Sure, let's solve each equation step-by-step for the given variable.
### Problem 5:
Equation: [tex]\( y = x + 12 \)[/tex]
Solve for: [tex]\( x \)[/tex]
1. To isolate [tex]\( x \)[/tex], we need to move the constant term (12) to the other side of the equation. We do this by subtracting 12 from both sides of the equation:
[tex]\[
y - 12 = x
\][/tex]
2. The final equation is:
[tex]\[
x = y - 12
\][/tex]
3. Hence, when given a specific value of [tex]\( y \)[/tex] (note in the problem we weren't given a specific value for [tex]\( y \)[/tex], so the solution is general):
[tex]\[
x = y - 12
\][/tex]
4. For example, if [tex]\( y = 0 \)[/tex]:
[tex]\[
x = 0 - 12 = -12
\][/tex]
### Problem 6:
Equation: [tex]\( n = \frac{4}{5} (m + 7) \)[/tex]
Solve for: [tex]\( m \)[/tex]
1. To isolate [tex]\( m \)[/tex], start by eliminating the fraction. Multiply both sides of the equation by the reciprocal of [tex]\( \frac{4}{5} \)[/tex], which is [tex]\( \frac{5}{4} \)[/tex]:
[tex]\[
\frac{5}{4} \cdot n = m + 7
\][/tex]
2. Next, move the constant term (7) to the other side of the equation. We do this by subtracting 7 from both sides:
[tex]\[
\frac{5}{4} n - 7 = m
\][/tex]
3. The final equation is:
[tex]\[
m = \frac{5}{4} n - 7
\][/tex]
4. Hence, when given a specific value of [tex]\( n \)[/tex] (note in the problem we weren't given a specific value for [tex]\( n \)[/tex], so the solution is general):
[tex]\[
m = \frac{5}{4} n - 7
\][/tex]
5. For example, if [tex]\( n = 0 \)[/tex]:
[tex]\[
m = \frac{5}{4} \cdot 0 - 7 = -7
\][/tex]
So, putting it all together, we have the following results for each equation:
1. [tex]\( x = -12 \)[/tex] when [tex]\( y = 0 \)[/tex]
2. [tex]\( m = -7 \)[/tex] when [tex]\( n = 0 \)[/tex]
