# Lesson 5

Solving Any Linear Equation

Let's solve linear equations.

### Problem 1

Solve each of these equations. Explain or show your reasoning.

$$2(x+5)=3x+1$$

$$3y-4=6-2y$$

$$3(n+2)=9(6-n)$$

### 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}

### Problem 3

Solve each equation, and check your solution.

$$\frac19(2m-16) = \frac13(2m+4)$$

$$\text-4(r+2)=4(2-2r)$$

$$12(5+2y)=4y-(6-9y)$$

### Problem 4

Here is the graph of a linear equation.

Select all true statements about the line and its equation.

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 $$y=\frac54 x +\frac14$$.

(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.

(From Unit 3, Lesson 9.)