# The Elimination Method

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## Solving a System of Equation using the Elimination Method

In algebra, one common challenge is solving systems of linear equations - sets of equations with multiple variables. The elimination method is a powerful tool used to find these solutions, especially in real-world scenarios like budgeting or when working with coordinates.

The elimination method is an algebraic technique used to solve systems of linear equations. This method involves adding or subtracting equations to eliminate one of the variables, making solving the remaining variables easier.

## The Elimination Method – Steps

The elimination method is particularly useful when equations in a system are not easily solvable using substitution. It involves aligning equations in such a way that adding or subtracting them results in one variable being eliminated, simplifying the system to one equation with one variable.

System of equations are sometimes referred to as simultaneous equations. This is because they represent a set of two or more equations that are solved together, sharing common variables, and solving them involves finding values for these variables that satisfy all equations in the system simultaneously.

Step Number Action Example
1 Write down the system of equations. $5x + y = 9$
$10x - 7y = -18$
2 Manipulate one or both equations to align one variable. Multiply the first equation by 7:
$7(5x + y) = 7 \times 9$
$35x + 7y = 63$
3 Rewrite the manipulated system. $35x + 7y = 63$
$10x - 7y = -18$
4 Add or subtract the equations to eliminate one variable. Add the equations:
$(35x + 7y) + (10x - 7y) = 63 - 18$
$45x = 45$
5 Solve for the remaining variable. $x = \frac{45}{45}$
$x = 1$
6 Substitute the solved value into one of the original equations. Substitute $x = 1$ into $5x + y = 9$:
$5(1) + y = 9$
$y = 9 - 5$
7 Solve for the second variable. $y = 4$
8 Write the final solution. $x = 1$, $y = 4$

## The Elimination Method – Examples

Let's solve a system of equations using the steps for the elimination method:

Find the solution for $x$ and $y$.

$\begin{array}{rcl} x + 2y & = & 6 \\ x - 2y & = & 2 \\ \end{array}$

Add the equations to eliminate $y$:

$(x + 2y) + (x - 2y) = 6 + 2$

$2x = 8$

Solve for $x$:

$x = \frac{8}{2}$

$x = 4$

Substitute $x = 4$ into the first equation to find $y$:

$4 + 2y = 6$

$2y = 2$

$y = \frac{2}{2}$

$y = 1$

The solution is:

$x = 4, y = 1$ and can be written as $(4, 1)$.

Let’s try another one!

Solve the system of equations.

$\begin{array}{rcl} 7x + 2y &=& 24\\ 4x + y &=& 15\\ \end{array}$

Manipulate one equation to align one variable:

Multiply the second equation by -2: $-2(4x + y = 15)$

$-8x - 2y = -30$

Rewrite the manipulated system:

$7x + 2y = 24$

$-8x - 2y = -30$

Add the equations to eliminate $y$:

$(7x + 2y) + (-8x - 2y) = 24 - 30$

$-x = -6$

Solve for $x$:

$x = \frac{-6}{-1}$

$x = 6$

Substitute $x = 6$ into one of the original equations to find $y$:

Substitute into $4x + y = 15$:

$4(6) + y = 15$

$24 + y = 15$

$y = 15 - 24$

$y = -9$

The solution is:

$x = 6, y = -9$ and can be written as $(6, -9)$.

## The Elimination Method – Checking the Solutions

After solving a system of equations using the elimination method, it's important to verify that your solutions are correct. This involves substituting the solutions back into the original equations to ensure they satisfy both equations.

Suppose we solved a system and found that $x = 3$ and $y = 2$. The original equations were:

$\begin{array}{rcl} x+y&=&5\\ 2x-y&=&4\\ \end{array}$

Check:

• For Equation 1: Substitute $x = 3$ and $y = 2$ into $x + y = 5$.

$3 + 2 = 5$ which simplifies to $5 = 5$. This is true, so the solution satisfies the first equation.

• For Equation 2: Substitute $x = 3$ and $y = 2$ into $2x - y = 4$.

$2(3) - 2 = 4$ which simplifies to $6 - 2 = 4$ and further to $4 = 4$. This is also true, confirming the solution is correct for the second equation as well.

Try some checks on your own!

Check the solution $x = 2$, $y = 3$ for the system of equations: $x + 2y = 8$, $3x - y = 3$.
Verify the solution $x = -4$, $y = 1$ for the system: $2x + 3y = -5$, $x + 4y = 0$.

## The Elimination Method – Summary

Key Learnings from this Text:

• The elimination method is used to solve systems of linear equations by eliminating one variable.
• It involves adding or subtracting equations to cancel out one of the variables.
• This method is especially useful when substitution is not straightforward.
• It's a practical skill in various real-world applications, like planning and geometry.

For a more, check out Writing and Solving Linear Equations.

## The Elimination Method – Frequently Asked Questions

When is the elimination method preferable over substitution?
Can the elimination method be used for any system of linear equations?
Is it necessary to use the elimination method in all cases?
What do I do if none of the variables cancel out immediately?
How do I check my solution is correct?
Can the elimination method be used for systems with more than two variables?
What happens if I add the equations and all variables are eliminated?
Are there real-world examples where the elimination method is used?
Can the elimination method be applied to inequalities?
What if the coefficients of the variables are fractions in the equations?
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