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 = 2x 6 4 Point of intersection 2 1 2 4/21/2017 Geometry Honors

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

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

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

1 Solution Point of intersection

Another type of solution How would you describe these lines? Y = 3x + 2 Y = 3x - 4 What do you think the solution (point of intersection) is? 4/21/2017 Geometry Honors

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 Slides 10 through 14 show how I explained the different solutions to their worksheet that the students worked on in class. I didn’t focus to much on the solution they found on the worksheet but rather on the type of solution or the concepts they the solutions involved. In each slide I explain each type of solution as well as how a system has these types of solutions. Not focusing to much on the powerpoint I referred back to their worksheet to one of the examples and asked students to remember how we knew the equations graphed were parallel by only looking at the equations. From their prior knowledge students knew that it was because the equations in the system contained the same slope; making connections with new and old material. 4/21/2017 Geometry Honors

Another type of solution What do you notice about the graphs and equations? y = -3x + 4 3x + y = 4 What do you think the solution (point of intersection) is? 4/21/2017 Geometry Honors

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 Again giving explanations as to how we have such a solution. Just like the last slide, I also questioned students as to how we can find the whether a system has infinite solutions by looking at the equation, triggering prior knowledge. 4/21/2017 Geometry Honors

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) 2y = 8 - x y = 2x + 4 y = -6x + 8 y + 6x = 8 3) x - 5y = 10 -5y = -x +6 4/21/2017 Geometry Honors

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) 3) 2y + x = 8 y = 2x + 4 y = -6x + 8 y + 6x = 8 x - 5y = 10 -5y = -x +6 ANS: One Solution ANS: Infinite Solutions ANS: No Solution 4/21/2017 Geometry Honors

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 16x + 10y = 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 16x + 10y = 240 What does this equation represent in the problem? 16x + 10y = 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 16x + 10y = 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: 16x + 10y = 240 x + y = 18 You must verify the solution into BOTH equations for x AND y. 16(10) + 10(8) = 240 160 + 80 = 240 240 = 240 10 + 8 = 18 18 = 18 Geometry CP

Where’s the solution? 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) SOLUTION

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: 16x + 10y = 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. y = -x – 5 y = x + 5 4x – y = - 4 3x – 2y = 7 2x + y = 14 x + y = 11 4/21/2017 Geometry Honors

Verifying Solutions Determine whether the point (3,8) is a solution to each system of equations. y = -x – 5 y = x + 5 4x – y = - 4 3x – 2y = 7 2x + y = 14 x + y = 11 2(3) + 8 = 14 (YES) 3 + 8 = 11 (YES) YES! Both equations are true 8 = -3 – 5 (NO) 8 = 3 + 5 (YES) NO! Only 1 equation is true 4(3) - 8 = - 4 (NO) 3(3) -2(8) = 7 (NO) NO! Neither equations are true 4/21/2017 Geometry Honors

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? 4/21/2017 Geometry Honors

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. 4/21/2017 Geometry Honors

GRAPHING CALCULATOR Rewrite equation in y = form Use the INTERSECT function to find the intersection point 4/21/2017 Geometry Honors

GRAPHING CALCULATOR EXAMPLES y = - 3x and 4x + y = 2 x + y = 1 and 2x + y = 4 3x + y = 1 and –x + 2y = 16 2x + y = 1 and 5x + 4y = 10 4/21/2017 Geometry Honors

GRAPHING CALCULATOR EXAMPLES (2, - 6) y = - 3x and 4x + y = 2 x + y = 1 and 2x + y = 4 3x + y = 1 and –x + 2y = 16 2x + y = 1 and 5x + 4y = 10 (3, -2) (-2, 7) (-2, 5) 4/21/2017 Geometry Honors

GRAPHING CALCULATOR EXAMPLES y = 3x + 2 and y = 3x – 4 y = -3x + 4 and 3x + y = 4 4/21/2017 Geometry Honors

GRAPHING CALCULATOR EXAMPLES Infinitely many y = 3x + 2 and y = 3x – 4 y = -3x + 4 and 3x + y = 4 No solution 4/21/2017 Geometry Honors

Daily Homework Quiz For use after Lesson 7.1 Use the graph to solve the linear system 1. 3x – y = 5 –x + 3y = 5 ANSWER (2, 1)

Daily Homework Quiz For use after Lesson 7.1 2. Solve the linear system by graphing. 2x + y = –3 –6x + 3y = 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