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 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

3.1

Ex: Solve by graphing

y = -x + 8

y = 4x - 7

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

Ex #2. Solve by graphing

x + 2y = 5

2x + 4y = 2

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

Substitution Method: used when one equation can be solved for a variable. When one variable has a coefficient of 1 or -1

Elimination Method: used when the two equations have the same coefficient for one of the variables (you can add or subtract right away to cancel one variable)

Elimination with Multiplication Method: used when substitution and elimination don’t work

3.2

Substitutionx + 5y = -33x – 2y = 8

Infinite or No solutions

6x – 2y = -4y = 3x + 2

Write and solve a System of Equations

The 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?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

Elimination5s + 2t = 69s + 2t = 22

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.

Multiply One Equation3x + 4y = 65x + 2y = -4

Multiply Two Equations3x + 4y = -252x – 3y = 6

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

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

1. get the inequality in slope-intercept from2. Graph the intercept and use slope to find

the next points3. Draw the line: < or > = dotted, or

= solid4. If needed: 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

5. Repeat steps 1-4 for the second inequality.

3.3

2

32

xy

xy

834

434

xy

xy

4

632

22

y

yx

xy