Solving Systems of Equations by Graphing Algebra 1 ~ Chapter 7.1

** A system of linear equations is a set of two or more linear equations containing two or more variables and connected with a bracket. ** 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.

All solutions of a linear equation are on its graph 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.

Checking to make sure you graphed the lines correctly, therefore checking for SURE your answer. In the previous slide we graphed the 2 lines and found (2, 3) to be the solution. y = 2x – 1 y = –x + 5 Check your answer by plugging in (2, 3) to each line. y = 2x – 1 3 = 2(2) – 1 3 = 4 – 1 3 = 3 y = -x + 5 3 = -(2) + 5 3 = -2 + 5 3 = 3  

Ex. 1 - Solve the system by graphing, then check your solution. The solution appears to be at (–1, –1). y = x y = –2x – 3 Check Substitute (–1, –1) into the system. y = x y = x (–1) (–1) –1 –1  y = –2x – 3 (–1) –2(–1) –3 –1 2 – 3 –1 – 1  • y = –2x – 3 The solution is (–1, –1).

Ex. 2 - Solve the system by graphing. Check your solution. The solution appears to be (–2, 3). y = –2x – 1 y = x + 5 Check Substitute (–2, 3) into the system. y = x + 5 y = –2x – 1 y = –2x – 1 3 –2(–2) – 1 3 4 – 1 3 3  y = x + 5 3 –2 + 5 3 3  The solution is (–2, 3).

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

Example 3 Continued …. CHECK Check Substitute (3, –2) into the system. Into the ORIGINAL equations. 2x + y = 4 –2 (3) – 3 –2 1 – 3 –2 –2  2x + y = 4 2(3) + (–2) 4 6 – 2 4 4 4  The solution is (3, –2).

Substitute –2 for x and 2 for y. Ex. 4 - Tell whether the ordered pair is a solution of the given system. x + 3y = 4 4 (–2, 2); –x + y = 2 x + 3y = 4 –x + y = 2 –2 + 6 4 4 4  4 –2 + (3)2 Substitute –2 for x and 2 for y. 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 this system.

Ex. 5 - Tell whether the ordered pair is a solution of the given system. (5, 2); 3x – y = 13 3x – y = 13 Substitute 5 for x and 2 for y. 2 – 2 0 0 0  3(5) – 2 13 15 – 2 13 13 13  The ordered pair (5, 2) makes both equations true, (5, 2) is the solution of this system.

Number of Solutions

a.) y = -x + 5 y = x – 3 b.) y = -x + 5 2x + 2y = -8 c.) 2x + 2y = -8 Ex. 6 – Number of solutions. Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions. a.) y = -x + 5 y = x – 3 b.) y = -x + 5 2x + 2y = -8 c.) 2x + 2y = -8 y = -x - 4 One solution Consistent/independent No solutions Inconsistent Infinitely many solutions Consistent/Dependent

Ex. 7: 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? Wren p = 2  n + 14 Jenni p = 3  n + 6

Example 7 Continued Graph p = 2n + 14 and p = 3n + 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. (8, 30) Nights

Example 7 Continued   Check (8, 30) using both equations. After 8 nights, Wren will have read 30 pages: 2(8) + 14 = 16 + 14 = 30  After 8 nights, Jenni will have read 30 pages: 3(8) + 6 = 24 + 6 = 30 

Example 8 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? Club A c = 3  + 10 Club B 2 15 r

Example 8 Continued Graph c = 3r + 10 and c = 2r + 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.

Example 8 Continued Check (5, 25) using both equations. Number of movie rentals for Club A to reach $25: 3(5) + 10 = 15 + 10 = 25  Number of movie rentals for Club B to reach $25: 2(5) + 15 = 10 + 15 = 25 

no yes Lesson Wrap Up 1. (–3, 1); 2. (2, –4); Tell whether the ordered pair is a solution of the given system. Remember you do NOT have to graph the lines to answer these questions. 1. (–3, 1); 2. (2, –4); no yes

(2, 5) Solve and CHECK the system by graphing. 3. y + 2x = 9