Question

Solve the system by elimination:$\left\{\begin{array}{l}7x+8y=4\\ 3x-5y=27\end{array}\right.$

Short answer

The solution of the system of equations is$(4,-3)$.

Step-by-step solution

Step 1. Given Information. 

We have been given system of equations$\left\{\begin{array}{l}7x+8y=4\\ 3x-5y=27\end{array}\right.$.

Step 2. Eliminate the y's.

To get opposite coefficients of y, we will multiply the first equation by 5 and the second equation by 8.

$\left\{\begin{array}{l}5(7x+8y)=5\left(4\right)\\ 8(3x-5y)=8\left(27\right)\end{array}\right.$

localid="1644381617193" $\left\{\begin{array}{l}35x+40y=20\\ 24x-40y=216\end{array}\right.$

Step 3. Add the equations resulting from Step 2 to eliminate one variable and solve it.

Add the $x\text{'}s,y\text{'}s$and constants.

$\left\{\begin{array}{l}35x+40y=20\\ 24x-40y=216\end{array}\right.$

$59x=236$

$x=4$

Step 4. Substitute the solution from Step 3 into one of the original equation and solve for the other variable.

Substitute $x=4$into the first equation.

$7x+8y=4$

$7\left(4\right)+8y=4$

$28+8y=4$

$8y=-24$

$y=-3$

Step 5. Write the solution as an ordered pair.

Write it as $(x,y)$.

$(4,-3)$

Step 6. Check that the ordered pair is a solution to both original equations. 

Substitute $x,y=4,-3$into both original equations.

$7x+8y=4$

$7\left(4\right)+8(-3)=4$

$28-24=4$

$4=4$

And, $3x-5y=27$

$3\left(4\right)-5(-3)=27$

$12+15=27$

$27=27$

Both equations are True.

The point$(4,-3)$is the solution to the system.