{ } Sec 3.1 Systems of Linear Equations in Two Variables

Transcription

1 Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination method. 1. Deciding Whether an Ordered Pair Is a Solution Sstem of Equation consists of at least two or more linear equations. Eample Solution of the sstem is the point(s) where the graphs intersect (give true for both equations) 4 Eample 1. Determine whether the ordered pairs are a solution to the sstem: 6, 0 a. ( ) Answer: 4, b. ( ) Answer:. Solve a sstem of linear equations using the graphing method Three tpes of the Sstem of Equations. 1. Consistent sstem with Independent equation Two lines intersect at one point(, ). Has one solution,(, ). m1 m When solve the sstem, get a number, a number.. Inconsistent Sstem Two lines are parallel. Has no solution, φ or {}. m and b1 b When solve, get false statement. 1 m. Consistent sstem with dependent equation Two lines lie on top of the others (same line). m b { } Has infinitel man solutions, (, ) or (, ) m 1 m and b 1 b When solve the sstem, get true statement. { a b c}

2 Steps to solve linear equations b graphing 1. Solve and graph each equation separatel.. Identif tpe of sstems (consistent, inconsistent, or dependent).. State number of solution (one solution, infinitel man solutions or no solution). Eample. Solve b graphing. Label at least two points for each graph on the graph grid Solution: Solve a sstem of linear equations using the elimination method Steps: 1. Write each equation in the form: A B C. Choose variable to eliminate.. If necessar, multipl one or both equations b appropriate number(s) so that the coefficients of the eliminated variable will have the sum of ero. 4. Add two equations together.. Solve for the variable. 6. Solve for the other variable. 7. State the final solution in ordered pair, if it eists. Eample. Solve linear equations using the elimination method Answer: 4. Solve a sstem of linear equations using the substitution method Steps: 1. Solve one of the equations for one of its variable: or.. Substitute the resulting found in step 1 into the other equation.. Solve the equation found in step to find the value of one variable. 4. Substitute the value found step in an original equations containing both variables to find the value of the other variable.. Check the solution b substituting the numerical values of the variables in both original equations.

3 Eample 4. Solve linear equations using the substitution method Answer: Eample. Solve linear equations using either substitution or elimination method Answer: Answer:

4 Sec. Problem Solving: Sstems of Two Linear Equations Learning Objectives: 1. Use a sstem of equations to solve problems. Problem-Solving Steps: 1. UNDERSTAND the problem b do the following: Read and reread the problem. Identif what is given and what is the question. Choose two variables to represent the two unknowns being asked. Construct a drawing if needed.. TRANSLATE the problem into two equations.. SOLVE the sstem of equations. 4. INTERPRET the results: Check the proposed solution in the stated problem and state our conclusion. 1. Finding Unknown Numbers Solve the following problems b (a) Choose the variables to represent the unknown. (b) Set up a sstem and solve. (c) Write the answer using a complete sentence. Eample 1. The sum of two numbers is 6. Their difference is 1. What are the numbers? First Number is Second Number is. Uniform Motion Problems: Formula: d rt where d distance; r rate or speed; t time Eample. With a tailwind, a small Piper aircraft can fl 600 miles in hours. Against this same wind, the Piper can fl the same distance in 4 hours. Find the speed of the Piper and speed of the wind. t r d With Wind Against Wind Piper speed is Wind speed is

5 . Solving a Problem about Prices: Formula: Total price Number of tickets price per ticket Eample. Admission prices at a local weekend fair were $ for children and $7 for adults. The total mone collected was $79, and 87 people attended the fair. How man children and how man adults attended the fair? Numbers of tickets Price per ticket Total price children adults There were adults and children attended the fair Coin Problems Formula: Total Value numbers of coins value of each coin Eample 4. Tim has $ 1.10 in quarters and nickels. How man quarters and nickels does he have if he has 14 coins in total? Numbers of coins Value of each coin Total value quarters nickels Tim has quarters and nickels.

6 . Investment Problems Formula: I Prt Where I interest earn, P principal, r interest rate, t time (in ear) Eample. Lit invested $6000, part at 6% and the rest at 4%. How much is invested at each rate if the annual income from the two investments is $90? P r t I interest Account 1 Account Lit invested at 6% and at 4% 6. Miture Problems: Formula: Amount of solution number of liters percent of the solution Eample 6. Nanc wants to make 0 liters of a 60% alcohol solution. She currentl has a 0% alcohol solution and a 70% alcohol solution. How man liters of a 0% alcohol solution and a 70% alcohol solution she needs to make 0 liters of a 60% alcohol solution? Number of liters Percent of solution Amount of solution Solution 1 Solution Miture Nanc needs liters of 0% and liters of 70%

7 Sec. Sstems of Linear Equations in Three Variables Learning Objectives: 1. Solve sstems of three linear equations containing three variables.. Model and solve problems involving three linear equations containing three variables. 1. Solve sstems of three linear equations containing three variables Definitions: 1. Linear Equations in Three Variables Algebraic equation of the forma b c d, where a, b, c and d are real numbers, with a, b and c are not all ero.,, that give true statement to all. A Solution to a sstem of equation is an ordered triple ( ) equations in the sstem. Eample 1. Determine whether the given ordered triple ( 1,, ) equation. 1 9 is solution of the sstem of linear Answer: Three tpes of the sstem 1. Consistent Sstem with independent equations (independent sstem)-has eactl one solution(,, ).. Inconsistent Sstem-has no solution, φ.. Consistent Sstem with dependent equations (dependent sstem) has infinitel man solutions. Steps for Solving Sstems of Linear Equations in Three Variables 1. Select two of the equations and eliminate one of the variables form one of the equations. Select an two other equations and eliminate the same variable from one of the equations.. You will have two equations that have onl two unknowns. Eliminate a second variable form the two linear equations in two unknown.. Solve the remaining variable.

8 Eample. Solve each sstem of equations Answer:. 14 Answer:. 4 7 Answer:

9 . Model and solve problems involving three linear equations containing three variables Eample. Curve Fitting The function ( ) f a b c is a quadratic function, where a, b, and c are constant. a. If f ( 1 ) 4, then 4 a ( 1) b( 1) c f ( 1) 6 and f ( ). or a b c 4. Find two additional linear equations if Sstem: b. Use the three linear equations found in part (a) to determine a, b and c. What is the 1, 6, 1, 4,? quadratic function that contains the points ( )( ) and ( ) a b c Function:

10 Sec.6 Sstems of Linear Inequalities Learning Objectives: 1. Graph a linear inequalit in two variables.. Determine whether an ordered pair is a solution to a sstem of linear inequalities.. Graph a sstem of linear inequalities. 1. Graph a linear inequalit in two variables Definition: Linear Inequalit is an equation of the form a b< c; a b> c; a b c; a b c Steps to solve linear inequalit: 1. Solve inequalit in the form > m b; < m b; m b; m b. If inequalit involving or, draw a solid line. If inequalit involving < or >, draw a dashed line.. Pick a test point. Substitute the values in the inequalit. If the result is true, shade the side that contain the test point. If a false statement, shade the other side. Shaded below the line if < m b or m b Shaded above the line if > m b or m b CAUTION! If multipl or divide b a negative number, the inequalit sign change to opposite Eample 1. Graph the following inequalit. Label at least two points on the graph grid. < Determine whether an ordered pair is a solution to a sstem of linear inequalities Solution to a sstem of linear inequalities is an ordered pair that satisfies a sstem of linear inequalities. (It makes each inequalit in the sstem a true statement) Eample. Is (, 6) a solution to the sstem of linear inequalities? 8 ; 9 Answer:

11 . Graph a Sstem of linear inequalities Recall: 1. Using solid line when the inequalit is nonstrict ( or ). Using a dashed line when the inequalit is strict (< or > ). If multipl or divide b a negative number, the inequalit switches. 4. The solution of the sstem of the inequalities is the overlap portion. If there is no overlap, then the sstem of inequalities has no solution. Eample. Graph the sstem. State the corner points and tell whether the graph is bounded or unbounded. 1. > Answer: Graph: Corner points:. < Graph: Corner points:

12 Eample 4. Write a sstem of linear inequalities that has the given graph. Sstem: Notes: