Sure, let's solve the system of linear equations step-by-step to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
The system of equations is:
[tex]\[
\begin{cases}
5x - 4y = 6 \quad \text{(Equation 1)} \\
2x - 5y = 3 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
### Step 1: Express one variable in terms of the other
Let's solve Equation 1 for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
5x - 4y = 6
\][/tex]
[tex]\[
5x = 6 + 4y
\][/tex]
[tex]\[
x = \frac{6 + 4y}{5}
\][/tex]
### Step 2: Substitute [tex]\(x\)[/tex] into the second equation
Now, we substitute [tex]\(x\)[/tex] from Equation 1 into Equation 2:
[tex]\[
2 \left(\frac{6 + 4y}{5}\right) - 5y = 3
\][/tex]
### Step 3: Clear the fraction by multiplying through by 5
[tex]\[
2 \cdot (6 + 4y) - 5 \cdot 5y = 3 \cdot 5
\][/tex]
[tex]\[
12 + 8y - 25y = 15
\][/tex]
### Step 4: Combine like terms
[tex]\[
12 - 17y = 15
\][/tex]
### Step 5: Isolate [tex]\(y\)[/tex]
Subtract 12 from both sides:
[tex]\[
-17y = 3
\][/tex]
Divide by -17:
[tex]\[
y = \frac{3}{-17}
\][/tex]
[tex]\[
y = -\frac{3}{17}
\][/tex]
### Step 6: Substitute [tex]\(y\)[/tex] back into the expression for [tex]\(x\)[/tex]
Using the [tex]\(y\)[/tex] value in the expression for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{6 + 4\left(-\frac{3}{17}\right)}{5}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{6 - \frac{12}{17}}{5}
\][/tex]
Convert 6 to a fraction with a denominator of 17:
[tex]\[
6 = \frac{102}{17}
\][/tex]
So,
[tex]\[
x = \frac{\frac{102}{17} - \frac{12}{17}}{5}
\][/tex]
[tex]\[
x = \frac{\frac{90}{17}}{5}
\][/tex]
Simplify the division:
[tex]\[
x = \frac{90}{17} \cdot \frac{1}{5}
\][/tex]
[tex]\[
x = \frac{90}{85}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{18}{17}
\][/tex]
### Final Answer
Thus, the solution to the system of equations is:
[tex]\[
x = \frac{18}{17}, \quad y = -\frac{3}{17}
\][/tex]
