We are given the following system of linear equations:

1. \( 5x - 6y = 52 \)
2. \( 3x + 7y = 10 \)

We will solve for \( x \) using the method of substitution or elimination. I'll use the elimination method here.

### Step 1: Multiply the equations to align the coefficients of \( x \).

We can multiply the first equation by 3 and the second equation by 5, so that the coefficients of \( x \) in both equations will be the same:

\[
(5x - 6y = 52) \times 3 \implies 15x - 18y = 156
\]
\[
(3x + 7y = 10) \times 5 \implies 15x + 35y = 50
\]

### Step 2: Subtract the equations to eliminate \( x \).

Now, subtract the second equation from the first:

\[
(15x - 18y) - (15x + 35y) = 156 - 50
\]
\[
0x - 53y = 106
\]
\[
-53y = 106
\]

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

Divide both sides by \(-53\):

\[
y = \frac{106}{-53} = -2
\]

### Step 4: Substitute \( y = -2 \) into one of the original equations to solve for \( x \).

Using the first equation \( 5x - 6y = 52 \):

\[
5x - 6(-2) = 52
\]
\[
5x + 12 = 52
\]
\[
5x = 52 - 12
\]
\[
5x = 40
\]
\[
x = \frac{40}{5} = 8
\]

### Final Solution:
The value of \( x \) is \( 8 \).

Would you like any further details or have questions?

Here are 5 related questions:
1. How would you solve the system using the substitution method?
2. What is the geometric interpretation of solving a system of two linear equations?
3. How can you check if the solution is correct?
4. What happens if the system has no solution or infinite solutions?
5. How does the determinant of the coefficient matrix help in solving linear systems?

**Tip:** Always double-check your calculations by substituting the values back into both original equations to verify the solution.

Learn how to solve the system of linear equations 5x - 6y = 52 and 3x + 7y = 10 using the elimination method. This problem covers important algebraic concepts, including the elimination of variables and solving linear systems. Step-by-step solution for high school students in Grades 9-12.

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