How to Solve Linear Systems of Equations by Elimination

Concept

In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of equations for which we sought the common solutions. A system of equations can be classified in a similar manner as single equations.

Any system of equations can be solved in different methods. To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.

Substitution Method
Elimination Method
Graphical Method

Using the elimination method to solve the system of equations, we eliminate one of the unknowns, by multiplying equations by suitable numbers, so as the coefficients of one of the variables become the same.

Rules

The elimination method eliminates one of the variables by adding the equations together.

If adding the equations together does not eliminate a variable, multiply one of the equations by a constant to make a coefficient in each equation be of opposite values.

Solve the new system by following the three steps:
Step 1: Eliminate one of the variables by adding the equations together.
Step 2: After eliminating one variable, solve for the variable that is left in the equation.
Step 3: After solving for one variable, substitute that value into one of the equations and find the value of the second variable. The values of the two variables is the solution to the system of equations (x,y).

Example

Solve the system of linear equations by elimination.
$\fn_phv -4x-15y=-17$
$\fn_phv x-5y=13$

Solution

$\fn_phv -4x-15y=-17$
$\fn_phv x-5y=13$
Adding the equations together does not eliminate a variable. Multiply one of the equations by a constant to make a coefficient in each equation be of opposite values. Choose to multiply by 4 or -3.
$\fn_phv -4x-15y=-17$
$\fn_phv 4(x-5y=13)$

The new system becomes:
$\fn_phv -4x-15y=-17$
$\fn_phv 4x-20y=52$

Eliminate one of the variables by adding the equations together.
$\fn_phv -4x-15y=-17$
$\fn_phv \underline{\;\;\;4x-20y=52}$
$\fn_phv \underline{\;\;\;\;\;\;\;\;-35y=35}$

$\fn_phv \frac{-35y}{-35}=-\frac{35}{35}$

$\fn_phv y=-1$

Substitute -1 in for y in either of the equation and solve for the second variable.
$\fn_phv -4x-15y=-17$
$\fn_phv -4x-15(-1)=-17$
$\fn_phv -4x+15=-17$
$\fn_phv -4x=-32$