Solving Systems by Elimination - Algebra I Notes

Solving Systems by Elimination

Welcome to the lesson on solving systems of equations by elimination. This method involves eliminating one of the variables by adding or subtracting the equations to simplify the system into a single equation with one variable. Let’s explore the process with examples, exercises, and solutions.

Outline

Introduction to Elimination Method
Examples
Exercises
Homework
Revision
Video Explanation

Examples

Example 1

Solve the system:

1. 2x + 3y = 12

2. 4x - 3y = 8

Solution: Add the equations to eliminate y:

(2x + 3y) + (4x - 3y) = 12 + 8

6x = 20

x = \frac{20}{6} = \frac{10}{3}

Substitute x = \frac{10}{3} into 2x + 3y = 12:

2 \times \frac{10}{3} + 3y = 12

\frac{20}{3} + 3y = 12

3y = 12 - \frac{20}{3}

3y = \frac{16}{3}

y = \frac{16}{9}

Solution: \left(\frac{10}{3}, \frac{16}{9}\right)

Example 2

Solve the system:

1. 5x - 2y = 3

2. 3x + 4y = 7

Solution: Multiply the first equation by 2 and the second equation by 1:

2(5x - 2y) = 2 \times 3

10x - 4y = 6

3x + 4y = 7

Add the equations:

(10x - 4y) + (3x + 4y) = 6 + 7

13x = 13

x = 1

Substitute x = 1 into 5x - 2y = 3:

5(1) - 2y = 3

5 - 2y = 3

-2y = -2

y = 1

Solution: (1, 1)

Example 3

Solve the system:

1. 3x + y = 7

2. 2x - y = 1

Solution: Add the equations to eliminate y:

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

5x = 8

x = \frac{8}{5}

Substitute x = \frac{8}{5} into 3x + y = 7:

3 \times \frac{8}{5} + y = 7

\frac{24}{5} + y = 7

y = 7 - \frac{24}{5}

y = \frac{11}{5}

Solution: \left(\frac{8}{5}, \frac{11}{5}\right)

Example 4

Solve the system:

1. 4x - 5y = 6

2. 3x + 2y = 7

Solution: Multiply the first equation by 2 and the second equation by 5:

2(4x - 5y) = 2 \times 6

8x - 10y = 12

5(3x + 2y) = 5 \times 7

15x + 10y = 35

Add the equations:

(8x - 10y) + (15x + 10y) = 12 + 35

23x = 47

x = 2

Substitute x = 2 into 4x - 5y = 6:

4(2) - 5y = 6

8 - 5y = 6

-5y = -2

y = \frac{2}{5}

Solution: (2, \frac{2}{5})

Example 5

Solve the system:

1. 7x - 3y = 1

2. 2x + 6y = 12

Solution: Multiply the first equation by 2 and the second equation by 7:

2(7x - 3y) = 2 \times 1

14x - 6y = 2

7(2x + 6y) = 7 \times 12

14x + 42y = 84

Subtract the equations:

(14x - 6y) - (14x + 42y) = 2 - 84

-48y = -82

y = \frac{82}{48} = \frac{41}{24}

Substitute y = \frac{41}{24} into 7x - 3y = 1:

7x - 3 \times \frac{41}{24} = 1

7x - \frac{123}{24} = 1

7x = 1 + \frac{123}{24}

x = \frac{147}{24} = \frac{49}{8}

Solution: \left(\frac{49}{8}, \frac{41}{24}\right)

Example 6

Solve the system:

1. 8x + 4y = 20

2. 6x - 2y = 4

Solution: Multiply the first equation by 1 and the second equation by 2:

1(8x + 4y) = 20

8x + 4y = 20

2(6x - 2y) = 2 \times 4

12x - 4y = 8

Add the equations:

(8x + 4y) + (12x - 4y) = 20 + 8

20x = 28

x = \frac{28}{20} = \frac{7}{5}

Substitute x = \frac{7}{5} into 8x + 4y = 20:

8 \times \frac{7}{5} + 4y = 20

\frac{56}{5} + 4y = 20

4y = 20 - \frac{56}{5}

4y = \frac{44}{5}

y = \frac{11}{5}

Solution: \left(\frac{7}{5}, \frac{11}{5}\right)

Exercises

Complete the following problems:

Solve the system: 3x + 2y = 10 and 5x - 2y = 8
Solve the system: 4x - y = 9 and 2x + 3y = 12
Solve the system: 6x + 4y = 14 and 3x - 2y = 5
Solve the system: 7x - 3y = 2 and 5x + 2y = 8
Solve the system: 2x + 5y = 15 and 3x - y = 7

Solutions: Check your answers using similar steps to the examples provided.

Homework

Complete the following problems for additional practice:

Solve the system: 5x + 2y = 7 and 3x - 4y = -5
Solve the system: 2x - 3y = 4 and 4x + y = 11
Solve the system: 7x + 3y = 12 and 5x - 2y = 9
Solve the system: 6x - y = 10 and 2x + 3y = 7
Solve the system: 3x - 2y = -1 and 4x + y = 6

Revision

To review solving systems by elimination:

Ensure you correctly align and multiply the equations to eliminate one variable.
Combine the equations by adding or subtracting them to simplify the system.
Substitute the solution back into one of the original equations to find the remaining variable.
Practice various systems to solidify your understanding of the method.

Video Explanation

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