Sure! Let's solve the system of equations step-by-step:
The given system of equations is:
[tex]\[
\begin{cases}
\frac{x}{3} + \frac{y}{4} = 4 & \quad \text{(1)} \\
\frac{5x}{6} - \frac{y}{a} = 4 & \quad \text{(2)}
\end{cases}
\][/tex]
Step 1: Multiply each equation to eliminate the denominators.
To eliminate the denominators in equation (1):
[tex]\[
\frac{x}{3} + \frac{y}{4} = 4
\][/tex]
Multiply through by 12 (the least common multiple of 3 and 4):
[tex]\[
4x + 3y = 48 \quad \text{(3)}
\][/tex]
To eliminate the denominators in equation (2):
[tex]\[
\frac{5x}{6} - \frac{y}{a} = 4
\][/tex]
Multiply through by 6a (the least common multiple of 6 and a):
[tex]\[
5ax - 6y = 24a \quad \text{(4)}
\][/tex]
So, our system of equations now looks like:
[tex]\[
\begin{cases}
4x + 3y = 48 & \quad \text{(3)} \\
5ax - 6y = 24a & \quad \text{(4)}
\end{cases}
\][/tex]
Step 2: Solve the system using substitution or elimination.
Let's solve for [tex]\( y \)[/tex] from equation (3):
[tex]\[
4x + 3y = 48 \implies 3y = 48 - 4x \implies y = \frac{48 - 4x}{3} \quad \text{(5)}
\][/tex]
Substitute equation (5) into equation (4):
[tex]\[
5ax - 6 \left( \frac{48 - 4x}{3} \right) = 24a
\][/tex]
Simplify this equation:
[tex]\[
5ax - 2 (48 - 4x) = 24a
\][/tex]
[tex]\[
5ax - 96 + 8x = 24a
\][/tex]
Combine like terms:
[tex]\[
5ax + 8x = 24a + 96
\][/tex]
Factor out [tex]\(x\)[/tex]:
[tex]\[
x (5a + 8) = 24a + 96
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{24a + 96}{5a + 8} \quad \text{(6)}
\][/tex]
Step 3: Solve for [tex]\(y\)[/tex] by substituting [tex]\(x\)[/tex] back into equation (5):
Substitute [tex]\(x\)[/tex] from equation (6) into equation (5):
[tex]\[
y = \frac{48 - 4 \left( \frac{24a + 96}{5a + 8} \right)}{3}
\][/tex]
Simplify this:
[tex]\[
y = \frac{48 - \frac{96a + 384}{5a + 8}}{3}
\][/tex]
[tex]\[
y = \frac{48(5a + 8) - (96a + 384)}{3(5a + 8)}
\][/tex]
[tex]\[
y = \frac{240a + 384 - 96a - 384}{3(5a + 8)}
\][/tex]
[tex]\[
y = \frac{144a}{3(5a + 8)}
\][/tex]
[tex]\[
y = \frac{48a}{5a + 8} \quad \text{(7)}
\][/tex]
Final Result:
The solution to the system of equations is:
[tex]\[
x = \frac{24a + 96}{5a + 8}
\][/tex]
[tex]\[
y = \frac{48a}{5a + 8}
\][/tex]
Thus, we have the values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] for any given [tex]\(a\)[/tex].
