Systems of Equations:

A solution to a system of equations intwo variables x and y is an ordered pair(x,y) of real numbers thatsatisfy each equation in the system.

7.1 Graphing Linear Systems

Solve the following system of equations:

y = -2x + 6

y = ½x + 1

6

5

4

3

2

1

01 2 3 4 5 6

7.1 Graphing Linear Systems

Systems of Equations

Solve the following system of equations:

y = -2x + 6

y = ½x + 1 6

5

4

3

2

1

01 2 3 4 5 6

Systems of EquationsExample 1.

Solve the following system of equations:

y = -2x + 6

y = ½x + 1 6

5

4

3

2

1

01 2 3 4 5 6

Solve the following system of equations:

y = -2x + 6

y = ½ + 1

6

5

4

3

2

1

01 2 3 4 5 6

7.1 Graphing Linear Systems

Solve the following system of equations:

y = -2x + 6

y = ½x + 16

5

4

3

2

1

01 2 3 4 5 6

Solution: (2, 2)

7.1 Graphing Linear Systems

Solve the following system of equations:

y = 2x + 1

y = ½x + 3

6

5

4

3

2

1

01 2 3 4 5 6

7.1 Graphing Linear Systems

Systems of Equations

Solve the following system of equations:

y = 2x + 1

y = ½x + 3 6

5

4

3

2

1

01 2 3 4 5 6

Solve the following system of equations:

y = 2x + 1

y = ½x + 3 6

5

4

3

2

1

01 2 3 4 5 6

7.1 Graphing Linear Systems

Solve the following system of equations:

y = 2x + 1

y = ½x + 3 6

5

4

3

2

1

01 2 3 4 5 6

7.1 Graphing Linear Systems

Solve the following system of equations:

y = 2x + 1

y = ½x + 3 6

5

4

3

2

1

01 2 3 4 5 6

Solution: (4/3, 11/3)

7.1 Graphing Linear Systems

Example:

x + y = 42x – y = 5

The point (3,1) is asolution to the systemof equations.

3 + 1 = 42(3) – 1 = 5

7.1 Graphing Linear Systems

x + y = 42x – y = 5

Solving by Graphing

Put in slope-intercept form

y = -x + 4

y = 2x - 5

(3,1)

7.1 Graphing Linear Systems

y = x - 2y = -2x + 1

Solving by Graphing

(1,-1)

7.1 Graphing Linear Systems

x + y = 42x + 3y = 6

Solving by Graphing

Put in slope-intercept form

y = -x + 4

(6,-2)

y 2

3x 2

7.1 Graphing Linear Systems

x + y = 42x – y = 5

Solving by Substitution

Solve one of theequations for x or y and substitute intothe other equation.

y = -x + 4

2x – (-x + 4) = 52x + x – 4 = 5

3x – 4 = 5

3x = 9

x = 3

y = -(3) + 4

y = 1

(3,1)

7.2 Substitution

x + y = 42x + 3y = 6

Solving by Substitution

Solve one of theequations for x or y and substitute intothe other equation.

y = -x + 4

2x +3(-x + 4) = 62x – 3x + 12 = 6

-x + 12 = 6

-x = -6

x = 6

y = -(6) + 4

y = -2

(6,-2)

7.2 Substitution

x + y = 42x – y = 5

Solving by Elimination

(3,1)

Add to eliminate y

3x = 9

x = 3 y = 1

Remember to substitute back in to find the other variable

7.3 Elimination

x + y = 42x + 3y = 6

Solving by Elimination

(6,-2)

Multiply the top equation by -3

-3x – 3y = -122x + 3y = 6-x = -6 x = 6

y = -2

7.3 Elimination

If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent.

Solution

y

x

7.5 Special Types of Systems

If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent.

x

y

7.5 Special Types of Systems

If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent.

x

y

7.5 Special Types of Systems