Question

1. which equations have the same solution as the equation $\frac{2}{3}m + 4 = \frac{1}{3}-2m$? (choose all that apply.

a. $2m + 12 = 1 - 6m$
b. $\frac{2}{3}m + 2m = \frac{1}{3}-4$
c. $2m + 4 = 1 - 2m$
d. $4 + 2m = \frac{1}{3}-\frac{2}{3}m$
e. $4m + 24 = 2 - 12m$

Explanation:

First, solve the original equation \(\frac{2}{3}m + 4=\frac{1}{3}-2m\) to find the value of \(m\).

Step1: Add \(2m\) to both sides

\(\frac{2}{3}m + 2m+ 4=\frac{1}{3}-2m + 2m\)
\(\frac{2}{3}m+\frac{6}{3}m + 4=\frac{1}{3}\)
\(\frac{8}{3}m + 4=\frac{1}{3}\)

Step2: Subtract 4 from both sides

\(\frac{8}{3}m + 4-4=\frac{1}{3}-4\)
\(\frac{8}{3}m=\frac{1}{3}-\frac{12}{3}\)
\(\frac{8}{3}m=-\frac{11}{3}\)

Step3: Multiply both sides by \(\frac{3}{8}\)

\(m = -\frac{11}{8}\)

Now check each option:

Option A: \(2m + 12 = 1- 6m\)

Step1: Add \(6m\) to both sides

\(2m+6m + 12=1-6m + 6m\)
\(8m + 12=1\)

Step2: Subtract 12 from both sides

\(8m+12 - 12=1-12\)
\(8m=-11\)

Step3: Divide by 8

\(m = -\frac{11}{8}\) (Same solution)

Option B: \(\frac{2}{3}m+2m=\frac{1}{3}-4\)

Step1: Combine like terms

\(\frac{2}{3}m+\frac{6}{3}m=\frac{1}{3}-\frac{12}{3}\)
\(\frac{8}{3}m=-\frac{11}{3}\)

Step2: Multiply by \(\frac{3}{8}\)

\(m = -\frac{11}{8}\) (Same solution)

Option C: \(2m + 4=1-2m\)

Step1: Add \(2m\) to both sides

\(2m+2m + 4=1-2m+2m\)
\(4m + 4=1\)

Step2: Subtract 4

\(4m=1 - 4=-3\)

Step3: Divide by 4

\(m=-\frac{3}{4}\) (Different solution)

Option D: \(4 + 2m=\frac{1}{3}-\frac{1}{3}m\)

Step1: Add \(\frac{1}{3}m\) to both sides

\(4 + 2m+\frac{1}{3}m=\frac{1}{3}-\frac{1}{3}m+\frac{1}{3}m\)
\(4+\frac{7}{3}m=\frac{1}{3}\)

Step2: Subtract 4

\(\frac{7}{3}m=\frac{1}{3}-4=\frac{1}{3}-\frac{12}{3}=-\frac{11}{3}\)

Step3: Multiply by \(\frac{3}{7}\)

\(m=-\frac{11}{7}\) (Different solution)

Option E: \(4m + 24=2-12m\)

Step1: Add \(12m\) to both sides

\(4m+12m + 24=2-12m+12m\)
\(16m + 24=2\)

Step2: Subtract 24

\(16m=2 - 24=-22\)

Step3: Divide by 16

\(m=-\frac{22}{16}=-\frac{11}{8}\) (Same solution)

Answer:

A. \(2m + 12 = 1- 6m\), B. \(\frac{2}{3}m+2m=\frac{1}{3}-4\), E. \(4m + 24=2-12m\)