Slide 1 / 179 Slide 2 / 179 Algebra I Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Slide 3 / 179 Slide 4 / 179 Table of Contents Click on the topic to go to that section 8th Grade Review 8th Grade Review of Systems of Equations · Solving Systems by Graphing · Teacher Note Solving Systems by Substitution · Solving Systems by Elimination · Choosing your Strategy · Writing Systems to Model Situations · Return to Table of Solving Systems of Inequalities · Contents Slide 5 / 179 Slide 6 / 179 To solve by GRAPHING, you must graph both lines and find the point where they intersect. When you have 2 or more linear equations that is called a system of equations, there will be two or more variables. The solution to the system of equations will be the ordered (3, 4) pair: (3, 4) To find the solution, you will need a set of two numbers (ordered pair) that makes all the equations true. You have previously learned how to solve a system using graphing, let's review.
Slide 7 / 179 Slide 8 / 179 Example Example: y = 2x + 3 Given two sets of coordinate points that represent a system of linear equations, determine whether the lines intersect to given a y = -1x - 2 solution to the system. 2 Linear Equation 1: Step 1: (1, 1) and (2, 3) Graph both lines from slope-intercept form on Linear Equation 2: (1, -2) and (4, 4) the same coordinate plane Step 2: Will the system of linear equations intersect into Write the intersection point a solution? as an ordered pair. Slide 9 / 179 Slide 9 (Answer) / 179 Example Example Decide if you will be able to find a solution to the system of equation Decide if you will be able to find a solution to the system of equation just by inspecting. Do not try to solve algebraically. just by inspecting. Do not try to solve algebraically. System: System: 6x + 3y = 10 6x + 3y = 10 Discuss with the 6x + 3y = 5 6x + 3y = 5 students how "6x + 3y" is identical in both Teacher Note equations. "6x + 3y" can not equal 10 and 5 at the same time. So, There can not be a solution found. [This object is a pull tab] Slide 10 / 179 Slide 11 / 179 Vocabulary A system of linear equations is two or more linear equations. Solving Systems by The solution to a system of linear inequalities is the ordered pair that will satisfy both equations. Graphing One way to find the solution to a system is to graph the equations on the same coordinate plane and find the point of intersection. There are 3 different types of solutions that are possible to get when solving a system. They are easiest to understand by looking at the graph. Return to Table of Contents Click here to watch a music video that introduces what we will learn about systems.
Slide 12 / 179 Slide 13 / 179 Type 1: One Solution Compare the Slopes This is the most common type of solution, it happens 6x + 2y = 4 y= 2x + 5 when two lines intersect in exactly ONE place - 6x - 6x m = 2 2y = -6x + 4 2 2 2 The slopes of the y = -3x + 2 lines will be m = -3 DIFFERENT What did we find out about the slopes? So, how many solutions will there be? Slide 14 / 179 Slide 15 / 179 Type 2: No Solution Compare the Slopes and Y-Intercepts This happens when the lines NEVER intersect! 10x + 2y = 6 The lines will be PARALLEL. y= -5x + 4 - 10x - 10x m = -5 2y = -10x + 6 b = 4 2 2 2 The slopes of the lines will be THE SAME y = -5x + 3 The y-intercepts will be m = -5 b = 3 DIFFERENT What did we find out about the slopes and the y-intercepts? So, how many solutions will there be? Slide 16 / 179 Slide 17 / 179 Type 3: Infinite Solutions Compare the Slopes and Y-Intercepts This happens when the lines overlap! The lines -4x + 2y = 2 y= 2x + 1 will be the SAME EXACT line! + 4x + 4x m = 2 b = 1 2y = 4x + 2 2 2 2 The slopes of the lines y = 2x + 1 will be THE SAME b = 1 The y-intercepts will m = 2 be THE SAME What did we find out about the slopes and the y-intercepts? So, how many solutions will there be?
Slide 18 / 179 Slide 19 / 179 1 How many solutions does the following system How can you quickly decide the number have: of solutions a system has? y = 2 x - 7 y = 3 x + 8 Different slopes 1 Solution A 1 solution Different lines B no solution Same slope No Solution Different y -intercept answer C infinitely many solutions Parallel Lines Same slope Infinitely Many Same y -intercept Same Line Slide 20 / 179 Slide 21 / 179 3 How many solutions does the following 2 How many solutions does the following system system have: have: 3 x + 3 y = 8 3 x - y = -2 1 y = 3 x + 2 y = x 3 A 1 solution A 1 solution B no solution answer answer B no solution C infinitely many solutions C infinitely many solutions Slide 22 / 179 Slide 23 / 179 4 How many solutions does the following system 5 How many solutions does the following system have: have: y = 4 x 3 x + y = 5 2 x - 0.5 y = 0 6 x + 2 y = 1 A 1 solution A 1 solution B no solution answer answer B no solution C infinitely many solutions C infinitely many solutions
Slide 24 / 179 Slide 25 / 179 Solution Consider this... First, make a table to represent the problem. Friend's Your distance Time distance from from your start (min.) your start (blocks) Suppose you are walking to school. Your friend is 5 blocks ahead (blocks) of you. You can walk two blocks per minute and your friend can walk one block per minute. 0 5 0 1 6 2 How many minutes will it take for you to catch up with your friend? 2 7 4 3 8 6 4 9 8 5 10 10 Slide 26 / 179 Slide 27 / 179 Solution Continued Solution Continued Next, plot the points on a graph. The point where the lines intersect is the solution to the system. Friend's Your distance Time distance from your (min.) from your (5,10) is the solution start start(blocks) (blocks) In the context of the 0 5 0 Blocks Blocks problem this means 1 6 2 after 5 minutes, you 2 7 4 will meet your friend at block 10. 3 8 6 4 9 8 5 10 10 Time (min.) Time (min.) Slide 28 / 179 Slide 29 / 179 Example Graphing Lines Solve the system of equations graphically: y = 2 x -3 Recall from Algebra I that you need a minimum of two points y = x - 1 to graph a line. answer Therefore, when solving a system of linear equations graphically, you will only need to plot two points for each equation.
Slide 30 / 179 Slide 31 / 179 Example Checking Your Work Solve the following system by graphing: Given the graph below, what is the point of intersection? y = 4 x + 6 y = -3 x - 1 (move the hand!) y = -3 x + 4 y = x - 4 (-1, 2) answer Slide 32 / 179 Slide 33 / 179 6 Solve the following system by graphing: Checking Your Work y = - x + 4 Now take the ordered pair we just found and substitute it into y = 2x + 1 answer the equations to prove that it is a solution for BOTH lines. (-1, 2) y = -3 x - 1 y = 4 x + 6 A (3, 1) 2 = -3(-1) - 1 2 = 4(-1) + 6 B (1, 3) Click for answer choices 2 = 3 - 1 2 = -4 + 6 AFTER students have graphed the system C (-1, 3) 2 = 2 2 = 2 D (1, -3) Slide 34 / 179 Slide 35 / 179 8 Solve the following system by graphing: 7 Solve the following system by graphing: y = x + 3 answer answer A (0, 4) A (0,-1) B (-4, 2) B (0,0) Click for answer choices AFTER students have graphed the system C (5, 6) C (-1, 0) D (2, 5) D (0, 1)
Slide 36 / 179 Slide 37 / 179 Example Graphing Quickly Solve the following system of linear equations by graphing: 2 x + y = 5 Recall from 8th grade that slope-intercept form of a linear equation is: - x + y = 2 y = m x + b Where m = the slope and b = the y -intercept Step 1: Rewrite the linear equation in slope-intercept form 2 x + y = 5 - x + y = 2 If you transform linear equations not in slope-intercept form to + x + x -2 x -2 x slope-intercept form, graphing them will be quicker. y = x + 2 y = -2 x + 5 Slide 38 / 179 Slide 39 / 179 Solution Continued Solution Continued Step 2: Plot the y -intercept and use the slope to plot the second Step 3: Locate the Point of Intersection and check your work: point (1, 3) y-intercept = (0, 5) y = -2 x + 5 slope = -2 y = x + 2 slope= (down 2, right 1) 3 = -2(1) + 5 3 = 1 + 2 3 = -2 + 5 3 = 3 y-intercept = (0, 2) slope = 1 3 = 3 slope= (up 1, right 1) Slide 40 / 179 Slide 41 / 179 Solution Continued Example Step 2: Plot y -intercept and use slope to plot second point Solve the system of equations graphically: y-intercept = (0, 3) 2 x + y = 3 slope = -2 x - 2 y = 4 slope= (down 2, right 1) Step 1: Rewrite in slope-intercept form y-intercept = (0, -2) slope = 2 x + y = 3 x - 2 y = 4 slope= (up 1, right 2) - x - x -2 x -2 x y = -2 x + 3 -2 y = - x + 4 -2 -2 1 y = x - 2 2 Step 3: Locate the Point of Intersection and check your work: (2, -1)
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