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5-1 Solving Systems by Graphing

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Geogebra Solving a System by Graphing Solving a System by Graphing (2)

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5-1 Videos Determine whether a point is a solution to a linear equation In this lesson you will learn how to determine whether a point is a solution to a linear equation. Standards: 8.EE.C.8 Determine whether 2 lines intersect In this lesson you will learn how to determine whether 2 lines intersect. Standards: 8.EE.C.8a Find the solution to a pair of linear equations by looking at their g... In this lesson you will learn how to find the solution to a pair of linear equations by looking at their graphs. Standards: 8.EE.C.8

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Solve systems of equations: graphing (1) This is 1 of a series of lessons in which you will learn how to solve systems of equations by graphing. This lesson teaches you how to use the method of graphing the slope- intercepts. Standards: 8.EE.C.8b Estimate solutions of equations: graphing In this lesson you will learn how to estimate solutions of equations by using graphing. Standards: 8.EE.C.8b Solve systems of equations: graphing (2) This is 1 of a series of lessons in which you will learn how to solve systems of equations by graphing. This lesson deals with systems of equations in which one equation is in the slope-int... Standards: 8.EE.C.8b

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Video Tutor Help Solving Systems by Graphing Khan Academy

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Find the solution to a pair of linear equations by using tables In this lesson you will learn how to find the solution to a pair of linear equations by using tables. Standards: 8.EE.C.8a Decide whether a point is a solution to a pair of linear equations In this lesson you will learn to decide whether a point is a solution to a pair of linear equations by substituting values. Standards: 8.EE.C.8a Find the solution to a pair of linear equations by looking at their g... In this lesson you will learn how to find the solution to a pair of linear equations by looking at their graphs. Standards: 8.EE.C.8 Find the number of solutions to a pair of linear equations using tabl... In this lesson you will learn how to find the number of solutions to a pair of linear equations by using tables and graphs. Standards: 8.EE.C.8 Solve pairs of linear equations using patterns In this lesson you will learn to solve pairs of linear equations by using patterns. Standards: 8.EE.C.8a

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Determine whether a point is a solution to a linear equation In this lesson you will learn how to determine whether a point is a solution to a linear equation. Standards: 8.EE.C.8 Determine whether 2 lines intersect In this lesson you will learn how to determine whether 2 lines intersect. Standards: 8.EE.C.8a Solve systems of equations: graphing (1) This is 1 of a series of lessons in which you will learn how to solve systems of equations by graphing. This lesson teaches you how to use the method of graphing the slope- intercepts. Standards: 8.EE.C.8b Estimate solutions of equations: graphing In this lesson you will learn how to estimate solutions of equations by using graphing. Standards: 8.EE.C.8b Solve systems of equations: graphing (2) This is 1 of a series of lessons in which you will learn how to solve systems of equations by graphing. This lesson deals with systems of equations in which one equation is in the slope-int... Standards: 8.EE.C.8b

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Worksheets 5-1 Note-Taking Guide 5-1 Practice 5-1 Guided Problem Solving

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Vocabulary Practice Chapter 5 Vocabulary (Electronic) Flash Cards Currently not available

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Additional Lesson Examples 5-1 Step-by-Step Examples Currently not available

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Lesson Readiness 5-1 Problem of the Day 5-1 Lesson Quiz Currently not available

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Solving a System by Graphing Solving a System by Graphing (2)

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What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair. When graphing, you will encounter three possibilities.

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Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) (1,2)

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Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y- intercepts.

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Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts.

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What is the solution of the system graphed below? 1.(2, -2) 2.(-2, 2) 3.No solution 4.Infinitely many solutions

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1) Find the solution to the following system: 2x + y = 4 x - y = 2 Graph both equations. I will graph using x- and y-intercepts (plug in zeros). Graph the ordered pairs. 2x + y = 4 (0, 4) and (2, 0) x – y = 2 (0, -2) and (2, 0)

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Graph the equations. 2x + y = 4 (0, 4) and (2, 0) x - y = 2 (0, -2) and (2, 0) Where do the lines intersect? (2, 0) 2x + y = 4 x – y = 2

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Check your answer! To check your answer, plug the point back into both equations. 2x + y = 4 2(2) + (0) = 4 x - y = 2 (2) – (0) = 2 Nice job…let’s try another!

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2) Find the solution to the following system: y = 2x – 3 -2x + y = 1 Graph both equations. Put both equations in slope-intercept or standard form. I’ll do slope-intercept form on this one! y = 2x – 3 y = 2x + 1 Graph using slope and y-intercept

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Graph the equations. y = 2x – 3 m = 2 and b = -3 y = 2x + 1 m = 2 and b = 1 Where do the lines intersect? No solution! Notice that the slopes are the same with different y- intercepts. If you recognize this early, you don’t have to graph them!

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Check your answer! Not a lot to check…Just make sure you set up your equations correctly. I double-checked it and I did it right…

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What is the solution of this system? 3x – y = 8 2y = 6x -16 1.(3, 1) 2.(4, 4) 3.No solution 4.Infinitely many solutions

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Solving a system of equations by graphing. Let's summarize! There are 3 steps to solving a system using a graph. Step 1: Graph both equations. Step 2: Do the graphs intersect? Step 3: Check your solution. Graph using slope and y – intercept or x- and y-intercepts. 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.

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A system of linear equations is a set of two or more linear equations containing two or more variables. A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. So, if an ordered pair is a solution, it will make both equations true.

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Tell whether the ordered pair is a solution of the given system. Additional Example 1B: Identifying Systems of Solutions (–2, 2); –x + y = 2 x + 3y = 4 Substitute –2 for x and 2 for y. x + 3y = 4 –2 + 6 4 4 4 –2 + (3)2 –x + y = 2 4 2 2 –(–2) + 2 The ordered pair (–2, 2) makes one equation true, but not the other. (–2, 2) is not a solution of the system.

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Partner Share! Example 1b Tell whether the ordered pair is a solution of the given system. (2, –1); x – 2y = 4 3x + y = 6 The ordered pair (2, –1) makes one equation true, but not the other. (2, –1) is not a solution of the system. 3x + y = 6 3(2) + (–1) 6 6 – 1 6 5 6 x – 2y = 4 2 – 2(–1) 4 2 + 2 4 4 Substitute 2 for x and –1 for y.

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All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection. y = 2x – 1 y = –x + 5 The point (2, 3) is where the two lines intersect and is a solution of both equations, so (2, 3) is the solution of the systems.

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Solve the system by graphing. Check your answer. Additional Example 2A: Solving a System Equations by Graphing y = x y = –2x – 3 Graph the system. The solution appears to be at (–1, –1). The solution is (–1, –1). Check Substitute (–1, –1) into the system. y = x y = –2x – 3 y = x (–1) –1 y = –2x – 3 (–1) –2(–1) –3 –1 2 – 3 –1 – 1

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Solve the system by graphing. Check your answer. Additional Example 2B: Solving a System Equations by Graphing y = x – 6 y + x = –1 Rewrite the second equation in slope-intercept form. y + x = –1 − x y = y = x – 6 Graph the system. y + 1313 x =– 1

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Additional Example 2B Continued y = x – 6 y + x = –1 + – 1 –1 –1 – 1 y = x – 6 – 6 The solution is Check Substitute into the system Solve the system by graphing. Check your answer.

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Partner Share! Example 2a y = –2x – 1 y = x + 5 Graph the system. The solution appears to be (–2, 3). Check Substitute (–2, 3) into the system. y = x + 5 3 –2 + 5 3 y = –2x – 1 3 –2(–2) – 1 3 4 – 1 3 The solution is (–2, 3). y = x + 5 y = –2x – 1

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Solve the system by graphing. Check your answer. Partner Share! Example 2b 2x + y = 4 Graph the system. Rewrite the second equation in slope-intercept form. 2x + y = 4 –2x – 2x y = –2x + 4 The solution appears to be (3, –2). y = –2x + 4

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Solve the system by graphing. Check your answer. Partner Share! Example 2b Continued 2x + y = 4 Check Substitute (3, –2) into the system. 2x + y = 4 2(3) + (–2) 4 6 – 2 4 4 –2 (3) – 3 –2 1 – 3 –2 The solution is (3, –2).

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Example 7-1a The graphs of the equations appear to intersect at (1, 0). Solve the system and by graphing. One Solution

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Example 7-1b Answer: The solution of the system is (1, 0). Check this estimate. Check

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Example 7-2a Write the equation. Add to each side. Both equations are the same. Write in slope-intercept form. Solve the system and by graphing. Infinitely Many Solutions

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Example 7-2b Answer: The solution of the system is all the coordinates of points on the graph of

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Additional Example 3: Problem-Solving Application Wren and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be?

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1 Understand the Problem The answer will be the number of nights it takes for the number of pages read to be the same for both girls. List the important information: Wren on page 14 Reads 2 pages a night Jenni on page 6 Reads 3 pages a night Additional Example 3 Continued

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2 Make a Plan Write a system of equations, one equation to represent the number of pages read by each girl. Let x be the number of nights and y be the total pages read. Total pages is number read every night plus already read. Wreny=2 x x + 14 Jenni y = 3 x x + 6 Additional Example 3 Continued

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Solve 3 Additional Example 3 Continued (8, 30) Nights Graph y = 2x + 14 and y = 3x + 6. The lines appear to intersect at (8, 30). So, the number of pages read will be the same at 8 nights with a total of 30 pages.

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Look Back4 Check (8, 30) using both equations. After 8 nights, Wren will have read 30 pages: After 8 nights, Jenni will have read 30 pages: 3(8) + 6 = 24 + 6 = 30 2(8) + 14 = 16 + 14 = 30 Additional Example 3 Continued

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Partner Share! Example 3 Video club A charges $10 for membership and $3 per movie rental. Video club B charges $15 for membership and $2 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost?

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Partner Share! Example 3 Continued 1 Understand the Problem The answer will be the number of movies rented for which the cost will be the same at both clubs. List the important information: Rental price: Club A $3 Club B $2 Membership: Club A $10 Club B $15

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2 Make a Plan Write a system of equations, one equation to represent the cost of Club A and one for Club B. Let x be the number of movies rented and y the total cost. Partner Share! Example 3 Continued Total cost isprice rentalsplus membership fee. times Club A y =3 +10 Club B y=2 +15 x x

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Solve 3 Graph y = 3x + 10 and y = 2x + 15. The lines appear to intersect at (5, 25). So, the cost will be the same for 5 rentals and the total cost will be $25. Partner Share! Example 3 Continued

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Look Back4 Check (5, 25) using both equations. Number of movie rentals for Club A to reach $25: Number of movie rentals for Club B to reach $25: 2(5) + 15 = 10 + 15 = 25 3(5) + 10 = 15 + 10 = 25 Partner Share! Example 3 Continued

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Lesson Review: Part I Tell whether the ordered pair is a solution of the given system. 1. (–3, 1); 2. (2, –4); yes no

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Lesson Review: Part II Solve the system by graphing. 3. 4. Joy has 5 collectable stamps and will buy 2 more each month. Ronald has 25 collectable stamps and will sell 3 each month. After how many months will they have the same number of stamps? How many will that be? (2, 5) 4 months y + 2x = 9 y = 4x – 3 13 stamps

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