Solving Systems by Substitution - Algebra I

Solving Systems by Substitution

In this lesson, we will learn how to solve systems of equations by substitution. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Outline

Definition of Systems of Equations
Substitution Method Steps
Examples of Solving Systems by Substitution
Exercises with Solutions
Homework
Revision

Definition of Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving such a system means finding the values for these variables that satisfy all the equations simultaneously.

Substitution Method Steps

Choose one of the equations and solve it for one variable in terms of the other variable.
Substitute this expression into the other equation.
Solve the resulting equation for the remaining variable.
Substitute this value back into the original equation to find the value of the first variable.
Check the solution in both original equations to ensure it is correct.

Examples

Example 1

Solve the system:

1. y = 2x + 1

2. x + y = 7

Solution: Substitute y = 2x + 1 into x + y = 7:

x + (2x + 1) = 7

3x + 1 = 7

3x = 6

x = 2

Substitute x = 2 into y = 2x + 1:

y = 2(2) + 1 = 5

Solution: (2, 5)

Example 2

Solve the system:

1. y = -x + 4

2. 2x + y = 3

Solution: Substitute y = -x + 4 into 2x + y = 3:

2x + (-x + 4) = 3

x + 4 = 3

x = -1

Substitute x = -1 into y = -x + 4:

y = -(-1) + 4 = 5

Solution: (-1, 5)

Example 3

Solve the system:

1. 2y = 4x - 6

2. y = x + 2

Solution: Substitute y = x + 2 into 2y = 4x - 6:

2(x + 2) = 4x - 6

2x + 4 = 4x - 6

10 = 2x

x = 5

Substitute x = 5 into y = x + 2:

y = 5 + 2 = 7

Solution: (5, 7)

Example 4

Solve the system:

1. 3x + y = 10

2. y = 2x - 1

Solution: Substitute y = 2x - 1 into 3x + y = 10:

3x + (2x - 1) = 10

5x - 1 = 10

5x = 11

x = 2.2

Substitute x = 2.2 into y = 2x - 1:

y = 2(2.2) - 1 = 3.4

Solution: (2.2, 3.4)

Example 5

Solve the system:

1. 4x - y = 7

2. y = 3x + 2

Solution: Substitute y = 3x + 2 into 4x - y = 7:

4x - (3x + 2) = 7

x - 2 = 7

x = 9

Substitute x = 9 into y = 3x + 2:

y = 3(9) + 2 = 29

Solution: (9, 29)

Example 6

Solve the system:

1. y = -2x + 8

2. 3x + y = 4

Solution: Substitute y = -2x + 8 into 3x + y = 4:

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

x + 8 = 4

x = -4

Substitute x = -4 into y = -2x + 8:

y = -2(-4) + 8 = 16

Solution: (-4, 16)

Exercises

Exercise 1

Solve the system:

1. y = 3x - 2

2. x + y = 4

Solution: Substitute y = 3x - 2 into x + y = 4:

x + (3x - 2) = 4

4x - 2 = 4

4x = 6

x = 1.5

Substitute x = 1.5 into y = 3x - 2:

y = 3(1.5) - 2 = 2.5

Solution: (1.5, 2.5)

Exercise 2

Solve the system:

1. 2x + y = 8

2. y = -x + 6

Solution: Substitute y = -x + 6 into 2x + y = 8:

2x + (-x + 6) = 8

x + 6 = 8

x = 2

Substitute x = 2 into y = -x + 6:

y = -2 + 6 = 4

Solution: (2, 4)

Exercise 3

Solve the system:

1. 3x - y = 5

2. y = 2x + 1

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

3x - (2x + 1) = 5

x - 1 = 5

x = 6

Substitute x = 6 into y = 2x + 1:

y = 2(6) + 1 = 13

Solution: (6, 13)

Exercise 4

Solve the system:

1. y = -3x + 7

2. 2x + y = 4

Solution: Substitute y = -3x + 7 into 2x + y = 4:

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

-x + 7 = 4

-x = -3

x = 3

Substitute x = 3 into y = -3x + 7:

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

Solution: (3, -2)

Exercise 5

Solve the system:

1. 5x - 2y = 4

2. y = x - 3

Solution: Substitute y = x - 3 into 5x - 2y = 4:

5x - 2(x - 3) = 4

5x - 2x + 6 = 4

3x + 6 = 4

3x = -2

x = -\frac{2}{3}

Substitute x = -\frac{2}{3} into y = x - 3:

y = -\frac{2}{3} - 3 = -\frac{11}{3}

Solution: \left(-\frac{2}{3}, -\frac{11}{3}\right)

Exercise 6

Solve the system:

1. 4x - 3y = -2

2. y = 2x + 5

Solution: Substitute y = 2x + 5 into 4x - 3y = -2:

4x - 3(2x + 5) = -2

4x - 6x - 15 = -2

-2x - 15 = -2

-2x = 13

x = -\frac{13}{2}

Substitute x = -\frac{13}{2} into y = 2x + 5:

y = 2\left(-\frac{13}{2}\right) + 5 = -13 + 5 = -8

Solution: \left(-\frac{13}{2}, -8\right)

Homework

Complete the following problems:

Solve the system: x + y = 6 and y = 4x - 2
Solve the system: 2x - y = 3 and y = -x + 7
Solve the system: 3x + 2y = 8 and y = x + 1
Solve the system: 4x - y = 5 and y = -2x + 3
Solve the system: 5x + y = 0 and y = -x - 1

Revision

To revise solving systems by substitution:

Ensure you correctly isolate one variable in one of the equations.
Accurately substitute this expression into the other equation.
Double-check your solution by substituting the values back into the original equations.
Practice with various systems to strengthen your understanding.

Video Explanation

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