Question

Elimination Method

Use the elimination method to find all solutions of the system of equations.

$\left\{\begin{array}{l}2x-5y=-18\\ 3x+4y=19\end{array}\right.$

Short answer

Hence, all solutions of the system of equations are $x=1\&y=4$

Step-by-step solution

Step 1. Given information

We are given the system of linear equations that is-

$\left\{\begin{array}{l}2x-5y=-18\\ 3x+4y=19\end{array}\right.$

Step 2. Concept used

The system of linear equations is the collection of one or more linear equations involving the same set of variables.

We need to find all solutions to the system of equations.

By the Elimination Method, we will calculate the result.

Step 3. Calculation

We multiply the first equation by $2$ and add the second equation to it to eliminate the x term.

$\begin{array}{l}3\times [2x-5y=-18]\\ 6x-15y=-54\\ 2\times 3x+4y=19\\ 6x+8y=38\end{array}$

Further solving,

$\begin{array}{l}(6x-6x)+(-15y-8y)=-54-38\\ -23y=-92\\ y=4\end{array}$

We substitute the value of y in any of the original equations to get back the value of x

$\begin{array}{l}3x+4\times 4=19\\ 3x+16=19\\ 3x=3\\ x=1\end{array}$