To solve the system of equations:

\[
\begin{align*}
1) & \quad 12x - 7y = 96 \\
2) & \quad 10x - 9y = 42
\end{align*}
\]

We can solve this system using the substitution or elimination method. Here, I'll use the **elimination method**.

### Step 1: Make the coefficients of \(x\) or \(y\) the same

First, let's eliminate \(x\). We multiply the first equation by 10 and the second equation by 12 to make the coefficients of \(x\) equal.

\[
\begin{align*}
10 \times (12x - 7y) &= 10 \times 96 \\
12 \times (10x - 9y) &= 12 \times 42
\end{align*}
\]

This gives us:

\[
\begin{align*}
120x - 70y &= 960 \quad \text{(Equation 3)} \\
120x - 108y &= 504 \quad \text{(Equation 4)}
\end{align*}
\]

### Step 2: Subtract the two equations

Subtract Equation 4 from Equation 3 to eliminate \(x\):

\[
(120x - 70y) - (120x - 108y) = 960 - 504
\]

This simplifies to:

\[
38y = 456
\]

### Step 3: Solve for \(y\)

Now, divide by 38:

\[
y = \frac{456}{38} = 12
\]

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

We can substitute \(y = 12\) into the first original equation:

\[
12x - 7(12) = 96
\]

Simplify:

\[
12x - 84 = 96
\]

Add 84 to both sides:

\[
12x = 180
\]

Divide by 12:

\[
x = 15
\]

### Solution
The solution to the system is:

\[
x = 15, \quad y = 12
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you solve a system of equations using substitution?
2. What is the elimination method in solving systems of equations?
3. How can you verify the solution of a system of equations?
4. What is the significance of consistent and inconsistent systems?
5. Can you solve the system using matrix methods (e.g., Cramer's Rule)?

**Tip:** Always check your solution by substituting the values back into the original equations to ensure they satisfy both equations.

Learn how to solve the system of equations 12x - 7y = 96 and 10x - 9y = 42 using the elimination method with detailed step-by-step solutions. This problem covers systems of linear equations and the elimination method, suitable for students in Grades 9-12.

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