Question

solve the following system using the elimination method. enter your answer as an ordered pair in the form ((x, y)) if there is one, unique solution. enter all if there are infinitely many solutions and enter none if there are no solutions. (\begin{cases}-5x + 4y = 14 \\ 5x + 2y = -68end{cases})

Explanation:

Step1: Eliminate x by adding the two equations

The first equation is \(-5x + 4y = 14\) and the second is \(5x + 2y = -68\). When we add them together, the \(x\)-terms will cancel out.
\((-5x + 4y)+(5x + 2y)=14+(-68)\)
Simplifying the left side: \(-5x + 5x + 4y + 2y = 6y\)
Simplifying the right side: \(14 - 68 = -54\)
So we get \(6y=-54\)

Step2: Solve for y

Divide both sides of \(6y = -54\) by 6:
\(y=\frac{-54}{6}=-9\)

Step3: Substitute y back to find x

Substitute \(y = -9\) into the second equation \(5x + 2y = -68\) (we could use either equation, but the second one might be simpler here).
\(5x + 2(-9)=-68\)
Simplify: \(5x - 18 = -68\)
Add 18 to both sides: \(5x=-68 + 18=-50\)
Divide both sides by 5: \(x=\frac{-50}{5}=-10\)

Answer:

\((-10, -9)\)