Question

Solve the system by elimination:$\left\{\begin{array}{l}3x+y=5\\ 2x-3y=7\end{array}\right.$

Short answer

The solution of the system of equations is $(2,-1)$.

Step-by-step solution

Step 1. Given Information.

We have been given system of equations$\left\{\begin{array}{l}3x+y=5\\ 2x-3y=7\end{array}\right.$

Step 2. Eliminate the y's by multiplying a number in first equation.

Multiply both sides of $3x+y=5$by 3,

$\left\{\begin{array}{l}3(3x+y)=3\left(5\right)\\ 2x-3y=7\end{array}\right.$

localid="1644378022840" $\left\{\begin{array}{l}9x+3y=15\\ 2x-3y=7\end{array}\right.$

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.

$\left\{\begin{array}{l}9x+3y=15\\ 2x-3y=7\end{array}\right.$

$11x=22$

$x=2$

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

Substitute $x=2$into the first equation.

$3x+y=5$

$3\left(2\right)+y=5$

$6+y=5$

$y=-1$

Step 5. Write the solution as an ordered pair.

Write it as $(x,y).$

$(2,-1)$

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

Substitute $x,y=2,-1$into both original equations.

$3x+y=5$

$3\left(2\right)-1=5$

$5=5$

And,

$2x-3y=7$

$2\left(2\right)-3(-1)=7$

$4+3=7$

$7=7$

Both equations are True.

The point $(2,-1)$is the solution to the system.