Algebra 2 Solving Systems Using Tables and Graphs Lesson 3-1

Goals Goal To solve a linear system using a graph or a table. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary System of Equations Linear System Solution of a System

Definitions System of equations - is a set of two or more equations containing two or more variables. –Example: Linear system - is a system of equations containing only linear equations. –Example: a) b)

Definitions Solution of a System - Recall that 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 makes all the equations true. 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 equations if the x- and y-values of the point satisfy both equations.

Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. (1, 3); x – 3y = –8 3x + 2y = 9 Substitute 1 for x and 3 for y in each equation. Because the point is a solution for both equations, it is a solution of the system. 3x + 2y = 9 3(1) +2(3) x – 3y = –8 (1) –3(3) –8–8 –8–8–8–8 Example:

Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. (–4, ); x + 6 = 4y 2x + 8y = 1 x + 6 = 4y (–4) Because the point is not a solution for both equations, it is not a solution of the system. 2x + 8y = 1 2(–4) + 1 –4–4 1 Substitute –4 for x and for y in each equation. x Example:

Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. (5, 3); 6x – 7y = 1 3x + 7y = 5 Substitute 5 for x and 3 for y in each equation. Because the point is not a solution for both equations, it is not a solution of the system. 5 3(5) + 7(3) 3x + 7y = x 6x – 7y = 1 6(5) – 7(3) x Your Turn:

Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. (4, 3); x + 2y = 10 3x – y = 9 Because the point is a solution for both equations, it is a solution of the system. Substitute 4 for x and 3 for y in each equation. x + 2y = 10 (4) + 2(3) (4) – (3) 3x – y = Your Turn:

Use substitution to determine if the given ordered pair is an element of the solution set of the system of equations. x + y = 2 y + 2x = 5 x + 3y = –9 y – 2x = 4 no yes 1. (4, –2) 2. (–3, –2) Your Turn:

Graphs & Tables Just as you can use graphs or tables to find some of the solutions to a linear equation. You can do the same to find solutions to linear systems.

x – y = –1 x + 2y = 5 How to Use Graphs to Solve Linear Systems x y Consider the following system: (1, 2) We must ALWAYS verify that your coordinates actually satisfy both equations. To do this, we substitute the coordinate (1, 2) into both equations. x – y = –1 (1) – (2) = –1  Since (1, 2) makes both equations true, then (1, 2) is the solution to the system of linear equations. x + 2y = 5 (1) + 2(2) = = 5 

Graphing to Solve a Linear System While there are many different ways to graph these equations, we will be using the slope - intercept form. To put the equations in slope intercept form, we must solve both equations for y. Start with 3x + 6y = 15 Subtracting 3x from both sides yields 6y = –3x + 15 Dividing everything by 6 gives us… Similarly, we can add 2x to both sides and then divide everything by 3 in the second equation to get Now, we must graph these two equations. Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3

Graphing to Solve a Linear System Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3 Using the slope intercept form of these equations, we can graph them carefully on graph paper. x y Start at the y - intercept, then use the slope. Label the solution! (3, 1) Lastly, we need to verify our solution is correct, by substituting (3, 1). Since and, then our solution is correct!

Graphing to Solve a Linear System Step 1: Put both equations in slope - intercept form. Step 2: Graph both equations on the same coordinate plane. Step 3: Find the point where the graphs intersect. Step 4: Check to make sure your solution makes both equations true. Solve both equations for y, so that each equation looks like y = mx + b. Use the slope and y - intercept for each equation in step 1. Be sure to use a ruler and graph paper! This is the solution! LABEL the solution! Substitute the x and y values into both equations to verify the point is a solution to both equations.

x y LABEL the solution! Graphing to Solve a Linear System Step 1: Put both equations in slope - intercept form. Step 2: Graph both equations on the same coordinate plane. Step 3: Estimate where the graphs intersect. LABEL the solution! Step 4: Check to make sure your solution makes both equations true. Solve the following system of equations by graphing. 2x + 2y = 3 x – 4y = -1

Use a graph and a table to solve the system. Check your answer. 2x – 3y = 3 y + 2 = x Solve each equation for y. y= x – 2 y= x – 1 Example:

Example: continued On the graph, the lines appear to intersect at the ordered pair (3, 1) y= x – 2 y= x – 1

Example: continued Make a table of values for each equation. Notice that when x = 3, the y- value for both equations is 1. xy 0 –2–2 1– –1–10 yx y= x – 2 y= x – 1 The solution to the system is (3, 1).

Use a graph and a table to solve the system. Check your answer. x – y = 2 2y – 3x = –1 Solve each equation for y. y = x – 2 y = Example: Graphing Calculator

Use your graphing calculator to graph the equations and make a table of values. The lines appear to intersect at (–3, –5). This is the confirmed by the tables of values. Check Substitute (–3, –5) in the original equations to verify the solution. The solution to the system is (–3, –5). 2(–5) – 3(–3) 2y – 3x = –1 –1 (–3) – (–5) 2 x – y = Example: Graphing Calculator

Use a graph and a table to solve the system. Check your answer. 2y + 6 = x 4x = 3 + y Solve each equation for y. y= 4x – 3 y= x – 3 Your Turn:

On the graph, the lines appear to intersect at the ordered pair (0, –3) Your Turn: continued y= 4x – 3 y= x – 3

Make a table of values for each equation. Notice that when x = 0, the y-value for both equations is –3. 3 –22 1 –3–3 0 yx Your Turn: continued xy 0 –3– The solution to the system is (0, –3). y = 4x – 3y = x – 3

Use a graph and a table to solve the system. Check your answer. x + y = 8 2x – y = 4 y = 2x – 4 y = 8 – x Solve each equation for y. Your Turn:

On the graph, the lines appear to intersect at the ordered pair (4, 4). Your Turn: continued y = 2x – 4 y = 8 – x

Make a table of values for each equation. Notice that when x = 4, the y-value for both equations is 4. xy 1 –2– xy y = 2x – 4y= 8 – x The solution to the system is (4, 4). Your Turn: continued

Use a graph and a table to solve each system. Check your answer. y – x = 5 3x + y = 1 y= –3x + 1 y= x + 5 Solve each equation for y. Your Turn:

Your Turn: continued On the graph, the lines appear to intersect at the ordered pair (–1, 4). y= –3x + 1 y= x + 5

Make a table of values for each equation. Notice that when x = –1, the y-value for both equations is 4. xy –1– –2–2 2–5–5 xy –1– The solution to the system is (–1, 4). y= –3x + 1y= x + 5 Your Turn: continued

City Park Golf Course charges $20 to rent golf clubs plus $55 per hour for golf cart rental. Sea Vista Golf Course charges $35 to rent clubs plus $45 per hour to rent a cart. For what number of hours is the cost of renting clubs and a cart the same for each course? Example: Application

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 Because the slopes are different, the system is independent and has exactly one solution. Step 1 Write an equation for the cost of renting clubs and a cart at each golf course. City Park Golf Course charges $20 to rent golf clubs plus $55 per hour for golf cart rental. Sea Vista Golf Course charges $35 to rent clubs plus $45 per hour to rent a cart. For what number of hours is the cost of renting clubs and a cart the same for each course? Example: continued

Step 2 Solve the system by using a table of values. xy xy y = 55x + 20y = 45x + 35 Use increments of to represent 30 min. When x =, the y- values are both The cost of renting clubs and renting a cart for hours is $ at either company. So the cost is the same at each golf course for hours. Example: continued

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 Card B: y = 0.08x Your Turn:

xy xy y = 0.05x y = 0.08x Step 2 Solve the system by using a table of values. When x = 10, the y-values are both The cost of using the phone cards of 10 minutes is $1.00 for either cards. So the cost is the same for each phone card at 10 minutes. Your Turn: continued

Definitions The systems of equations in the past Examples have had exactly one solution. However, linear systems may also have infinitely many or no solutions. Consistent system - is a system of equations that has at least one solution. Inconsistent system - is a system of equations that has no solutions. You can classify linear systems by comparing the slopes and y-intercepts of the equations. Independent system - has equations with different slopes. Dependent system - has equations with equal slopes and equal y-intercepts.

Graphical Solutions of Linear Systems

Classify the system and determine the number of solutions. x = 2y + 6 3x – 6y = 18 Solve each equation for y. The system is consistent and dependent with infinitely many solutions. The equations have the same slope and y-intercept and are graphed as the same line. y = x – 3 Example:

Classify the system and determine the number of solutions. 4x + y = 1 y + 1 = –4x Solve each equation for y. The system is inconsistent and has no solution. y = –4x + 1 y = –4x – 1 The equations have the same slope but different y-intercepts and are graphed as parallel lines. Example:

Check A graph shows parallel lines. Example: continued

Classify the system and determine the number of solutions. 7x – y = –11 3y = 21x + 33 Solve each equation for y. The system is consistent and dependent with infinitely many solutions. The equations have the same slope and y-intercept and are graphed as the same line. y = 7x + 11 Your Turn:

Classify each system and determine the number of solutions. x + 4 = y 5y = 5x + 35 Solve each equation for y. The system is inconsistent with no solution. The equations have the same slope but different y-intercepts and are graphed as parallel lines. y = x + 4 y = x + 7 Your Turn:

Classify each system and determine the number of solutions –4x = 2y – 10 y + 2x = –10 y - 2x = –1 inconsistent; none consistent, independent; one solution Your Turn:

Review

What type of system of equations is shown? x + y = 5 2x = y – 5 A.consistent and independent – one solution B.consistent and dependent – infinite solutions C.Inconsistent – no solution D.none of the above Your Turn:

What type of system of equations is shown? x + y = 3 2x = –2y + 6 A.consistent and independent – one solution B.consistent and dependent – infinite solutions C.Inconsistent – no solution D.none of the above

Your Turn: What type of system of equations is shown? y = 3x + 2 –6x + 2y = 10 A.consistent and independent – one solution B.consistent and dependent – infinite solutions C.Inconsistent – no solution D.none of the above

Your Turn: A.f(x) and g(x) are consistent and dependent – infinite solutions. B.f(x) and g(x) are inconsistent – no solution. C.f(x) and h(x) are consistent and independent – one solution. D.g(x) and h(x) are consistent – at least one solution. Graph the system of equations below. Which statement is not true? f(x) = x + 2 g(x) = x + 4

Essential Question Big Idea: Solving Equations and Inequalities What does a solution of a system of linear equations represent? A solution represents values that make both equations true. To find the solution, graph the equations and find the point where the lines intersect. You can also make a table and find the value of x that makes the y–values equal.

Assignment Section 3-1, Pg ; #1 – 6 all, even, 18 – 50 even.