To solve the system of equations and find the value of [tex]\(x - y\)[/tex] for both given sets of equations, we'll proceed with solving each system step-by-step.
### Part (i):
Given equations:
[tex]\[
\begin{cases}
5x + 4y = 14 \quad \text{(1)} \\
4x + 5y = 13 \quad \text{(2)}
\end{cases}
\][/tex]
First, let's solve the system:
1. Multiply equation (1) by 5 and equation (2) by 4 to make the coefficients of [tex]\( y \)[/tex] the same:
[tex]\[
\begin{cases}
25x + 20y = 70 \quad \text{(3)} \\
16x + 20y = 52 \quad \text{(4)}
\end{cases}
\][/tex]
2. Subtract equation (4) from equation (3):
[tex]\[
(25x + 20y) - (16x + 20y) = 70 - 52 \\
25x - 16x = 18 \\
9x = 18 \\
x = 2
\][/tex]
3. Substitute [tex]\( x = 2 \)[/tex] back into equation (1) to find [tex]\( y \)[/tex]:
[tex]\[
5(2) + 4y = 14 \\
10 + 4y = 14 \\
4y = 4 \\
y = 1
\][/tex]
4. Now, calculate [tex]\( x - y \)[/tex]:
[tex]\[
x - y = 2 - 1 = 1
\][/tex]
### Part (ii):
Given equations:
[tex]\[
\begin{cases}
2x - 5y = 5 \quad \text{(1)} \\
5x - 8y = 14 \quad \text{(2)}
\end{cases}
\][/tex]
First, let's solve the system:
1. Multiply equation (1) by 5 and equation (2) by 2 to make the coefficients of [tex]\( x \)[/tex] the same:
[tex]\[
\begin{cases}
10x - 25y = 25 \quad \text{(3)} \\
10x - 16y = 28 \quad \text{(4)}
\end{cases}
\][/tex]
2. Subtract equation (4) from equation (3):
[tex]\[
(10x - 25y) - (10x - 16y) = 25 - 28 \\
-25y + 16y = -3 \\
-9y = -3 \\
y = \frac{1}{3}
\][/tex]
3. Substitute [tex]\( y = \frac{1}{3} \)[/tex] back into equation (1) to find [tex]\( x \)[/tex]:
[tex]\[
2x - 5\left(\frac{1}{3}\right) = 5 \\
2x - \frac{5}{3} = 5 \\
2x = 5 + \frac{5}{3} \\
2x = \frac{15}{3} + \frac{5}{3} \\
2x = \frac{20}{3} \\
x = \frac{10}{3}
\][/tex]
4. Now, calculate [tex]\( x - y \)[/tex]:
[tex]\[
x - y = \frac{10}{3} - \frac{1}{3} = \frac{9}{3} = 3
\][/tex]
### Conclusion
After solving both systems, we find:
[tex]\[
\begin{aligned}
(i) &\quad x - y = 1 \\
(ii) &\quad x - y = 3
\end{aligned}
\][/tex]
Thus, the values of [tex]\( x - y \)[/tex] are [tex]\( 1 \)[/tex] and [tex]\( 3 \)[/tex] respectively for the two systems of equations.
