Sure! Let's solve the given system of linear equations using the substitution method step-by-step.
### Part i:
Solve the following system of equations using the substitution method:
[tex]\[
\begin{cases}
3x - 2y = 5 \\
2x + y = 8
\end{cases}
\][/tex]
1. Solve the second equation for [tex]\( y \)[/tex]:
[tex]\[
2x + y = 8 \quad \Rightarrow \quad y = 8 - 2x
\][/tex]
2. Substitute [tex]\( y = 8 - 2x \)[/tex] into the first equation:
[tex]\[
3x - 2(8 - 2x) = 5
\][/tex]
Simplify the equation:
[tex]\[
3x - 16 + 4x = 5 \quad \Rightarrow \quad 7x - 16 = 5 \quad \Rightarrow \quad 7x = 21 \quad \Rightarrow \quad x = 3
\][/tex]
3. Substitute [tex]\( x = 3 \)[/tex] back into the equation [tex]\( y = 8 - 2x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 8 - 2(3) \quad \Rightarrow \quad y = 8 - 6 \quad \Rightarrow \quad y = 2
\][/tex]
Therefore, the solution to the first system of equations is:
[tex]\[
(x, y) = (3, 2)
\][/tex]
### Part ii:
Solve the following system of equations using the substitution method:
[tex]\[
\begin{cases}
7x - y = 14 \\
8x - 2y = 16
\end{cases}
\][/tex]
1. Solve the second equation for [tex]\( y \)[/tex]:
[tex]\[
8x - 2y = 16 \quad \Rightarrow \quad -2y = 16 - 8x \quad \Rightarrow \quad y = \frac{8x - 16}{2} \quad \Rightarrow \quad y = 4x - 8
\][/tex]
2. Substitute [tex]\( y = 4x - 8 \)[/tex] into the first equation:
[tex]\[
7x - (4x - 8) = 14
\][/tex]
Simplify the equation:
[tex]\[
7x - 4x + 8 = 14 \quad \Rightarrow \quad 3x + 8 = 14 \quad \Rightarrow \quad 3x = 6 \quad \Rightarrow \quad x = 2
\][/tex]
3. Substitute [tex]\( x = 2 \)[/tex] back into the equation [tex]\( y = 4x - 8 \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 4(2) - 8 \quad \Rightarrow \quad y = 8 - 8 \quad \Rightarrow \quad y = 0
\][/tex]
Therefore, the solution to the second system of equations is:
[tex]\[
(x, y) = (2, 0)
\][/tex]
### Summary of Solutions:
- For the first system: [tex]\((x, y) = (3, 2)\)[/tex]
- For the second system: [tex]\((x, y) = (2, 0)\)[/tex]
These solutions are found by following the step-by-step substitution method.
