To solve the system of linear equations:

\[
\begin{align*}
1) & \quad 3x - 5y = 8 \\
2) & \quad 4x - 7y = 12
\end{align*}
\]

We can use either the substitution method, elimination method, or matrix method. I'll use the elimination method here:

### Step 1: Multiply the equations to align coefficients for elimination.

To eliminate \(x\) or \(y\), we'll multiply the first equation by 4 and the second equation by 3 to align the coefficients of \(x\):

\[
\begin{align*}
4(3x - 5y) & = 4(8) \\
3(4x - 7y) & = 3(12)
\end{align*}
\]

This gives us:

\[
\begin{align*}
12x - 20y & = 32 \quad \text{(Equation 3)} \\
12x - 21y & = 36 \quad \text{(Equation 4)}
\end{align*}
\]

### Step 2: Subtract the equations.

Now subtract Equation 4 from Equation 3 to eliminate \(x\):

\[
(12x - 20y) - (12x - 21y) = 32 - 36
\]

Simplifying:

\[
y = 4
\]

### Step 3: Substitute \(y = 4\) back into one of the original equations.

Let's substitute into the first equation \(3x - 5y = 8\):

\[
3x - 5(4) = 8
\]

Simplifying:

\[
3x - 20 = 8 \implies 3x = 28 \implies x = \frac{28}{3}
\]

### Solution:
\[
x = \frac{28}{3}, \quad y = 4
\]

The solution to the system is \( \left(\frac{28}{3}, 4\right) \).

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. How would you solve the system using the substitution method?
2. Can you graph these equations and find the point of intersection?
3. What happens if the system has no solution or infinite solutions?
4. How does the elimination method differ from Gaussian elimination?
5. Can you solve the same system using matrix methods like Cramer's rule?

**Tip:** When solving systems of equations, always check the final solution by substituting the values back into the original equations to ensure accuracy.

Learn how to solve the system of linear equations 3x - 5y = 8, 4x - 7y = 12 using the elimination method. Step-by-step solution provided with detailed algebraic operations. Suitable for high school students.

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