Chapter 5

Solving Systems of Linear Equations

5.1 Graphing Systems of Equations

Systems of equations- two equations together A solution of a system of equations is an ordered pair that satisfies both

equations

Consistent- the graphs of the equations intersect (at least one solution) If consistent with exactly 1 solution = independent If consistent with infinite solutions = dependent

Inconsistent- the graphs of the equations are parallel No ordered pair solutions

Exactly one solution

Infinite solutions

No solutions

Consistent and Independent

Consistent and Dependent

Inconsistent

a. Solve a system of Equations by graphingy = -x + 8

y = 4x - 7

a. Solve a system of Equations by graphingx + 2y = 5

2x + 4y = 2

*See teacher or other student class for work on these examples

6.8 Graphing Systems of Inequalities

1. get the inequality in slope-intercept from2. State the slope and y-intercept3. Graph the intercept and use slope to find the next

points4. Draw the line:

< or > = dotted, or = solid5. Test an ordered pair not on the line

-if true, shade that side of the line-if false, shade the other side of the line

6. Repeat steps 1-5 for the second inequality.

Ex: graph the system of inequalities

y<-x+4

y 2x+3

Ex: Graph the system of inequalities

x-y<-1

x-y>3

*See teacher or other student class for work on these examples

5.2 Substitution

Solve Using Substitution

y = 3x

x + 2y = -21

Solve using Substitution

x + 5y = -3

3x – 2y = 8

x + 2(3x)= -21x + 6x = -217x = -21/7 /7x = -3

y = 3xy = 3(-3)

y = -9

Solution = (-3, -9)

Substitute 3x for y

Solve for x

Plug in -3 for x and solve for y

x + 5y = -3

-5y -5y

x = -3 – 5y

Solve one equation for a variable

3(-3 – 5y) – 2y = 8-9 -15y – 2y = 8-9 -17y = 8

+9 +9-17y = 17

/-17 /-17y = -1

x = -3 -5(-1)x = -3 +5

x = 2Solution = (2, -1)

Infinite or No solutions6x – 2y = -4

y = 3x + 2

Write and solve a System of EquationsThe New York Yankees and Cincinnati Reds together

have won a total 31 World Series. The Yankees have won 5.2 times as many as the Reds. How many have each team won?

Yankees = x Reds = y

Total games x + y = 31Times games x = 5.2y

5.2y + y = 316.2y = 31

/6.2 /6.2y = 5

x = 5.2(5)x = 26

Yankees = 26

Reds = 5

6x – 2(3x +2) = -46x – 6x - 4 = -4

-4 = -4

When all variables cancel, if:

the statement is true = infinite solutions

the statement is false = no solutions

Infinite solutions

5.3 Elimination Using Addition and Subtraction

Elimination: Addition3x – 5y = -16

2x + 5y = 31

Elimination: Subtraction5s + 2t = 6

9s + 2t = 22

3x – 5y = -16

+ 2x + 5y = 315x = 15

/5 /5x = 3

3(3) – 5y = -169 – 5y = -16-9 -9

-5y = -25/-5 /-5

y = 5

Solution:

(3, 5)

5s + 2t = 6

- 9s + 2t = 22

-4s = -16/-4 /-4

s = 4

5(4) + 2t = 620 + 2t = 6

-20 -20 2t = -14

/2 /2t =-7

Solution:

(4, -7)

Add to eliminate because the y’s are the same number opposite signs

Subtract to eliminate because the t’s are the same number same sign

Write and solve a system of equationsTwice one number added to another number is 18. Four times the first

number minus the other number is 12. Find the numbers.

2x + y = 184x - y = 12

Add because the y’s are the same number opposite signs

2x + y = 184x - y = 12+

6x = 30 /6 /6

x = 5

2 (5) + y = 1810 + y = 18-10 -10

y = 8

Solution:

5 and 8

5.4 Elimination Using Multiplication

Multiply One Equation3x + 4y = 6

5x + 2y = -4

Multiply Two Equations3x + 4y = -25

2x – 3y = 6 3x +4y = 6

-2[5x + 2y = -4]

3x +4y = 6

-10x + -4y = 8+

-7x = 14/-7 /-7

x = -2

3(-2) + 4y = 6-6 + 4y = 6+6 +6

4y = 12 /4 /4

y = 3

Solution:

(-2, 3)

Multiply one equation to make a variable have the same number and opposite sign

3[3x +4y = -25]

4[2x – 3y = 6]

9x +12y = -75

8x – 12y = 24+

17x = -51/17 /17

x = -3

3(-3) + 4y = -25-9 + 4y = -25+9 +9

4y = -16 /4 /4

y = -4

Solution:

(-3, -4)

Multiply both equations to make a variable have the same number and opposite sign

5.5 Applying Systems of Equations

Method The Best Time to Use

Graphing To estimate the solution. When both equations are in Slope-Intercept Form

Substitution If one variable in either equation has a coefficient of 1

Elimination:Addition

If one variable has coefficients with the same number and opposite signs

Elimination:

Subtraction

If one variable has coefficients with the same number and same sign

Elimination:

Multiplication

If none of the coefficients are the same number