Chapter 3 Systems of Equations Graphing Systems of

Chapter 3: Systems of Equations

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 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 3. 1 Exactly one solution Infinite solutions No solutions Consistent and Independent Consistent and Dependent Inconsistent

*See teacher or other student class for work on these examples Ex: Solve by graphing y = -x + 8 y = 4 x - 7

*See teacher or other student class for work on these examples Ex #2. Solve by graphing x + 2 y = 5 2 x + 4 y = 2

Methods for Solving Systems of Equations Algebraically �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

Substitution x + 5 y = -3 3 x – 2 y = 8

Infinite or No solutions 6 x – 2 y = -4 y = 3 x + 2 6 x – 2(3 x +2) = -4 6 x – 6 x - 4 = -4 -4 = -4 Infinite solutions When all variables cancel, if: the statement is true = infinite solutions the statement is false = no solutions 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?

Elimination 5 s + 2 t = 6 9 s + 2 t = 22

�Write and solve a system of equations Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers.

Multiply One Equation 3 x + 4 y = 6 5 x + 2 y = -4 Multiply one equation to make a variable have the same number and opposite sign Multiply Two Equations 3 x + 4 y = -25 2 x – 3 y = 6 Multiply both equations to make a variable have the same number and opposite sign

Graphing Systems of Inequalities 1. get the inequality in slope-intercept from 2. Graph the intercept and use slope to find the next points 3. Draw the line: < or > = dotted, or = solid 4. 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

Example 1

Find the coordinates of the vertices formed by the system.