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KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities 4.1 Solving SYSTEMS of Linear Equations by y GRAPHING 10 4.1.1 Determine if an ordered pair is a solution of a system of equations in two variables 5 A System of Linear Equations consists of 2 or more linear -10 -5 5 10 x equations. -5 A Solution of this system consists of a point(s) that are common to all linear equations. We will only study -10 linear equations in two variables. The 'solution' to the system of equations The solution to these systems consists of a common is the common point of intersection intersection point. This intersection point is an ordered pair that is common to both equations. 1 2 (4.1.1)To determine if a given ordered pair 4.1.1 Examples: Which of the ordered pairs are solutions? is a solution: 2x y 5 Order Pairs: Substitute the ordered pair into both (a) System: (i) (5,0) x 3y 5 equations and determine if it is a solution (ii) (1,2) to both equations. (ii) (2,1) 3x y 5 (b) System: x 2y 11 Order Pairs: (i) (2-1) (ii) (3,4) (ii) (0, -5) 3 4 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities 4.1.2 Solve a system of linear equations by graphing 'Solution Point' for the system: Since a of a system of two linear y = -2x + 6 equations is a common ordered pair, or point, y = 3x - 4 this will be the ordered pair of the point of 10 y intersection of the graphs. 5 By drawing the two graphs, the can be . -10 -5 5 10 x -5 -10 5 6 (4.1.2) A system that has at LEAST one intersection point is defined as a consistent system. A system with NO intersection point is an nconsistent Line 1 Line 2 Solution consistency dependency system. Point a. y = 3x - 4 y=x+2 Systems with graphs are termed b. 2x + y=0 3x + y=1 c. 2x + y = 4 x+y=2 d. x - 2y =2 3x + 2y = -2 Systems with IDENTICAL, overlapping are termed (Compare your solution graphs to the following slides) 7 8 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities (4.1.2 solutions for examples (a) and (b) (4.1.2 solutions for examples (c) and (d) y 10 y 10 ( c) 5 (a) 2x + y = 4 y = 3x – 4 y=x +2 5 x+y=2 -10 -5 5 10 x -10 -5 5 10 x -5 (b) -5 2x + y = 0 (d) 3x + y = 1 x – 2y = 2 -10 3x + 2y = -2 -10 9 10 4.1.3 Without graphing, you can also determine the (4.1.3)There are also methods of determining number of solutions of a system a system will have. Note: Write the two equations in slope-intercept Graphing is not an accurate method of form: y = mx + b determining the point of intersection. If both have the same slope and different There are other methods of accurately intercepts, then the system has NO determining the intersection point. SOLUTION and is INCONSISTENT. The two lines are Parallel! If the two equations have different slopes, then they must intersect and therefore has 11 ONE SOLUTION and is CONSISTENT 12 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities The 'solution point' can also be determined, more accurately, by using one of two 'algebraic methods'. Line 1 Line 2 Orientation Number of The two algebraic methods are: Solutions a. 3x + y = 1 3x + 2y = 6 4.2: The 'substitution method' b. 2x + y = 0 2y = 6 - 4x c. 3y - 2x = 6 x + 2y = 9 4.3: The 'addition method' d. 8y + 6x = 4 4y - 2 = 3x Both methods use algebraic methods using two equations to find the common point (x,y) that they have in common. 13 This point is their intersection point. 14 4.2 Solving Systems of Linear Equations by the (4.2.1) Examples: Solve each of the following system of equations using the substitution method 4.2.1 To determine the solution point: a. x y 20 Solution Point: Solve one of the equations for either 'x' or x 3y 'y' (which ever seems faster) y 3x 1 Substitute this expression for the b. 4y 8x 12 corresponding 'x' or 'y' variable in the other equation. c. x 2y 6 Solve for the other variable 2x 3y 8 Substitute this value of the other variable x 2y 10 into either of the original equations to find d. 15 2x 3y 18 16 the value for the remaining variable. E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities (4.3.1) 4.3 Solving Systems of Linear Equations by the Multiply one of the equations by a positive ADDITION METHOD or negative number which will result in eliminating one of the variables if the two equations are added. 4.3.1Another method of determining the solution (One equation's coefficient of 'x' or 'y' point is by the addition or 'elimination' method. will be the negative of that in the other equation) This is based on the addition property of equality: Find the value for the remaining variable. If A = B and S = T then A + S = B + T Substitute this back into one of the original Also: equations to find the other coordinate. If A = B S=T A+S=B+T 17 18 4.4 Systems of Linear equations and Problem Solving x y a. 4x y 13 1 Write each true statement as an equation in d. 3 6 2x y 5 variables which define the unknowns. x y 0 b. 2x y 5 2 4 Use any of the system solution processes to find 4x 2y 12 the solution for this system. x 5 y 14 e. 2 4 c. 4x 2y 2 Note: the Substitution or Addition Method are the 3x 2y 12 x 2y 2 most accurate methods of determining the intersection 3 6 point, while graphing is the least accurate method. 19 20 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities (4.4.1 )Examples: Without solving each problem, choose each correct solution by deciding which choice satisfies the given conditions by substituting the choices into the equations. Which choice will make the equation true. 4. Two numbers total 83 and have a difference of 17. 1 Two CDs and 4 cassettes cost a total of $40. However, 3 CD's Find the numbers and 5 cassettes cost $55. Find the price of each. a. CD = $12; cassette = $4 5. One number is 4 more than twice the second. Their b. CD = $15; cassette = $2 total is 25. Find the numbers. c. CD = $10; cassette = $5 2 A chemistry lab stores 28 gallons of saline solution in two 6. In investigating train fares from Cleveburg to NYC, a containers. One container holds three times the capacity of customer finds that 3 adults and 4 children must pay the other. Find the capacity of each container. $159, while 2 adults and 3 children must pay $112. a. 15 gallons: 5 gallons Find the price of the adult and child's tickets. b. 20 gallons: 8 gallons c. 21 gallons: 7 gallons 21 22 (4.5.1)Inequalities have 'half planes' sets of points that 4.5 Graphing Linear INEQUALITIES satisfy the original linear inequality: 4.5.1 Graphing a linear inequality in two variables y -1.5x + 5 The solution for include all points Use 'Test Points' to find the correct half plane. the line. The 'solution' for all points TP1: (0,2) or the line, depending on the type TP2: (5,0) of inequality. This results in a 'half-plane' of solution points. Terms: half-planes, boundary 23 24 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities Graph the inequality as if it were a linear equation to find the boundary line. (i.e. y = mx+b) 4.5.1 Examples: Use a dotted boundary line if the inequality is < or > Determine which of the ordered pairs are solutions Use a solid boundary line if the inequality contains the = of the linear inequality in two variables. sign ( , or ). 1. y – x < -2 (2, 1) or (5, -1) Choose a test point (TP), that is not on the line. Substitute the coordinates of the TP back into the original inequality. 2. 3x – 5y <= -4 (-1, 4) or (4, 0) If the substitution results in a true condition, shade this side of the boundary line, otherwise shade the other side of the boundary line. You may also use arrows on the boundary line ends, in place of shading, to denote the correct half-plane of solution points 25 26 4.5.1 Examples: Graph the following inequalities: 10y 10y Shade the correct the half-plane that is the 'solution'. 5. 6x – 2y 0 5 3. 2x + y 4 5 TP=(1,0) -10 -5 5 10 -10 -5 5 10 x x 3. 2x y 4 -5 -5 -10 -10 4. x 2y 3 5. 6x 2y 0 10 y 10y 4. x + 2y 3 5 5 1 1 6. x y 1 2 3 -10 -5 5 10 x -10 -5 5 10 x -5 TP=(1,0) -5 6. ½ x – 1/3 y -1 -10 -10 (Compare your graphs to the following slide) 27 28 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities 4.6.1 To find the 'solution' to a set of inequalities, 4.6 Graphing SYSTEMS of Linear Note: Graph each inequality as in Section 4.5, both on the same graph. Finding the 'solution' to a system of linear EQUATIONS finds their Graph the boundary lines (dashed or solid). Use a test point for each inequality and determine Finding the 'solution' to a system of linear which half-plane is the solution for each inequality INEQUALITIES finds the Shade that side of the boundary line, or use arrows on that satisfy each of the two inequalities. the end of each line to denote the solution half-plane for each inequality. The 'solution' to the 'system' is the overlapping shaded regions. 29 30 4.6.1 Graphing 'systems' of inequalities: 4.6.1 Examples: Graph the solution to the y<x+4 following systems of inequalities. Compare your y > -1.5x + 5 solutions to the following slides. Use Test Points for each inequality Find half plane of each 1. y x 4 3. y x 4 5. x 2y 6 Determine their intersection region 10 y y 2x 5 1 x 2y 4 y -1.5x + 5 y x 2 5 2 -10 -5 5 10 TP=(2,0) x 2 4. 2x y 4 2. y x 5 -5 3 y x+4 x y 5 1 -10 y x 3 31 4 32 E. Millspaw KSU M10005 Chapter 4: Systems of Linear Equations and Inequalities 10 y 1. 3. 10 y y -x -4 y x–4 5 5 y 2x + 5 y ½x+2 -10 -5 5 10 -10 -5 5 10 x x TP -5 -5 TP=(6,0) -10 10 y 2. -10 10 y y 2/3 x – 5 TP=(0,4) 5 5 y ¼x-3 TP = (4,0) -10 -5 5 10 -10 -5 5 10 x x 4. -5 -5 2x + y 4 -10 -10 x+y 5 33 34 y 10 5 -10 -5 5 10 x -5 6. x + 2y 6 -10 x – 2y 4 35 E. Millspaw

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