Answers (Lesson 3-1) Study Guide and Intervention. Study Guide and Intervention (continued) Solving Systems of Equations by Graphing

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1 Glencoe/McGraw-Hill A Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations b Graphing Graph Sstems of Equations A sstem of equations is a set of two or more equations containing the same variables. You can solve a sstem of linear equations b graphing the equations on the same coordinate plane. If the lines intersect, the solution is that intersection point. Eample Solve the sstem of equations b graphing. Write each equation in slope-intercept form. The graphs appear to intersect at (0, ). CHECK Substitute the coordinates into each equation. 0 ( ) 0 ( ) The solution of the sstem is (0, ). Eercises Solve each sstem of equations b graphing.... (6, ) (, ) (, ) (, ) (, ) (, ) (6, ) (, ) (, ) (, ) Glencoe/McGraw-Hill 9 Glencoe Algebra (, ) (, ) (0, ) Lesson - - Classif Sstems of Equations The following chart summarizes the possibilities for graphs of two linear equations in two variables. Graphs of Equations Slopes of Lines Classification of Sstem Number of Solutions Lines intersect Different slopes Consistent and independent ne Lines coincide (same line) Lines are parallel Same slope, same -intercept Same slope, different -intercepts Consistent and dependent Inconsistent Infinitel man Glencoe/McGraw-Hill 0 Glencoe Algebra None Eample Graph the sstem of equations 6 and describe it as consistent and independent, consistent and dependent, or inconsistent. Write each equation in slope-intercept form. 6 The graphs intersect at (, ). Since there is one solution, the sstem is consistent and independent. Eercises Graph the sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent consistent 6 inconsistent and dependent inconsistent consistent consistent 6 consistent and independent and dependent and independent NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Equations b Graphing (, ) Answers (Lesson -)

2 Glencoe/McGraw-Hill A5 Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations Algebraicall Substitution To solve a sstem of linear equations b substitution, first solve for one variable in terms of the other in one of the equations. Then substitute this epression into the other equation and simplif. Eample Use substitution to solve the sstem of equations. 9 6 Solve the first equation for in terms of. 9 First equation 9 Subtract from both sides. 9 Multipl both sides b. Substitute the epression 9 for into the second equation and solve for. 6 Second equation ( 9) 6 Substitute 9 for Distributive Propert Simplif. 7 Add 7 to each side. Divide each side b 7. Now, substitute the value for in either original equation and solve for. 9 First equation () 9 Replace with. 6 9 Simplif. Subtract 6 from each side. Multipl each side b. The solution of the sstem is (, ). Eercises Solve each sstem of linear equations b using substitution (, 0) (, ) (, 7) no solution (, ) (, 9) (, ), (, 6) , infinitel man, Glencoe/McGraw-Hill 5 Glencoe Algebra Lesson - - NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Equations Algebraicall Elimination To solve a sstem of linear equations b elimination, add or subtract the equations to eliminate one of the variables. You ma first need to multipl one or both of the equations b a constant so that one of the variables has the same (or opposite) coefficient in one equation as it has in the other. Eample Use the elimination method to solve the sstem of equations. 6 Multipl the second equation b. Then subtract the equations to eliminate the variable. 6 6 Multipl b Replace with 7 and solve for. 6 ( 7) 6 6 The solution is ( 7, ). Eample Use the elimination method to solve the sstem of equations. 5 5 Multipl the first equation b and the second equation b. Then add the equations to eliminate the variable. Multipl b Multipl b Eercises Solve each sstem of equations b using elimination. Replace with and solve for. ( ) The solution is (, 5) (, ) (, ) (, ) (, 0) no solution (, ) infinitel man (, ) m n m n 5 (6, ) (, ), (, 6) Glencoe/McGraw-Hill 6 Glencoe Algebra Answers (Lesson -) Answers

3 Glencoe/McGraw-Hill A Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention Solving Sstems of Inequalities b Graphing Graph Sstems of Inequalities To solve a sstem of inequalities, graph the inequalities in the same coordinate plane. The solution set is represented b the intersection of the graphs. Eample Solve the sstem of inequalities b graphing. and The solution of is Regions and. The solution of is Regions and. The intersection of these regions is Region, which is the solution set of the sstem of inequalities. Eercises Solve each sstem of inequalities b graphing Glencoe/McGraw-Hill Glencoe Algebra Region Region Region Lesson - - NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Inequalities b Graphing Find Vertices of a Polgonal Region Sometimes the graph of a sstem of inequalities forms a bounded region. You can find the vertices of the region b a combination of the methods used earlier in this chapter: graphing, substitution, and/or elimination. Eample Find the coordinates of the vertices of the figure formed b 5 0,, and. Graph the boundar of each inequalit. The intersections of the boundar lines are the vertices of a triangle. The verte (, 0) can be determined from the graph. To find the coordinates of the second and third vertices, solve the two sstems of equations and 5 0 For the first sstem of equations, rewrite the first equation in standard form as. Then multipl that equation b and add to the second equation. Multipl b. 5 0 ( ) 5 0 Then substitute in one of the original equations and solve for The coordinates of the second verte are,. Eercises Glencoe/McGraw-Hill Glencoe Algebra For the second sstem of equations, use substitution. Substitute for in the second equation to get ( ) Then substitute in the 5 first equation to solve for. 6 The coordinates of the third verte are,. 5 5 Thus, the coordinates of the three vertices are (, 0),,, and,. Find the coordinates of the vertices of the figure formed b each sstem of inequalities (, ), (, ), (, ), (, ), (, ), (, ) (, ), (, ) 0, 5 Answers (Lesson -)

4 Glencoe/McGraw-Hill A Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention Linear Programming Maimum and Minimum Values When a sstem of linear inequalities produces a bounded polgonal region, the maimum or minimum value of a related function will occur at a verte of the region. Eample Graph the sstem of inequalities. Name the coordinates of the vertices of the feasible region. Find the maimum and minimum values of the function f(, ) for this polgonal region. 6 6 First find the vertices of the bounded region. Graph the inequalities. The polgon formed is a quadrilateral with vertices at (0, ), (, ), (5, ), and (, ). Use the table to find the maimum and minimum values of f(, ). (, ) f(, ) (0, ) (0) () (, ) () () (5, ) (5) () 7 (, ) ( ) ( ) 7 The maimum value is 7 at (5, ). The minimum value is 7 at (, ). Eercises Graph each sstem of inequalities. Name the coordinates of the vertices of the feasible region. Find the maimum and minimum values of the given function for this region.... f(, ) f(, ) f(, ) vertices: (, ), (, ), vertices: ( 5, ), vertices (0, ), (, ), (5, ), (5, ); ma: ; (, ), (, ); 7, ; ma: 5; min: 6 min; 5 ma: 0; min: Glencoe/McGraw-Hill 7 Glencoe Algebra Lesson - - Real-World Problems When solving linear programming problems, use the following procedure.. Define variables.. Write a sstem of inequalities.. Graph the sstem of inequalities.. Find the coordinates of the vertices of the feasible region. 5. Write an epression to be maimized or minimized. 6. Substitute the coordinates of the vertices in the epression. 7. Select the greatest or least result to answer the problem. Eample A painter has eactl units of ellow de and 5 units of green de. He plans to mi as man gallons as possible of color A and color B. Each gallon of color A requires units of ellow de and unit of green de. Each gallon of color B requires unit of ellow de and 6 units of green de. Find the maimum number of gallons he can mi. Step Define the variables. the number of gallons of color A made the number of gallons of color B made Step Write a sstem of inequalities. Since the number of gallons made cannot be negative, 0 and 0. There are units of ellow de; each gallon of (6, ) color A requires units, and each gallon of (0, 9) 5 (, 0) color B requires unit So. Color A (gallons) Similarl for the green de, 6. Steps and Graph the sstem of inequalities and find the coordinates of the vertices of the feasible region. The vertices of the feasible region are (0, 0), (0, 9), (6, ), and (, 0). Steps 5 7 Find the maimum number of gallons,, that he can make. The maimum number of gallons the painter can make is, 6 gallons of color A and gallons of color B. (, ) (0, 0) (0, 9) (6, ) (, 0) f(, ) 0 9 Eercises NAME DATE PERID Stud Guide and Intervention (continued) Linear Programming. FD A delicatessen has pounds of plain sausage and 0 pounds of garlic-flavored sausage. The deli wants to make as much bratwurst as possible. Each pound of bratwurst requires pound of plain sausage and pound of garlic-flavored sausage. Find the maimum number of pounds of bratwurst that can be made. 0 lb. MANUFACTURING achine A can produce 0 steering wheels per hour at a cost of $6 per hour. Machine B can produce 0 steering wheels per hour at a cost of $ per hour. At least 60 steering wheels must be made in each -hour shift. What is the least cost involved in making 60 steering wheels in one shift? $9 Glencoe/McGraw-Hill Glencoe Algebra Color B (gallons) Answers (Lesson -) Answers

5 Glencoe/McGraw-Hill A Glencoe Algebra -5 NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations in Three Variables Sstems in Three Variables Use the methods used for solving sstems of linear equations in two variables to solve sstems of equations in three variables. A sstem of three equations in three variables can have a unique solution, infinitel man solutions, or no solution. A solution is an ordered triple. Eample Solve this sstem of equations. z 6 z z Step Use elimination to make a sstem of two equations in two variables. z 6 First equation z Second equation ( ) z Second equation ( ) z Third equation 5 z Add to eliminate. 6 z Add to eliminate. Step Solve the sstem of two equations. 5 z ( ) 6 z Add to eliminate z. Divide both sides b. Substitute for in one of the equations with two variables and solve for z. 5 z Equation with two variables 5( ) z Replace with. 5 z Multipl. z 7 Add 5 to both sides. The result so far is and z 7. Step Substitute for and 7 for z in one of the original equations with three variables. z 6 riginal equation with three variables ( ) 7 6 Replace with and z with Multipl. Simplif. The solution is (,, 7). Eercises Solve each sstem of equations.. z 0. z. z z 6z z 0 z 7 0z 6 z (,, ), 5, infinitel man solutions. z 5. z z z z 6 z 6 6 z z,, 5 no solution 6,, Glencoe/McGraw-Hill Glencoe Algebra Lesson -5-5 NAME DATE PERID Real-World Problems Eample The Laredo Sports Shop sold 0 balls, bats, and bases for $99 on Monda. n Tuesda the sold balls, bats, and bases for $7. n Wednesda the sold balls, bats, and base for $.60. What are the prices of ball, bat, and base? First define the variables. price of ball price of bat z price of base Translate the information in the problem into three equations. 0 z 99 z 7 z.60 Stud Guide and Intervention (continued) Solving Sstems of Equations in Three Variables Subtract the second equation from the first equation to eliminate z. 0 z 99 ( ) z 7 6 Multipl the third equation b and subtract from the second equation. z 7 ( ) 6 z So a ball costs $, a bat $5.0, and a base $.0. Eercises Substitute 5.0 for in the equation (5.0) 6 Substitute for and 5.0 for in one of the original equations to solve for z. 0 z 99 0() (5.0) z z 99 z.0 z.0. FITNESS TRAINING Carl is training for a triathlon. In her training routine each week, she runs 7 times as far as she swims, and she bikes times as far as she runs. ne week she trained a total of miles. How far did she run that week? 56 miles. ENTERTAINMENT At the arcade, Ran, Sara, and Tim plaed video racing games, pinball, and air hocke. Ran spent $6 for 6 racing games, pinball games, and game of air hocke. Sara spent $ for racing games, pinball games, and 5 games of air hocke. Tim spent $.5 for racing games, 7 pinball games, and games of air hocke. How much did each of the games cost? Racing game: $0.50; pinball: $0.75; air hocke: $.50. FD A natural food store makes its own brand of trail mi out of dried apples, raisins, and peanuts. ne pound of the miture costs $.. It contains twice as much peanuts b weight as apples. ne pound of dried apples costs $., a pound of raisins $.0, and a pound of peanuts $.. How man ounces of each ingredient are contained in pound of the trail mi? oz of apples, 7 oz of raisins, 6 oz of peanuts Glencoe/McGraw-Hill Glencoe Algebra Answers (Lesson -5)