Find the solution of the system of equations: -7x + 4y = -32 and 9x - 2y = 38.

Understand the Problem

The question is asking us to find the solution to a system of two linear equations. The equations given are -7x + 4y = -32 and 9x - 2y = 38. To solve this, we typically look for values of x and y that satisfy both equations simultaneously.

Answer

The solution to the system of equations is $ (4, -1) $.

Steps to Solve

Rewrite the equations for easier manipulation

The equations are: $$ -7x + 4y = -32 $$ $$ 9x - 2y = 38 $$

Rearrange the first equation to isolate y

From the first equation, solve for ( y ): $$ 4y = 7x - 32 $$ $$ y = \frac{7x - 32}{4} $$

Substitute y in the second equation

Substitute ( y ) into the second equation: $$ 9x - 2\left(\frac{7x - 32}{4}\right) = 38 $$

Clear the fraction

Multiply every term by 4 to eliminate the fraction: $$ 4(9x) - 2(7x - 32) = 4(38) $$ This gives: $$ 36x - 14x + 64 = 152 $$

Combine like terms

Combine the ( x ) terms and simplify: $$ 22x + 64 = 152 $$

Isolate x

Subtract 64 from both sides: $$ 22x = 152 - 64 $$ $$ 22x = 88 $$

Now divide by 22: $$ x = \frac{88}{22} $$ $$ x = 4 $$

Substitute x back to find y

Now substitute ( x = 4 ) back into the rearranged equation for ( y ): $$ y = \frac{7(4) - 32}{4} $$ $$ y = \frac{28 - 32}{4} $$ $$ y = \frac{-4}{4} $$ $$ y = -1 $$

Final solution

Thus, the solution to the system of equations is: $$ (x, y) = (4, -1) $$

The solution to the system of equations is ( (4, -1) ).

More Information

This type of problem is a classic example of solving simultaneous linear equations, where both equations are true for the same ( (x, y) ) pair. It's often solved using substitution or elimination methods.

Tips

Forgetting to distribute correctly when clearing fractions.
Mistakenly combining constants and variables when simplifying.
Not checking the solution by substituting back into the original equations.

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