Question

question
solve the system of equations -3x - 2y = -14 and -x - 7y = -11 by combining the equations.
( -3x - 2y = -14 )
3 ( -x - 7y = -11 )
-3x - 2y = -14
-3x - 21y = -33
overline{quadquadquadquadquadquadquad}
x + box y = box

Explanation:

Step1: Multiply the second equation

We have the system:

$$
LATEXBLOCK0
$$

Multiply the second equation by 3 to make the coefficients of \(x\) opposite:
\(3\times(-x - 7y)=3\times(-11)\)
\(-3x - 21y = -33\)

Step2: Subtract the first equation

Now we have:

$$
LATEXBLOCK1
$$

Subtract the first equation from the second equation:
\((-3x - 21y)-(-3x - 2y)=-33 - (-14)\)
Simplify the left side: \(-3x - 21y + 3x + 2y=-19y\)
Simplify the right side: \(-33 + 14=-19\)
So, \(-19y=-19\)

Step3: Solve for \(y\)

Divide both sides by \(-19\):
\(y = \frac{-19}{-19}=1\)

Step4: Substitute \(y\) into first equation

Substitute \(y = 1\) into \(-3x - 2y = -14\):
\(-3x - 2\times1=-14\)
\(-3x - 2=-14\)

Step5: Solve for \(x\)

Add 2 to both sides:
\(-3x=-14 + 2=-12\)
Divide both sides by \(-3\):
\(x=\frac{-12}{-3}=4\)

Answer:

The solution to the system is \(x = 4\), \(y = 1\)