Answer :

GinnyAnswer
To solve the system of linear equations given by:

[tex]\[ 3x + 4y = -22 \][/tex]
[tex]\[ 2x - 5y = 39 \][/tex]

we can proceed using the method of elimination or substitution. Here, I will use the elimination method:

1. Multiplying Equations for Elimination:

To eliminate one of the variables, let's aim to make the coefficients of [tex]\( y \)[/tex] the same (in magnitude) in both equations.

Let's multiply the first equation by 5 and the second equation by 4:

[tex]\[ 5(3x + 4y) = 5(-22) \][/tex]
[tex]\[ 4(2x - 5y) = 4(39) \][/tex]

This results in:

[tex]\[ 15x + 20y = -110 \][/tex]
[tex]\[ 8x - 20y = 156 \][/tex]

2. Adding Equations to Eliminate y:

Now, add these two equations together to eliminate [tex]\( y \)[/tex]:

[tex]\[ (15x + 20y) + (8x - 20y) = -110 + 156 \][/tex]

Simplifying this, we get:

[tex]\[ 15x + 8x = 46 \][/tex]
[tex]\[ 23x = 46 \][/tex]

3. Solving for x:

Divide both sides by 23 to solve for [tex]\( x \)[/tex]:

[tex]\[ x = \frac{46}{23} \][/tex]
[tex]\[ x = 2 \][/tex]

4. Substituting x back into one of the original equations:

Now, substitute [tex]\( x = 2 \)[/tex] back into the original first equation:

[tex]\[ 3(2) + 4y = -22 \][/tex]
[tex]\[ 6 + 4y = -22 \][/tex]

5. Solving for y:

Subtract 6 from both sides:

[tex]\[ 4y = -22 - 6 \][/tex]
[tex]\[ 4y = -28 \][/tex]

Divide by 4:

[tex]\[ y = \frac{-28}{4} \][/tex]
[tex]\[ y = -7 \][/tex]

So, the solution to the system of equations is:

[tex]\[ (x, y) = (2, -7) \][/tex]

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