Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Quadratic equation wikipedia , lookup
Basis (linear algebra) wikipedia , lookup
Bra–ket notation wikipedia , lookup
Elementary algebra wikipedia , lookup
Signal-flow graph wikipedia , lookup
System of polynomial equations wikipedia , lookup
History of algebra wikipedia , lookup
3-1 3-1,2 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Warm Up (Slide #2-3) Objective and Standards (Slide #4) Vocab (Slide #5–8) Lesson Presentation (Slide #9–31) Text Questions (NONE) Worksheets 3.1A, 3.2A (Slide #32) Lesson Quiz (Slide #33-34) Holt Algebra Holt Algebra 22 3-1 3-1,2 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Warm Up Use substitution to determine if (1, –2) is an element of the solution set of the linear equation. 1. y = 2x + 1 no 2. y = 3x – 5 yes Write each equation in slope-intercept form. 4. 4y – 3x = 8 3. 2y + 8x = 6 y = –4x + 3 Holt Algebra 2 3-1 3-1,2 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Warm Up Determine if the given ordered pair is an element of the solution set of 2x – y = 5 3y + x = 6 1. (3, 1) yes 2. (–1, 1) no Solve each equation for y. 3. x + 3y = 2x + 4y – 4 y = –x + 4 4. 6x + 5 + y = 3y + 2x – 1 y = 2x + 3 Holt Algebra 2 3-1 3-1,2 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Objectives 1. Solve systems of linear equations with: •Graphs and tables •Substitution •Elimination 2. Determine whether there will be one, none, or an infinite number of solutions by noting characteristics of each equation. Holt Algebra 2 3-1 3-1,2 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Vocabulary system of equations linear system substitution elimination linear combinations Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations. A line is an infinite set of points that are solutions to a linear equation. The solution of a system of equations is the set of all points that satisfy each equation. Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems There are two aspects of substitution: In one, a possible solution (ordered pair) is given and you simply substitute its x and y values into each equation to see if that point satisfies both. In the other, you substitute the equivalent expression for a variable from one equation into the other equation, solve for one variable, then use that value to solve for the other variable. (I know…it sounds all so confusing, but it’s really easy.) Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems You can also solve systems of equations with the elimination method. With elimination, you get rid of one of the variables by adding or subtracting equations. You may have to multiply one or both equations by a number to create variable terms that can be eliminated. Reading Math The elimination method is sometimes called the addition method or linear combinations. Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Points to remember about linear equations and systems: On the graph of the system of two equations, the solution is the set of points where the lines intersect. A point is a solution to a system of equation if the x- and y-values of the point satisfy both equations. Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Example 1A: Verifying Solutions of Linear Systems To see if a given point is a solution to a linear system, substitute the (x,y) values into both equations. For example… “Is (1,3) the solution to this linear system?” (1, 3); x – 3y = –8 3x + 2y = 9 Ans: YES Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Check It Out! Example 1b Is (5,3) an element of the solution set for the system of equations? (5, 3); 6x – 7y = 1 3x + 7y = 5 Ans: NO Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Example 2A: Solving Linear Systems by Using Graphs and Tables on your graphing calculator Solve the system. Check your answer. First, solve each equation for y, then graph both: 2x – 3y = 3 y+2=x On the graph, the lines appear to intersect at the ordered pair (3, 1) Use the calculator’s “Table” or “Trace” function to verify. Holt Algebra 2 Sol’n: (3,1) 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Check It Out! Example 2b Use a graph and a table to solve the system. Check your answer. First, solve each equation for y. x+y=8 2x – y = 4 Sol’n: (4, 4) Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Different Slopes will have ONE solution Holt Algebra 2 Same Slopes and Same Y-int. will have INFINITE Sol’ns. Same Slopes but Different Y-int. will have NO Solutions. 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Remember! An identity, such as 0 = 0, 8 = 8, -7 = -7, etc… is always true and indicates infinite solutions. A contradiction, such as 1 = 3, 5 = 9, -8 = 8, etc… is never true and indicates no solution. Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Example 4: Summer Sports Application One golf course charges $20 to rent golf clubs plus $55 per hour for golf cart rental. A different course charges $35 to rent clubs plus $45 per hour to rent a cart. Both places allow rentals in ½hr. increments. Q: For what number of hours is the cost of renting clubs and a cart the same for each course? Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Example 4 Continued Step 1 Write an equation for the cost of renting clubs and a cart at each golf course. Let x represent the number of hours and y represent the total cost in dollars. City Park Golf Course: y = 55x + 20 Sea Vista Golf Course: y = 45x + 35 Sol’n: 1.5hrs Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Check It Out! Example 4 Ravi is comparing the costs of long distance calling cards. To use card A, it costs $0.50 to connect and then $0.05 per minute. To use card B, it costs $0.20 to connect and then $0.08 per minute. For what number of minutes does it cost the same amount to use each card for a single call? Step 1 Write an equation for the cost for each of the different long distance calling cards. Let x represent the number of minutes and y represent the total cost in dollars. Card A: y = 0.05x + 0.50 Holt Algebra 2 Card B: y = 0.08x + 0.20 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Check It Out! Example 4 Card A: y = 0.05x + 0.50 Card B: y = 0.08x + 0.20 Step 2 Here’s a situation where, since both are equal to “y”, you can set the two equations equal to each other. .05x + .50 = .08x + .20 -.05x -.05x -.20 -.20 .30 = .03x ÷.03 ÷.03 10 = x So, both plans are the same for a 10minute call Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 1A: Solving Linear Systems by Substitution Use variable substitution to solve the system: y= x–1 x+y=7 Step 1: Substitute the equivalent expression for “y” from the first equation in place of “y” in the second equation and solve for “x”. x+y=7 x + (x – 1) = 7 2x – 1 = 7 2x = 8 x=4 Then… Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 1A Continued Step 2: Substitute the x-value into one of the original equations to solve for y. y=x–1 y = (4) – 1 y=3 The solution is the ordered pair (4, 3). Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 1A Continued Check A graph or table supports your answer. Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 1B: Solving Linear Systems by Substitution Use substitution to solve the systems of equations. 2y + x = 4 y = 2x – 1 5x + 6y = –9 3x – 4y = 7 3x + 2y = 26 Sol’n: (3, 1/2) Sol’n: (4,7) Holt Algebra 2 2x – 2 = –y Sol’n: (3,-4) Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Check It Out! Example 1a Continued Check A graph or table supports your answer. Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 2A: Solving Linear Systems by Elimination 3x + 2y = 4 3x + 5y = –16 4x – 2y = –18 2x + 3y = –9 Sol’n: (–2, 5) Holt Algebra 2 Sol’n: (3,-5) 4x + 7y = –25 –12x –7y = 19 Sol’n: (.75, -4) Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Check It Out! Example 2b Use elimination to solve the system of equations. 5x – 3y = 42 8x + 5y = 28 Sol’n: (6,-4) Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 3: Solving Systems with Infinitely Many or No Solutions 3x + y = 1 56x + 8y = –32 6x + 3y = –12 2y + 6x = –18 Sol’n: NONE Holt Algebra 2 7x + y = –4 Sol’n: INFINITE 2x + y = –6 Sol’n: NONE Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 4: Zoology Application A veterinarian needs 60 pounds of dog food that is 15% protein. He will combine a beef mix that is 18% protein with a bacon mix that is 9% protein. How many pounds of each does he need to make the 15% protein mixture? Let x present the amount of beef mix in the mixture. Let y present the amount of bacon mix in the mixture. Write one equation based on the amount of dog food Write another equation based on the amount of protein Holt Algebra 2 THEN… Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Example 4 Continued Solve the system. x + y = 60 0.18x +0.09y = 9 Sol’n: (40, 20) Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-1,2 3-2 to Solve Linear Systems Check It Out! Example 4 A coffee blend contains Sumatra beans which cost $5/lb, and Kona beans, which cost $13/lb. If the blend costs $10/lb, how much of each type of coffee is in 50 lb of the blend? Let x represent the amount of the Sumatra beans in the blend. Let y represent the amount of the Kona beans in the blend. Write one equation based on the amount of each bean Write another equation based on cost of the beans: Holt Algebra 2 THEN… Solving Linear Systems Using Algebraic Methods 3-2 to Solve Linear Systems 3-1,2 Check It Out! Example 4 Continued Solve the system. x + y = 50 5x + 13y = 500 Sol’n: (18.75, 31.25) Holt Algebra 2 Solving Linear Systems Using Algebraic Methods 3-2 to Solve Linear Systems 3-1,2 Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Lesson Quiz: Part I Use substitution to determine if the given ordered pair is an element of the solution set of the system of equations. x + 3y = –9 x+y=2 2. (–3, –2) 1. (4, –2) y – 2x = 4 y + 2x = 5 Solve the system using a table and graph. Check your answer. x+y=1 3. 3x –2y = 8 Holt Algebra 2 3-1,2 3-1 Using Graphs and Tables Solving Linear to Solve Linear SystemsSystems Lesson Quiz: Part II Which system has NO solution and which has INFINITE solutions. y + 2x = –10 –4x = 2y – 10 5. 4. y + 2x = –10 y + 2x = –10 6. Kayak Kottage charges $26 to rent a kayak plus $24 per hour for lessons. Power Paddles charges $12 for rental plus $32 per hour for lessons. Both places allow rentals in 15min (1/4hr) For what number of hours is the cost of equipment and lessons the same for each company? Holt Algebra 2