5 Ways to Solve Systems by Elimination

Understanding Systems of Equations

Systems of equations are a fundamental concept in algebra, where two or more equations are used to solve for multiple variables. These equations can be linear or nonlinear, and solving them can be a complex task. One popular method for solving systems of equations is the elimination method. In this article, we will explore five ways to solve systems by elimination.

Method 1: Addition and Subtraction

The addition and subtraction method involves adding or subtracting the equations to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations, but with opposite signs.

For example, consider the following system of equations:

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

To solve this system, we can add the two equations to eliminate the y-variable:

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

Now, we can solve for x:

x = 1

Once we have the value of x, we can substitute it into one of the original equations to solve for y.

Method 2: Multiplication and Addition/Subtraction

The multiplication and addition/subtraction method involves multiplying one or both equations by a constant to make the coefficients of one variable the same, and then adding or subtracting the equations to eliminate that variable.

For example, consider the following system of equations:

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

To solve this system, we can multiply the first equation by 3 and the second equation by 1 to make the coefficients of x the same:

3(x + 2y) = 3(3) 3x + 6y = 9

Now, we can subtract the second equation from the first equation to eliminate the x-variable:

(3x + 6y) - (3x - 4y) = 9 - 5 10y = 4

Now, we can solve for y:

y = ²⁄₅

Once we have the value of y, we can substitute it into one of the original equations to solve for x.

Method 3: Elimination Using Fractions

The elimination using fractions method involves multiplying one or both equations by a fraction to make the coefficients of one variable the same, and then adding or subtracting the equations to eliminate that variable.

For example, consider the following system of equations:

1/2x + 3y = 2 1/4x - 2y = -1

To solve this system, we can multiply the first equation by 2 and the second equation by 4 to make the coefficients of x the same:

(1/2x + 3y) x 2 = 2 x 2 x + 6y = 4

Now, we can multiply the second equation by -1 to make the coefficients of x the same:

-(1/4x - 2y) x 4 = -(-1) x 4 -x + 8y = 4

Now, we can add the two equations to eliminate the x-variable:

(x + 6y) + (-x + 8y) = 4 + 4 14y = 8

Now, we can solve for y:

y = ⁴⁄₇

Once we have the value of y, we can substitute it into one of the original equations to solve for x.

Method 4: Elimination Using Decimals

The elimination using decimals method involves multiplying one or both equations by a decimal to make the coefficients of one variable the same, and then adding or subtracting the equations to eliminate that variable.

For example, consider the following system of equations:

0.5x + 2y = 1 1.5x - 3y = 2

To solve this system, we can multiply the first equation by 3 and the second equation by 1 to make the coefficients of x the same:

3(0.5x + 2y) = 3(1) 1.5x + 6y = 3

Now, we can subtract the second equation from the first equation to eliminate the x-variable:

(1.5x + 6y) - (1.5x - 3y) = 3 - 2 9y = 1

Now, we can solve for y:

y = ¹⁄₉

Once we have the value of y, we can substitute it into one of the original equations to solve for x.

Method 5: Elimination Using Complex Numbers

The elimination using complex numbers method involves multiplying one or both equations by a complex number to make the coefficients of one variable the same, and then adding or subtracting the equations to eliminate that variable.

For example, consider the following system of equations:

x + 2yi = 3 + 4i 2x - 3yi = 1 - 2i

To solve this system, we can multiply the first equation by 2 and the second equation by 1 to make the coefficients of x the same:

2(x + 2yi) = 2(3 + 4i) 2x + 4yi = 6 + 8i

Now, we can add the two equations to eliminate the x-variable:

(2x + 4yi) + (2x - 3yi) = (6 + 8i) + (1 - 2i) 4x + yi = 7 + 6i

Now, we can solve for y:

y = 6 + 7i

Once we have the value of y, we can substitute it into one of the original equations to solve for x.

📝 Note: The elimination method is a powerful tool for solving systems of equations. However, it requires careful attention to the coefficients and constants in the equations. Make sure to multiply and add/subtract the equations correctly to avoid errors.

To summarize, there are five ways to solve systems by elimination: addition and subtraction, multiplication and addition/subtraction, elimination using fractions, elimination using decimals, and elimination using complex numbers. Each method has its own strengths and weaknesses, and the choice of method depends on the specific system of equations being solved.

In conclusion, solving systems of equations by elimination is a fundamental skill in algebra. By mastering the five methods outlined above, students can develop a strong foundation in algebra and improve their problem-solving skills. Remember to always check your work and use the correct method for the specific system of equations you are solving.