9y)\)
Solution
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Problem 4
Here is the graph of a linear equation.
Select all true statements about the line and its equation.
![The graph of a line in the x y plane. The line crosses the y axis at 1 point 25. The line also crosses through the point negative 1 comma 1.](https://cms-im.s3.amazonaws.com/wzDoFQhw83BhnJn8hjh7HhhW?response-content-disposition=inline%3B%20filename%3D%228-8.3.D13.PP.graph1.png%22%3B%20filename%2A%3DUTF-8%27%278-8.3.D13.PP.graph1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181557Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fa2e9c539d310397c51c26cdd27b6a4e792de1e166cc2792c0e985b31ff3762e)
One solution of the equation is \((3,2)\).
One solution of the equation is \((\text-1,1)\).
One solution of the equation is \(\left(1,\frac32\right)\).
There are 2 solutions.
There are infinitely many solutions.
The equation of the line is \(y=\frac14 x +\frac54\).
The equation of the line is \(y=\frac54 x +\frac14\).
Solution
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(From Unit 3, Lesson 13.)Problem 5
A participant in a 21-mile walkathon walks at a steady rate of 3 miles per hour. He thinks, “The relationship between the number of miles left to walk and the number of hours I already walked can be represented by a line with slope \(\text-3\).” Do you agree with his claim? Explain your reasoning.
Solution
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(From Unit 3, Lesson 9.)