To solve the system of linear equations using the elimination method, we have the following equations:

1. $10x + 7y = 12$
2. $8x + 7y = 18$

Step 1: Eliminate one variable

Notice that both equations have the same $7y$ term. We can eliminate $y$ by subtracting the second equation from the first. This will cancel out $y$:

$(10x + 7y) - (8x + 7y) = 12 - 18$

Simplifying the equation:

$(10x - 8x) + (7y - 7y) = -6$

$2x = -6$

Step 2: Solve for $x$

Now solve for $x$:

$x = \frac{-6}{2} = -3$

Step 3: Substitute $x = -3$ into one of the original equations

Substitute $x = -3$ into the first equation, $10x + 7y = 12$:

$10(-3) + 7y = 12$

$-30 + 7y = 12$

Step 4: Solve for $y$

Now solve for $y$:

$7y = 12 + 30$

$7y = 42$

$y = \frac{42}{7} = 6$

Final Answer:

The solution to the system is:

$x = -3, \quad y = 6$

Want more details? Or have any questions?

Here are five related questions:

1. What is the graphical interpretation of solving a system of linear equations by elimination?
2. How can the substitution method be used to solve this system instead of elimination?
3. What would happen if the coefficients of $y$ in both equations were different?
4. How can you solve a system of equations with three variables using the elimination method?
5. How can we check if our solution is correct using substitution?

Tip: Always check your solution by substituting the values back into the original equations to verify they satisfy both equations!