Sure! Let's solve the given linear equation step-by-step and verify the result.
The equation is:
[tex]\[
\frac{4m + 3}{7} + 2 = \frac{m + 5}{2}
\][/tex]
### Step 1: Eliminate the fractions
To eliminate the fractions, we can multiply every term by the least common multiple (LCM) of the denominators. The denominators are 7 and 2, so the LCM is 14.
Multiply every term by 14:
[tex]\[
14 \cdot \left(\frac{4m + 3}{7}\right) + 14 \cdot 2 = 14 \cdot \left(\frac{m + 5}{2}\right)
\][/tex]
This simplifies to:
[tex]\[
2(4m + 3) + 28 = 7(m + 5)
\][/tex]
### Step 2: Distribute and simplify both sides
Distribute the constants inside the parentheses:
[tex]\[
2 \cdot 4m + 2 \cdot 3 + 28 = 7 \cdot m + 7 \cdot 5
\][/tex]
This simplifies to:
[tex]\[
8m + 6 + 28 = 7m + 35
\][/tex]
Combine like terms:
[tex]\[
8m + 34 = 7m + 35
\][/tex]
### Step 3: Isolate the variable
To isolate [tex]\(m\)[/tex], subtract [tex]\(7m\)[/tex] from both sides:
[tex]\[
8m - 7m + 34 = 35
\][/tex]
This simplifies to:
[tex]\[
m + 34 = 35
\][/tex]
Now, subtract 34 from both sides:
[tex]\[
m = 35 - 34
\][/tex]
This gives:
[tex]\[
m = 1
\][/tex]
### Step 4: Verify the solution
Substitute [tex]\(m = 1\)[/tex] back into the original equation to verify it satisfies the equation:
[tex]\[
\frac{4(1) + 3}{7} + 2 = \frac{1 + 5}{2}
\][/tex]
Simplify inside each fraction:
[tex]\[
\frac{4 + 3}{7} + 2 = \frac{6}{2}
\][/tex]
So,
[tex]\[
\frac{7}{7} + 2 = 3
\][/tex]
This simplifies to:
[tex]\[
1 + 2 = 3
\][/tex]
Indeed, [tex]\(3 = 3\)[/tex] is true.
Therefore, the solution to the equation is:
[tex]\[
m = 1
\][/tex]
And we have verified that this value satisfies the original equation.
