Sure, let's solve the pair of linear equations by the substitution method. We'll address the equations one by one.
### Equation 7:
Given the pair of linear equations:
1. [tex]\( x + v = 8 \)[/tex]
2. [tex]\( x - v = 4 \)[/tex]
#### Step-by-Step Solution:
1. Isolate one variable in one of the equations:
From the first equation:
[tex]\[ x + v = 8 \][/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[ x = 8 - v \][/tex]
2. Substitute this expression into the second equation:
Now substitute [tex]\( x = 8 - v \)[/tex] into the second equation [tex]\( x - v = 4 \)[/tex]:
[tex]\[ (8 - v) - v = 4 \][/tex]
3. Simplify the equation:
Combine like terms:
[tex]\[ 8 - v - v = 4 \][/tex]
[tex]\[ 8 - 2v = 4 \][/tex]
4. Solve for [tex]\( v \)[/tex]:
Isolate [tex]\( v \)[/tex] by first subtracting 8 from both sides of the equation:
[tex]\[ 8 - 8 - 2v = 4 - 8 \][/tex]
[tex]\[ -2v = -4 \][/tex]
Divide both sides by -2:
[tex]\[ v = \frac{-4}{-2} \][/tex]
[tex]\[ v = 2 \][/tex]
5. Substitute back to find [tex]\( x \)[/tex]:
Use the value of [tex]\( v \)[/tex] to find [tex]\( x \)[/tex] by substituting [tex]\( v = 2 \)[/tex] back into the expression [tex]\( x = 8 - v \)[/tex]:
[tex]\[ x = 8 - 2 \][/tex]
[tex]\[ x = 6 \][/tex]
6. Solution:
The solution to the system of equations is:
[tex]\[
\begin{cases}
x = 6 \\
v = 2
\end{cases}
\][/tex]
### Verifying the Solution:
Let's verify the solution by substituting [tex]\( x = 6 \)[/tex] and [tex]\( v = 2 \)[/tex] back into the original equations:
For the first equation [tex]\( x + v = 8 \)[/tex]:
[tex]\[ 6 + 2 = 8 \][/tex]
[tex]\[ 8 = 8 \][/tex]
For the second equation [tex]\( x - v = 4 \)[/tex]:
[tex]\[ 6 - 2 = 4 \][/tex]
[tex]\[ 4 = 4 \][/tex]
Both equations are satisfied, hence, the solution is correct.
Therefore, the values that solve the system of equations [tex]\( x + v = 8 \)[/tex] and [tex]\( x - v = 4 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( v = 2 \)[/tex].
