Introduction to Systems of Equations - Algebra I

Introduction to Systems of Equations

In this lesson, we will explore systems of equations, which are sets of two or more equations with the same variables. The goal is to find the values of the variables that satisfy all the equations in the system.

Outline

Definition of a System of Equations
Methods to Solve Systems of Equations
Examples of Solving Systems of Equations
Exercises with Solutions
Homework
Revision

Definition of a System of Equations

A system of equations is a set of equations with the same variables. The solution to the system is the set of values that satisfy all the equations simultaneously. Systems can be solved using various methods including graphing, substitution, and elimination.

Methods to Solve Systems of Equations

Three common methods for solving systems of equations are:

Graphing: Plot each equation on the same coordinate plane and identify the intersection point(s).
Substitution: Solve one equation for one variable and substitute that expression into the other equation.
Elimination: Add or subtract equations to eliminate one variable, then solve the resulting equation.

Examples

Example 1

Solve the following system using substitution:

1. y = 2x + 3

2. x + y = 8

Solution: Substitute y = 2x + 3 into x + y = 8 to get x + (2x + 3) = 8. Simplify to find 3x + 3 = 8, so x = \frac{5}{3}. Substitute x = \frac{5}{3} back into y = 2x + 3 to find y = \frac{16}{3}.

Example 2

Solve the following system using elimination:

1. 3x + 2y = 12

2. 4x - 2y = 8

Solution: Add the equations to eliminate y: (3x + 2y) + (4x - 2y) = 12 + 8. This simplifies to 7x = 20, so x = \frac{20}{7}. Substitute x = \frac{20}{7} into one of the original equations to find y.

Example 3

Solve the following system by graphing:

1. y = -x + 4

2. y = 2x - 1

Solution: Graph both equations on the same coordinate plane. The point where the lines intersect is the solution to the system.

Example 4

Solve the following system using substitution:

1. y = 3x - 2

2. 2x + y = 6

Solution: Substitute y = 3x - 2 into 2x + y = 6 to get 2x + (3x - 2) = 6. Simplify to find 5x - 2 = 6, so x = \frac{8}{5}. Substitute x = \frac{8}{5} back into y = 3x - 2 to find y = \frac{14}{5}.

Example 5

Solve the following system using elimination:

1. 5x + 4y = 20

2. 3x - 4y = 2

Solution: Add the equations to eliminate y: (5x + 4y) + (3x - 4y) = 20 + 2. This simplifies to 8x = 22, so x = \frac{11}{4}. Substitute x = \frac{11}{4} into one of the original equations to find y.

Example 6

Solve the following system by graphing:

1. y = x - 2

2. y = -2x + 6

Solution: Graph both equations on the same coordinate plane. The intersection point is the solution to the system.

Exercises

Exercise 1

Solve the following system using substitution:

1. y = 4x + 1

2. 3x - y = 7

Solution: Substitute y = 4x + 1 into 3x - y = 7 to find x = 2. Substitute x = 2 back into y = 4x + 1 to find y = 9.

Exercise 2

Solve the following system using elimination:

1. 2x + 3y = 13

2. 4x - 3y = 5

Solution: Add the equations to eliminate y: (2x + 3y) + (4x - 3y) = 13 + 5. This simplifies to 6x = 18, so x = 3. Substitute x = 3 into one of the original equations to find y = 3.

Exercise 3

Solve the following system by graphing:

1. y = 2x + 1

2. y = -x + 4

Solution: Graph both equations on the same coordinate plane. The intersection point is the solution to the system.

Exercise 4

Solve the following system using substitution:

1. y = -3x + 5

2. x + y = 1

Solution: Substitute y = -3x + 5 into x + y = 1 to find x = 2. Substitute x = 2 back into y = -3x + 5 to find y = -1.

Exercise 5

Solve the following system using elimination:

1. 3x - y = 7

2. 5x + y = 11

Solution: Add the equations to eliminate y: (3x - y) + (5x + y) = 7 + 11. This simplifies to 8x = 18, so x = \frac{9}{4}. Substitute x = \frac{9}{4} into one of the original equations to find y = \frac{7}{4}.

Exercise 6

Solve the following system by graphing:

1. y = -2x + 3

2. y = x + 1

Solution: Graph both equations on the same coordinate plane. The intersection point is the solution to the system.

Homework

Complete the following problems to practice solving systems of equations:

Solve the system by substitution:

y = 3x - 4
2x + y = 10

Solve the system by elimination:

4x - 3y = 12
2x + y = 5

Solve the following system by graphing:

y = 2x + 2
y = -x + 6

Revision

Review the key points from this lesson:

A system of equations is a set of two or more equations with the same variables.
The solution to a system of equations is the set of values that satisfy all equations simultaneously.
Common methods for solving systems include graphing, substitution, and elimination.
Practice with different types of systems to gain a deeper understanding of the methods and solutions.