Presentation on theme: "Solving Systems by Graphing"— Presentation transcript:
1 Solving Systems by Graphing 6-1Solving Systems by GraphingWarm UpLesson PresentationLesson QuizHolt Algebra 1
2 Warm Up – this week on Graph Paper Evaluate each expression for x = 1 and y =–3.1. x – 4y –2x + yWrite each expression in slope-intercept form.3. y – x = 14. 2x + 3y = 65. 0 = 5y + 5x13–5y = x + 1y = x + 2y = –x
3 ObjectivesIdentify solutions of linear equations in two variables.Solve systems of linear equations in two variables by graphing.
4 Vocabulary systems of linear equations solution of a system of linear equations
5 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.
6 Example 1A: Identifying Systems of Solutions Tell whether the ordered pair is a solution of the given system.(5, 2);3x – y = 132 – 2 00 03(5) –15 –3x – y 13Substitute 5 for x and 2 for y in each equation in the system.The ordered pair (5, 2) makes both equations true.(5, 2) is the solution of the system.
7 If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations.Helpful Hint
8 Example 1B: Identifying Systems of Solutions Tell whether the ordered pair is a solution of the given system.x + 3y = 4(–2, 2);–x + y = 2–2 + 3(2) 4x + 3y = 4–4 4–x + y = 2–(–2)4 2Substitute –2 for x and 2 for y in each equation in the system.The ordered pair (–2, 2) makes one equation true but not the other.(–2, 2) is not a solution of the system.
9 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 – 1y = –x + 5The 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.
10 Sometimes it is difficult to tell exactly where the lines cross when you solve by graphing. It is good to confirm your answer by substituting it into both equations.Helpful Hint
11 Example 2A: Solving a System Equations by Graphing Solve the system by graphing. Check your answer.y = xGraph the system.y = –2x – 3The solution appears to be at (–1, –1).CheckSubstitute (–1, –1) into the system.y = xy = –2x – 3(–1) –2(–1) –3– – 3–1 – 1y = x(–1) (–1)–1 –1•(–1, –1)y = –2x – 3(–1, –1) is the solution of the system.
12 Example 2B: Solving a System Equations by Graphing Solve the system by graphing. Check your answer.y = x – 6Graph using a calculator and then use the intercept command.y + x = –1Rewrite the second equation in slope-intercept form.y = x – 6y + x = –1− x − xy =
13 Example 2B ContinuedSolve the system by graphing. Check your answer.Check Substitute into the system.– 1–1–1 – 1y = x – 6– 6y = x – 6The solution is
14 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?
15 Understand the Problem Example 3 Continued1Understand the ProblemThe 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 nightJenni on page Reads 3 pages a night
16 Example 3 Continued2Make a PlanWrite 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.Totalpagesisnumber readeverynightplusalready read.Wreny=2 x+14Jenniy=3 x+6
17 Example 3 Continued 3 Solve 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.(8, 30)Nights
18 Example 3 Continued Look Back 4 Check (8, 30) using both equations.Number of days for Wren to read 30 pages.2(8) + 14 = = 30Number of days for Jenni to read 30 pages.3(8) + 6 = = 30
19 Assignment: L6-1 pg 386 #2-30 evens, 34-42 evens, Skip #26 (#20 & 22 use a graphing calculator)ON GRAPH PAPER – USE A STRAIGHT EDGE TO MAKE LINESNote: use of “graphing” calculators will not be allowed on the Chapter 6 test.
20 NOTES: Solving By Graphing System of Equations: two or more equationsEX: y= 2x – The solution of system isy= -x + 5 the point that makes both equations true.Solution of a System of Equations: the point of intersection
21 Lesson Quiz: Part ITell whether the ordered pair is a solution of the given system.1. (–3, 1);2. (2, –4);noyes
22 Lesson Quiz: Part IISolve 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?y + 2x = 9(2, 5)y = 4x – 34 months13 stamps