Solving Systems of Linear Equations 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

*See teacher or other student class for work on these examples a. Solve a system of Equations by graphing a. Solve a system of Equations by graphing y = -x + 8 y = 4x - 7 x + 2y = 5 2x + 4y = 2

6.8 Graphing Systems of Inequalities 1. get the inequality in slope-intercept from 2. State the slope and y-intercept 3. Graph the intercept and use slope to find the next points 4. Draw the line: < or > = dotted, or = solid 5. 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.

*See teacher or other student class for work on these examples Ex: graph the system of inequalities y<-x+4 y 2x+3 Ex: Graph the system of inequalities x-y<-1 x-y>3

5.2 Substitution x + 2(3x)= -21 Substitute 3x for y Solve Using Substitution y = 3x x + 2y = -21 Solve using Substitution x + 5y = -3 3x – 2y = 8 x + 5y = -3 -5y -5y x = -3 – 5y Solve one equation for a variable x + 2(3x)= -21 Substitute 3x for y 3(-3 – 5y) – 2y = 8 x + 6x = -21 -9 -15y – 2y = 8 7x = -21 Solve for x -9 -17y = 8 /7 /7 +9 +9 -17y = 17 x = -3 /-17 /-17 y = -1 y = 3x Plug in -3 for x and solve for y y = 3(-3) x = -3 -5(-1) y = -9 x = -3 +5 Solution = (2, -1) x = 2 Solution = (-3, -9)

Infinite or No solutions Write and solve a System of Equations 6x – 2y = -4 y = 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) = -4 Yankees = x Reds = y 6x – 6x - 4 = -4 -4 = -4 Total games x + y = 31 Infinite solutions Times games x = 5.2y 5.2y + y = 31 When all variables cancel, if: the statement is true = infinite solutions the statement is false = no solutions 6.2y = 31 Yankees = 26 Reds = 5 /6.2 /6.2 y = 5 x = 5.2(5) x = 26

5.3 Elimination Using Addition and Subtraction Elimination: Addition 3x – 5y = -16 2x + 5y = 31 Elimination: Subtraction 5s + 2t = 6 9s + 2t = 22 Subtract to eliminate because the t’s are the same number same sign Add to eliminate because the y’s are the same number opposite signs 3x – 5y = -16 + 2x + 5y = 31 5s + 2t = 6 - 9s + 2t = 22 5x = 15 -4s = -16 /5 /5 Solution: (4, -7) Solution: (3, 5) /-4 /-4 x = 3 s = 4 3(3) – 5y = -16 5(4) + 2t = 6 9 – 5y = -16 20 + 2t = 6 -9 -9 -20 -20 -5y = -25 2t = -14 /-5 /-5 /2 /2 y = 5 t =-7

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. 2x + y = 18 2x + y = 18 Add because the y’s are the same number opposite signs 4x - y = 12 + 4x - y = 12 6x = 30 /6 /6 x = 5 Solution: 5 and 8 2 (5) + y = 18 10 + y = 18 -10 -10 y = 8

5.4 Elimination Using Multiplication Multiply One Equation 3x + 4y = 6 5x + 2y = -4 Multiply Two Equations 3x + 4y = -25 2x – 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 3x +4y = 6 -2[5x + 2y = -4] 3[3x +4y = -25] 4[2x – 3y = 6] 9x +12y = -75 8x – 12y = 24 3x +4y = 6 -10x + -4y = 8 + + 17x = -51 -7x = 14 /17 /17 /-7 /-7 x = -3 x = -2 Solution: (-2, 3) Solution: (-3, -4) 3(-3) + 4y = -25 3(-2) + 4y = 6 -9 + 4y = -25 -6 + 4y = 6 +9 +9 +6 +6 4y = -16 4y = 12 /4 /4 /4 /4 y = -4 y = 3

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 Multiplication If none of the coefficients are the same number