Lesson 5

Solving Any Linear Equation

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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.
A:

One solution of the equation is \((3,2)\).

B:

One solution of the equation is \((\text-1,1)\).

C:

One solution of the equation is \(\left(1,\frac32\right)\).

D:

There are 2 solutions.

E:

There are infinitely many solutions.

F:

The equation of the line is \(y=\frac14 x +\frac54\).

G:

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.)