Unit 1 Lesson 3 Systems of Linear Equations

Unit 1 – Lesson 3: Systems of Linear Equations Systems of Equations Two or more linear equations that have to be solved at the same time.

Systems of Equations • Let’s take a look at a few examples. • Your task: I Notice, I Wonder. – Discuss and write down some things that you NOTICE about the examples. – Discuss and write down some things that you WONDER about the examples.

Systems of Equations: Example 1

Systems of Equations: Example 2

Systems of Equations: Example 3

How do we “solve” a system of equations? ? ? • By finding the point where two or more equations, intersect. This is called the SOLUTION. x+y=6 y = 2 x 6 4 Point of intersection 2 11/25/2020 1 2 Geometry Honors 6

“All I do is Solve” video • http: //www. youtube. com/watch? v= 1 q. HTmxla. ZWQ&feature=related

3 Methods to Solve System of Equations • Graphing • Substitution Method • Elimination Method

3 Possible Solution Types • 1 Solution • No Solution • Infinitely Many Solutions • http: //www. algebra-class. com/graphing-systems-ofequations. html

1 Solution Point of int ersec tion

Another type of solution • How would you describe these lines? Y = 3 x + 2 Y = 3 x - 4 11/25/2020 What do you think the solution Honors (point of. Geometry intersection) is? 11

No Solution Parallel Lines Will not intersect

PARALLEL LINES No Solution: • when lines of a graph are parallel • since they do not intersect, there is no solution • Parallel lines have the same slope but different y-intercepts 11/25/2020 Geometry Honors 13

Another type of solution • What do you notice about the graphs and equations? y = -3 x + 4 3 x + y = 4 What do you think the solution (point of intersection) is? 11/25/2020 Geometry Honors 14

Infinitely Many Solutions SAME LINE

INFINITELY MANY SOLUTIONS Infinite Solutions: • a pair of equations that have the same slope and same y-intercept. • They are the SAME equation (just written in different forms) • Since they are the SAME EQUATION, they have the SAME LINE 11/25/2020 Geometry Honors 16

Does it have a solution? Determine whether the following have one, none, or infinite solutions by identifying the slope and y-intercept. Explain your reasoning. 1) 2 y = 8 - x y = 2 x + 4 11/25/2020 2) y = -6 x + 8 y + 6 x = 8 Geometry Honors 3) x - 5 y = 10 -5 y = -x +6 17

Does it have a solution? Determine whether the following have one, none, or infinite solutions by just looking at the slope and y-intercepts 1) 2 y + x = 8 2) 3) y = -6 x + 8 x - 5 y = 10 y = 2 x + 4 y + 6 x = 8 -5 y = -x +6 ANS: One Solution ANS: Infinite Solutions 11/25/2020 Geometry Honors ANS: No Solution 18

Systems of Equations Video • Systems of Equations: Part 01 – Watch carefully as this video explains what a system of equations are and gives a fantastic real-world example of how systems are used in the business world.

The Goal of Solving Systems • To find one pair (x, y) of values that satisfies both linear equations. –The one pair of values that makes both equations true.

Hamilton High School

Hamilton High School 16 x + 10 y = 240 x + y = 18 • What does the x represent? x: # of outdoor workers • What does the y represent? • y: # of indoor workers

Hamilton High School 16 x + 10 y = 240 • What does this equation represent in the problem? • 16 x + 10 y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor workers

Hamilton High School x + y = 18 • What does this equation represent in the problem? • x + y = 18 shows that the # of club members who will work

Hamilton High School 16 x + 10 y = 240 • Determine three combination of outdoor (x) and indoor (y) workers so that the club earns exactly $240. • SHOW ALL WORK!!

Hamilton High School x + y = 18 • Do any of the combinations from part d work for the 18 workers that are needed? • SHOW ALL WORK!!

Let’s verify • How can we verify that (10, 8) is the solution to the system of equation: 16 x + 10 y = 240 x + y = 18 • You must verify the solution into BOTH equations for x AND y. 16(10) + 10(8) = 240 160 + 80 = 240 Geometry CP 10 + 8 = 18 18 = 18

Where’s the solution? N O I SO T U L • Use the graph to estimate a solution for the system of equations (basically what x and y value works for BOTH equations) • SOLUTION: POINT OF INTERSECTION (10, 8)

Hamilton High School • Plugging and chugging numbers is exhausting and very time consuming. • What other strategies could you use to find a pair of values (x, y) that satisfy BOTH equations at the same time?

How Do We Graph a Linear Equation? ? ? In order to graph a linear equation it HAS to be in the form y = mx + b, where m is the slope and b is the y-intercept

How Do We Graph a Linear Equation? ? ? Let’s Practice: 16 x + 10 y = 240

How Do We Graph a Linear Equation? ? ? Let’s Practice: x + y = 18

Let’s Look at the Solution Complete Problem 1 Parts a, b, & c.

A Better Deal When the date for the work project was set, it turned out that only 13 science club members could participate. The club president talked again with the PTA president and got a new pay deal - $20 per outdoor worker and $15 per indoor worker.

MATCHING ACTIVITY

Verifying Solutions • Determine whether the point (3, 8) is a solution to each system of equations. 2 x + y = 14 x + y = 11 11/25/2020 y = -x – 5 y=x+5 Geometry Honors 4 x – y = - 4 3 x – 2 y = 7 36

Verifying Solutions • Determine whether the point (3, 8) is a solution to each system of equations. 2 x + y = 14 x + y = 11 2(3) + 8 = 14 (YES) 3 + 8 = 11 (YES) y = -x – 5 y=x+5 8 = -3 – 5 (NO) 8 = 3 + 5 (YES) YES! Both equations are NO! Only 1 equation true is true 11/25/2020 Geometry Honors 4 x – y = - 4 3 x – 2 y = 7 4(3) - 8 = - 4 (NO) 3(3) -2(8) = 7 (NO) NO! Neither equations are true 37

REMIND YOUR SHOULDER BUDDY How do I know if my answer is correct? What do you do if the equation is not in y= form? 11/25/2020 Geometry Honors 38

REMIND YOUR SHOULDER BUDDY How do I know if my answer is correct? Always replace x and y for BOTH equations to verify your solution. What do you do if the equation is not in y= form? You have to rewrite it solving for y so that it can be graphed. 11/25/2020 Geometry Honors 39

GRAPHING CALCULATOR • Rewrite equation in y = form • Use the INTERSECT function to find the intersection point 11/25/2020 Geometry Honors 40

GRAPHING CALCULATOR EXAMPLES • y = - 3 x and 4 x + y = 2 • x + y = 1 and 2 x + y = 4 • 3 x + y = 1 and –x + 2 y = 16 • 2 x + y = 1 and 5 x + 4 y = 10 11/25/2020 Geometry Honors 41

GRAPHING CALCULATOR EXAMPLES (2, - 6) • y = - 3 x and 4 x + y = 2 (3, -2) • x + y = 1 and 2 x + y = 4 (-2, 7) • 3 x + y = 1 and –x + 2 y = 16 (-2, 5) • 2 x + y = 1 and 5 x + 4 y = 10 11/25/2020 Geometry Honors 42

GRAPHING CALCULATOR EXAMPLES • y = 3 x + 2 and y = 3 x – 4 • y = -3 x + 4 and 3 x + y = 4 11/25/2020 Geometry Honors 43

GRAPHING CALCULATOR EXAMPLES Infinitely • y = 3 x + 2 and y = 3 x – 4 many No • y = -3 x + 4 and 3 x + y = 4 solution 11/25/2020 Geometry Honors 44

Daily Homework Quiz 1. For use after Lesson 7. 1 Use the graph to solve the linear system 3 x – y = 5 –x + 3 y = 5 ANSWER (2, 1)

Daily Homework Quiz For use after Lesson 7. 1 2. Solve the linear system by graphing. 2 x + y = – 3 – 6 x + 3 y = 3 ANSWER (– 1, – 1)

Summarize 3– 2– 1 3 methods to solve systems of equations 2 important items to identify when graphing a linear equation 1 way to identify the solution of a graphed systems of equations