Transcript Chapter 5
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 y = -x + 8 y = 4x - 7 a. Solve a system of Equations by graphing 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 Ex: Graph the system of inequalities x-y<-1 y 2x+3 x-y>3 5.2 Substitution Solve Using Substitution y = 3x x + 2y = -21 Solve using Substitution x + 5y = -3 x + 5y = -3 3x – 2y = 8 -5y -5y Solve one equation for a variable x + 2(3x)= -21 x + 6x = -21 7x = -21 /7 /7 x = -3 y = 3x y = 3(-3) Substitute 3x for y Solve for x Plug in -3 for x and solve for y y = -9 Solution = (-3, -9) x = -3 – 5y 3(-3 – 5y) – 2y = 8 -9 -15y – 2y = 8 -9 -17y = 8 +9 +9 -17y = 17 /-17 /-17 y = -1 x = -3 -5(-1) x = -3 +5 x=2 Solution = (2, -1) Infinite or No solutions 6x – 2y = -4 y = 3x + 2 6x – 2(3x +2) = -4 6x – 6x - 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? Yankees = x Total games Times games 5.2y + y = 31 6.2y = 31 /6.2 /6.2 y=5 x = 5.2(5) x = 26 Reds = y x + y = 31 x = 5.2y Yankees = 26 Reds = 5 5.3 Elimination Using Addition and Subtraction Elimination: Addition 3x – 5y = -16 2x + 5y = 31 3x – 5y = -16 + 2x + 5y = 31 5x = 15 /5 /5 x=3 3(3) – 5y = -16 9 – 5y = -16 -9 -9 -5y = -25 /-5 /-5 y=5 Add to eliminate because the y’s are the same number opposite signs Solution: (3, 5) Elimination: Subtraction 5s + 2t = 6 9s + 2t = 22 5s + 2t = 6 - 9s + 2t = 22 -4s = -16 /-4 /-4 s=4 5(4) + 2t = 6 20 + 2t = 6 -20 -20 2t = -14 /2 /2 t =-7 Subtract to eliminate because the t’s are the same number same sign Solution: (4, -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 4x - y = 12 + 2x + y = 18 4x - y = 12 6x = 30 /6 /6 x=5 2 (5) + y = 18 10 + y = 18 -10 -10 y=8 Add because the y’s are the same number opposite signs Solution: 5 and 8 5.4 Elimination Using Multiplication Multiply One Equation 3x + 4y = 6 5x + 2y = -4 3x +4y = 6 -2[5x + 2y = -4] Solution: (-2, 3) Multiply one equation to make a variable have the same number and opposite sign 3x +4y = 6 + -10x + -4y = 8 Multiply Two Equations 3x + 4y = -25 2x – 3y = 6 3[3x +4y = -25] 4[2x – 3y = 6] Multiply both equations to make a variable have the same number and opposite sign 9x +12y = -75 + 8x – 12y = 24 -7x = 14 /-7 /-7 17x = -51 /17 /17 x = -2 x = -3 3(-2) + 4y = 6 -6 + 4y = 6 +6 +6 4y = 12 /4 /4 y=3 Solution: (-3, -4) 3(-3) + 4y = -25 -9 + 4y = -25 +9 +9 4y = -16 /4 /4 y = -4 5.5 Applying Systems of Equations Method Graphing The Best Time to Use 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 Elimination: Subtraction Elimination: Multiplication If one variable has coefficients with the same number and opposite signs If one variable has coefficients with the same number and same sign If none of the coefficients are the same number