Chapter 5: Systems of Equations and Inequalities. Section 5.4. Check Point Exercises

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1 Chapter : Systems of Equations and Inequalities Section. Check Point Eercises. = y y = Solve the first equation for y. y = + Substitute the epression + for y in the second equation and solve for. ( + ) = = ( ) or = If, y = ( ) + =. If =, y = ( ) + = 7. {(, ), (, 7). + y ( ) + ( y ) = Solve the first equation for. = y Substitute the epression y for in the second equation and solve for y. ( y ) + ( y ) = y + y + + y y + = y + y ( y )( y + ) y or y + = y = or y = If y = = =,. If y =, = ( ) =.,, (, ).. + y = + y = 8 Eliminate the y -term by multiplying the first equation by and the second equation by. Add the resulting equations. 9 y = 8 + y = 9 = 9 = 9 =± If =, () + y = y = y =± If =, ( ) + y = y = y =± {(,),(, ),(,),(, ). y = + + y = Arrange the first equation so that variable terms appear on the left, and constants appear on the right. Add the resulting equations to eliminate the -terms and solve for y. + y = + y = y + y = y + y ( y + )( y ) y + or y y = or y = If y =, 9

2 Chapter : Systems of Equations and Inequalities + ( ) = = no real solution If y =, + () = {(, ). + y y = Solve the second equation for. = 7 Substitute the epression for in the y first equation and solve for y. y y + = + y y y y + ( y 7)( y ) y 7 or y y = 7 or y = If y = 7, = =. 7 If y =, = = 7. The dimensions are 7 feet by feet. Eercise Set.. + y = y = Solve the first equation for y. y =. Substitute the epression for y in the second equation and solve for. = + = ( + )( ) + or = or = If =, y = ( ) =. If =, y =. {(, ), (, ). + y = y = + Substitute the epression + for y in the first equation and solve for. + + = + ( )( ) = = Substitute = and then = into the equation + y = and solve for each value of y. + y = + y = y = y {(, ), (, ). y = y = + Substitute the epression for y in the second equation and solve for. = + ( ) ( + ) or + = or = 9

3 Chapter : Systems of Equations and Inequalities If =, y =( ) ( ) =. If =, y = ( ) ( ) =. {(, ), (, ) 7. + y = y = Solve the second equation for y. y = Substitute the epression for y in the first equation and solve for. + ( ) = + + = ( )( + ) or + = or = If =, y = =. If =, y = =. {(, ), (, ) 9. y = y = Solve the first equation for y. y = Substitute the epression for y in the second equation and solve for. = = ( + )( ) + or = or = If =, y = =. If =, y = =.,, (, ).. y = 9 y = Solve the second equation for y. y = Substitute the epression for y in the first equation and solve for. = = = + + ( + ) ( ) + or = or = If = y =, =. If = y =,.,,,. {( ) ( )}. y = + y = Solve the second equation for y. y = 9

4 Chapter : Systems of Equations and Inequalities Substitute the epression If =, y = =. for y in the If =, y = ( ) =. second equation and solve for. {(, ),(, ) + = 7. + y = 9 + ( ) + ( y + ) = Solve the first equation for y. + 9 y = ( 9) ( )= Substitute the epression for y in the second equation and solve for. ( ) ( + ) ( ) ( + ) ( ) + ( + ) = or + or or + = = or = or = or = ( ) + ( ) = If =, y = =. 8 If =, y = =. ( ) If =, y = =. or = If =, y = =. If, y = =. If =, y = =. {(, ),(, ),(, ),(, ) {(, ),(, ). + y = + y y = Solve the first equation for y. y = Substitute the epression for y in the second equation and solve for. + ( ) ( ) = + ( + )= + = ( ) ( + ) or + = = or = 9. Eliminate the y terms by adding the equations. + y = y = = 8 = 9 =± If =, () + y = y = y =± If =, 9

5 Chapter : Systems of Equations and Inequalities ( ) + y y = = y =± {(, ), (, ), (, ), (, ). y = 7 + y = Eliminate the terms by multiplying the first equation by and adding the resulting equations. + y = + y = y = y = y =± If y =, 7 = 9 =± If y =, ( ) = 7 = 9 =±,,,,,,,. {( ) ( ) ( ) ( )}. Arrange the equations so that variable terms appear on the left and constants appear on the right. + y = y = Eliminate the y terms by multiplying the first equation by and the second equation by. Add the resulting equations. 9 + y = 8 8 y 7 = 8 = =± If =, ( ) + y = y = y =± If =, ( ) + = y y =± {(, ), (, ), (, ), (, ). + y = ( 8) + y = Epand the second equation and eliminate and y terms by multiplying the first equation by and adding the resulting equations. + + y = y = + = = 8 = If =, () + y = y = y =± {(, ), (, ) 7. y = + y = Eliminate the y terms by multiplying the first equation by and adding the resulting equations. 9

6 Chapter : Systems of Equations and Inequalities y = + y = + ( + ) or + = If, y = y= ± If =, y ( ) = y = y =± (, ),(, ),,,,. 9. The addition method is used here to solve the system. + y = y = Eliminate the y terms by multiplying the first equation by and the second equation by. Add the resulting equations. 9 + y = 8 8 y 7 = 8 = =± If =, ( ) + y = y = y =± If =, ( ) + = y y =± {(, ), (, ), (, ), (, ). The substitution method is used here to + y = 8 y = Solve the second equation for y. y = Substitute the epression for y in the first equation and solve for. + 8 = + = 8 + = ( 8) ( )= 8 or = = 8 or = =± or =± If =, y = =. If =, y = =. If =, y = =. If =, y = =.,,,, (, ), (, ).. The substitution method is used here to 9

7 Chapter : Systems of Equations and Inequalities + y + y = Solve the second equation for. = y Substitute the epression y for in the first equation and solve for y. y y ( ) + = y + y + y 8y y + = y y + ( y ) ( y ) y or y = y = or y = If y =, =. If y =, = ()=. {(, ),(, ). Eliminate y by adding the equations. + y y + ( + ) or + = or = If, ( ) + y y If =, ( ) + y y = {(, ),(, ) 7. The substitution method is used here to + y y Solve the second equation for. = y Substitute the epression y for in the first equation and solve for y. y + y y + y y + = y y yy ( ) y or y y = If y, ( ) If y =, = ( ) = =± {(, ), (, ), (, ) 9. The substitution method is used here to ( ) y = + + y = Solve the first equation for. = y Substitute the epression y for in the first equation and solve for y. ( + ) y = y + y y = y y + y y + = ( y )( y ) y or y = y = or y = 9

8 Chapter : Systems of Equations and Inequalities If y = = =,. If y =, = () =. (, ),,.. The substitution method is used here to + y + y = + y = Solve the second equation for y. y = Substitute the epression for y in the first equation and solve for. + ( ) + ( ) ( )( + ) or + = or = If = y = = 9,. If =, y = ( ) =. 9,, (, ).. The substitution method is used here to + y = y = Solve the first equation for y. y = Substitute the epression for y in the second equation and solve for. = + ( ) ( ) or = or = If =, y = =. If =, y = =. The numbers are and.. Eliminate the y terms by adding the equations. y = + y = 9 = = =± If =, ( ) + y = 9 y = y =± If =, ( ) + y = 9 y = y =± The numbers are and, and, and, or and y = y = Substitute the epression for y in the first equation and solve for. + 97

9 Chapter : Systems of Equations and Inequalities + ( 8 + )= + + = ( )= or or = =± If, y = ( ) =. If =, y = ( ). If =, y = ( ). It is possible for the comet to intersect the orbiting body at (, ), (, ), (, ). 9. L+ W = LW = 77 Divide each term in the first equation by and solve L. L+ W = 8 L = 8 W Substitute the epression 8 W for L in the second equation and solve for W. 8 WW 77 8W W = 77 W 8W + 77 ( W ) ( W 7) W or W 7 W = or W = 7 If W =, L = 8 = 7. If W = 7, L = 8 7 =. The dimensions are feet by 7 feet.. L + W = = LW = 8 Solve the second equation for L. L = 8 W Substitute the epression 8 for L in the W 98 first equation and solve for W. 8 + W = W + W W + W W W W + ( W ) ( W )= W or W W = or W = W =± or W =± 8 The width cannot be or 8 inches. If W =, 8 L = = 8 If W = 8, 8 L = = 8 The dimensions are 8 inches by inches.. y = + y = Divide each term in the second equation by and solve for y. + y = y = Substitute the epression for y in the first equation and solve for. ( 8 + )= ( )( ) or = or = If =, y =. If =, y = () =. The dimensions of the floor are meters by

10 Chapter : Systems of Equations and Inequalities meters and the dimensions of the square that will accomodate the pool are meters by meters.. 7. Answers may vary. 9. Eercise Eercise Eercise Eercise 9 Eercise. a. False; a circle and a line will have at most intersection points. b. True; a parabola can intersect a circle in points. c. False; It is possible for two circles to not intersect. 99

11 Chapter : Systems of Equations and Inequalities d. False; A circle can intersect a parabola at one point. See Check Point for an eample. (b) is true.. By the Pythagorean Theorem: a + b = = a + b+ 9 7 Epand the second equation. a + b + 8b + 8 = 89 Eliminate the a and b terms by multiplying the first equation by and adding the resulting equations. a + b + 8b 8 a b = 8b = 8 b = a + () = a = a = 8. log = y + log = y y + = y = Substitute the epression y for in the equation y + = and solve for y. y+ y = ( ) y+ y = y + = y y = = =, {(,, ) Section. Check Point Eercises. y < 8 Graph y = 8 as a dashed line using its -intercept (, ), and its y-intercept (, ). Test (, ): ( ) ( ) < 8? < 8true Shade the half-plane containing (, ). y. y Graph y = as a solid line by using its slope,, and its y-intercept (, ). Test (, ): ()? true Shade the half plane containing (, ). y. + y Graph + y = as a solid circle with