Solve the system of equations: -5x + 6y = 12 and -x - y = -13.

Understand the Problem

The question is asking to solve a system of linear equations. We have two equations: -5x + 6y = 12 and -x - y = -13. We will find the values of x and y that satisfy both equations.

Answer

The solution is $(6, 7)$.

Steps to Solve

Rearranging the Second Equation

Start by rearranging the second equation, $-x - y = -13$.

We can add $x$ and $y$ to both sides: $$ y = -x + 13 $$
Substituting into the First Equation

Substitute $y$ from the rearranged second equation into the first equation $-5x + 6y = 12$:

$$ -5x + 6(-x + 13) = 12 $$
Expanding the Equation

Now, expand the equation: $$ -5x - 6x + 78 = 12 $$

Combine like terms: $$ -11x + 78 = 12 $$
Isolating x

Next, isolate $x$ by subtracting $78$ from both sides: $$ -11x = 12 - 78 $$ $$ -11x = -66 $$

Now, divide both sides by $-11$: $$ x = 6 $$
Finding y

Substitute $x = 6$ back into the equation for $y$: $$ y = -6 + 13 $$ $$ y = 7 $$

The solution to the system of equations is $(6, 7)$.

More Information

This solution indicates that the two lines represented by the equations intersect at the point $(6, 7)$. Linear equations can often be solved using substitution, elimination, or graphing methods to find such intersection points.

Tips

Not isolating y correctly: Failing to properly rearrange the second equation before substitution can lead to incorrect results.
Errors in arithmetic during the expansion or combining like terms can result in wrong values for $x$ and $y$.
Incorrect sign handling: Neglecting to properly manage negative signs during subtraction or division.

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