-10x + 7y = 5; 5x - 7y = -20. Find (x, y).

Understand the Problem

The question is asking to solve the system of equations for the variables x and y. We'll find the values of x and y that satisfy both equations.

Answer

The solution is $(3, 5)$.

Steps to Solve

Rearrange the equations for elimination or substitution

Let's take the two equations: $$ -10x + 7y = 5 \quad (1) $$ $$ 5x - 7y = -20 \quad (2) $$

We can add both equations to eliminate $y$, since they have opposite coefficients.

Add the equations

Adding equations (1) and (2): $$ (-10x + 7y) + (5x - 7y) = 5 + (-20) $$

This simplifies to: $$ -5x = -15 $$

Solve for x

Dividing both sides by -5: $$ x = 3 $$

Substitute x back into one of the original equations

Let's substitute $x = 3$ back into equation (1): $$ -10(3) + 7y = 5 $$

This simplifies to: $$ -30 + 7y = 5 $$

Solve for y

Adding 30 to both sides: $$ 7y = 35 $$

Now, divide by 7: $$ y = 5 $$

Final solution

The solution to the system of equations is $(x, y) = (3, 5)$.

The final answer is $(3, 5)$.

More Information

The solution means that when $x = 3$ and $y = 5$, both original equations are satisfied. This method can also be called the elimination method, which is useful for solving systems of linear equations.

Tips

Forgetting to check back in original equations: It's good practice to substitute found values back into the original equations to ensure they satisfy both.
Sign errors when adding or subtracting equations: Double-check the signs carefully when combining or rearranging equations.

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