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Solve the following system of equations:

[tex]\[
\begin{cases}
6x - 54y = -42 \\
-8x - 3y = 56
\end{cases}
\][/tex]

Sagot :

GinnyAnswer
To solve the given system of linear equations:

[tex]\[ \begin{cases} 6x - 54y = -42 \\ -8x - 3y = 56 \\ \end{cases} \][/tex]

we will follow a detailed, step-by-step approach.

### Step 1: Simplify the Equations (if possible)

First, let's see if the coefficients can be simplified to make the equations easier to work with.

For the first equation:
[tex]\[ 6x - 54y = -42 \][/tex]

Divide every term by 6:
[tex]\[ x - 9y = -7 \][/tex]

So, the first equation simplifies to:
[tex]\[ x - 9y = -7 \quad \text{(Equation 1)} \][/tex]

For the second equation:
[tex]\[ -8x - 3y = 56 \][/tex]

This equation does not simplify any further. Thus, we have:
[tex]\[ \begin{cases} x - 9y = -7 \\ -8x - 3y = 56 \\ \end{cases} \][/tex]

### Step 2: Solve One Equation for One Variable

From Equation 1:
[tex]\[ x - 9y = -7 \][/tex]

Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 9y - 7 \quad \text{(Equation 3)} \][/tex]

### Step 3: Substitute the Expression from Step 2 into the Other Equation

Now substitute Equation 3 into the second original equation:
[tex]\[ -8x - 3y = 56 \][/tex]

Substituting [tex]\( x = 9y - 7 \)[/tex]:
[tex]\[ -8(9y - 7) - 3y = 56 \][/tex]

Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ -72y + 56 - 3y = 56 \][/tex]

Combine like terms:
[tex]\[ -75y + 56 = 56 \][/tex]

Subtract 56 from both sides:
[tex]\[ -75y = 0 \][/tex]

Solve for [tex]\( y \)[/tex]:
[tex]\[ y = 0 \][/tex]

### Step 4: Substitute Back to Find the Other Variable

Substitute [tex]\( y = 0 \)[/tex] back into Equation 3:
[tex]\[ x = 9(0) - 7 \][/tex]

Simplify:
[tex]\[ x = -7 \][/tex]

### Conclusion

The solution to the system of equations is:
[tex]\[ (x, y) = (-7, 0) \][/tex]
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