Question

Solve the system by elimination:

$\left\{\begin{array}{l}3x-4y=-9\\ 5x+3y=14\end{array}\right.$

Short answer

Solution of the system of equations is$(1,3)$.

Step-by-step solution

Step 1. Given Information.

We have been given system of equations:

$\left\{\begin{array}{l}3x-4y=-9\\ 5x+3y=14\end{array}\right.$

Step 2. Eliminate the y.

To get opposite coefficients of $y$, we will multiply the first equation by 3 and the second equation by 4.

$\left\{\begin{array}{l}3(3x-4y)=3(-9)\\ 4(5x+3y)=4\left(14\right)\end{array}\right.$

And,

localid="1644380631474" $\left\{\begin{array}{l}9x-12y=-27\\ 20x+12y=56\end{array}\right.$

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.

$\left\{\begin{array}{l}9x-12y=-27\\ 20x+12y=56\end{array}\right.$

$29x=29$

$x=1$

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

Substitute $x=1$into the first equation.

$3x-4y=-9$

$3\left(1\right)-4y=-9$

$3-4y=-9$

$4y=12$

$y=3$

Step 5. Write the solution as an ordered pair. 

Write it as $(x,y).$

$(1,3)$

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

Substitute $x,y=1,3$into both original equations.

$3x-4y=-9$

$3\left(1\right)-4\left(3\right)=-9$

$3-12=-9$

$-9=-9$

And,

$5x+3y=14$

$5\left(1\right)+3\left(3\right)=14$

$5+9=14$

$14=14$

Both equations are True.

The point$(1,3)$is the solution to the system.