5 Ways to Solve Systems of Equations by Elimination

What is the Elimination Method?

The elimination method is a technique used to solve systems of linear equations by eliminating one of the variables. This method involves adding or subtracting the equations in a way that causes one of the variables to cancel out, leaving a single equation with one variable that can be solved. In this blog post, we will explore five ways to solve systems of equations by elimination.

Method 1: Elimination by Adding or Subtracting

This method involves adding or subtracting the equations to eliminate one of the variables. For example, consider the following system of equations:

2x + 3y = 7 x - 2y = -3

To eliminate the variable x, we can subtract the second equation from the first equation:

(2x + 3y) - (x - 2y) = 7 - (-3) 2x - x + 3y + 2y = 10 x + 5y = 10

Now we have a new equation with only one variable, y. We can solve for y by isolating it on one side of the equation.

Solution:

x + 5y = 10 5y = 10 - x y = (10 - x) / 5

📝 Note: When adding or subtracting equations, make sure to line up the variables and constants correctly to avoid errors.

Method 2: Elimination by Multiplying by a Constant

This method involves multiplying one or both of the equations by a constant to make the coefficients of one of the variables the same. For example, consider the following system of equations:

x + 2y = 4 3x - 2y = 5

To eliminate the variable y, we can multiply the first equation by 3 and the second equation by 1:

3(x + 2y) = 3(4) 3x + 6y = 12

Now we have a new equation with the same coefficients for the variable y. We can subtract the second equation from the first equation to eliminate y:

(3x + 6y) - (3x - 2y) = 12 - 5 6y + 2y = 7 8y = 7

Now we can solve for y by isolating it on one side of the equation.

Solution:

8y = 7 y = ⁷⁄₈

📝 Note: When multiplying equations by a constant, make sure to multiply all terms by the same constant to avoid errors.

Method 3: Elimination by Using Fractions

This method involves using fractions to make the coefficients of one of the variables the same. For example, consider the following system of equations:

1/2x + 3y = 5 2x - 3y = 1

To eliminate the variable y, we can multiply the first equation by 2 and the second equation by 1:

(1/2x + 3y) * 2 = 5 * 2 x + 6y = 10

Now we have a new equation with the same coefficients for the variable y. We can add the second equation to the first equation to eliminate y:

(x + 6y) + (2x - 3y) = 10 + 1 3x + 3y = 11

Now we can solve for x by isolating it on one side of the equation.

Solution:

3x + 3y = 11 3x = 11 - 3y x = (11 - 3y) / 3

📝 Note: When using fractions, make sure to simplify the equations to avoid errors.

Method 4: Elimination by Using Decimals

This method involves using decimals to make the coefficients of one of the variables the same. For example, consider the following system of equations:

0.5x + 2y = 3 2x - 2y = 1

To eliminate the variable y, we can multiply the first equation by 2 and the second equation by 1:

(0.5x + 2y) * 2 = 3 * 2 x + 4y = 6

Now we have a new equation with the same coefficients for the variable y. We can add the second equation to the first equation to eliminate y:

(x + 4y) + (2x - 2y) = 6 + 1 3x + 2y = 7

Now we can solve for x by isolating it on one side of the equation.

Solution:

3x + 2y = 7 3x = 7 - 2y x = (7 - 2y) / 3

📝 Note: When using decimals, make sure to line up the decimal points correctly to avoid errors.

Method 5: Elimination by Using Tables

This method involves using tables to organize the equations and make it easier to eliminate one of the variables. For example, consider the following system of equations:

x + 2y = 4 3x - 2y = 5

We can create a table to organize the equations:

To eliminate the variable y, we can add the two equations:

Now we have a new equation with only one variable, x. We can solve for x by isolating it on one side of the equation.

Solution:

4x = 9 x = ⁹⁄₄

📝 Note: When using tables, make sure to line up the variables and constants correctly to avoid errors.

By using these five methods, you can solve systems of equations by elimination. Remember to choose the method that best fits the problem and to always check your work to avoid errors.