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KSU M10005 4.1 Solving SYSTEMS of Linear Equations by GRAPHING 1. Determine if an ordered pair is a solution of a system of equations in two variables A System of Linear Equations consists of 2 or more linear equations. A Solution of this system consists of a point(s) that are common to all linear equations. We will only study linear equations in two variables. A solution to these systems consists of a common intersection point. This intersection point is an ordered pair that is common to both equations. To determine if a given ordered pair is a solution: Substitute the ordered pair into both equations and determine if it is a solution to both equations. Examples: Determine whether the given ordered pairs satisfy the system of linear equations. (a) System: 2x + y = 5 and x + 3y = 5 ordered pairs : (i) (5,0) (ii) (1,2) (iii) ( 2,1) (b) System: 3x - y = 5 and x + 2y = 11 ordered pairs : (i) 2, -1) (ii) (3,4) (iii) (0, -5) (c) System: 2x - 3y = 8 and x - 2y = 6 ordered pairs: (i) (4,0) (ii) (-2, -4) (iii) (7, 2) 2. Solve a system of linear equations by graphing Since a solution of a system of two linear equations is a common ordered pair, or point, this will be the ordered pair of the point of intersection. By drawing the two graphs, the solution can be estimated. A system that has at LEAST one intersection point is defined as a consistent system. A system with no intersection point is an inconsistent system. Systems with different graphs are termed independent and systems with identical, overlapping graphs are termed dependent. M10005_C4_notes.sxw 1/8 rev. 10/17/05 KSU M10005 Examples: Solve each system by graphing the equations on the same set of axes. Determine whether the system is consistent or inconsistent. Determine whether the equations are dependent or independent. Line 1 Line 2 Solution consistency dependency Point a. y = 3x - 4 y=x+2 b. 2x + y=0 3x + y=1 c. 2x + y = 4 x+y=2 d. x - 2y =2 3x + 2y = -2 3. Without graphing, determine the number of solutions of a system Graphical is not an accurate method of determining the point of intersection. There are other method of accurately determining the intersection point. There are methods of determining how many solutions a system will have. Write the two equations in slope-intercept form: • If both have the same slope and different intercepts, then the system has NO SOLUTION and is INCONSISTENT • If the two equations have different slopes, then they must intersect and therefore has ONE SOLUTION and is CONSISTENT Examples: Without graphing, decide; (a) the orientation of the lines (if intersecting, determine the common point) (b) how many solutions are true for each system. Line 1 Line 2 Orientation Number of Solutions a. 3x + y = 1 3x + 2y = 6 b. 2x + y = 0 2y = 6 - 4x c. 3y - 2x = 3 x + 2y = 9 d. 8y + 6x = 4 4y - 2 = 3x M10005_C4_notes.sxw 2/8 rev. 10/17/05 KSU M10005 4.2 Solving Systems of Linear Equations by SUBSTITUTION 1. Using the substitution method to solve a system of linear equations. For accuracy in determining the solution point: • Solve one of the equations for either 'x' or 'y' (which ever seems faster) • Substitute this expression for the corresponding 'x' or 'y' variable in the other equation. • Solve for the other variable • Substitute this value of the other variable into either of the original equations to find the value for the remaining variable. Examples: Solve each of the following system of equations using the substitution method Line 1 Line 2 Solution Point a. x + y = 20 x = 3y b. y = 3x + 1 4y - 8x = 12 c. x + 2y = 6 2x + 3y = 8 d. -x + 2y = 10 -2x + 3y = 18 4.3 Solving Systems of Linear Equations by ADDITION 1. Using the addition method to solve a system of linear equations Another method of determining the solution point is by the addition or 'elimination' method. This is based on the addition property of equality: If A = B and S = T then A + S = B + T • Align the two equations vertically. • Multiply one of the equations by a positive or negative number which will result in eliminating one of the variables if the two equations are added. (One equation's coefficient will be the negative of that coefficient in the other equation) • Find the value for the remaining variable, substitute this back into one of the original equations to find the other coordinate. M10005_C4_notes.sxw 3/8 rev. 10/17/05 KSU M10005 Examples: Solve each of the equations by the addition method. Line 1 Line 2 Solution Point a. 4x + y = 13 2x - y = 5 b. 2x + y = 6 4x + 2y = 12 c. 4x + 2y = 2 3x - 2y =12 d. x/3 + y /6 = 1 x/2 - y /4 =0 e. (x + 5)/2 = (y + 14) / 4 x/3 = (2y + 2) /6 4.4 Systems of Linear equations and Problem Solving 1. Using a system of equations to solve problems Write each true statement as an equation in variables which define the unknowns. Use the system solution processes to find the solution for this system. Examples: Without solving each problem, choose each correct solution by deciding which choice satisfies the given conditions. 1 Two CDs and 4 cassettes cost a total of $40. However, 3 CD's and 5 cassettes cost $55. Find the price of each. a. CD = $12; cassette = $4 b. CD = $15; cassette = $2 c. CD = $10; cassette = $5 2 A chemistry lab stores 28 gallons of saline solution in two containers. One container holds three times the capacity of the other. Find the capacity of each container. a. 15 gallons: 5 gallons b. 20 gallons: 8 gallons c 21 gallons: 7 gallons Solve the following situations: 4 Two numbers total 83 and have a difference of 17. Find the numbers 5 One number is 4 more than twice the second. Their total is 25. Find the numbers. 6 In investigating train fares from Cleveburg to NYC, a customer finds that 3 adults and 4 children must pay $159, while 2 adults and 3 children must pay $112. Find the price of the adult and child's tickets. M10005_C4_notes.sxw 4/8 rev. 10/17/05 KSU M10005 4.5 Graphing Linear INEQUALITIES 1. Graphing a linear equality in two variables The equations include all points ON the line, inequalities are all point below or above the line. Terms: half-planes, boundary • Graph the inequality by replacing the inequality sign with an equal sign. Use a dotted line if the inequality is < or >, use a solid line if it could be =. • Chose a test point, not on the line. Substitute theses coordinates back into the original inequality. • If the substitution results in a true condition, shade this side of the boundary line, otherwise shade the other side of the boundary line. Examples: • Determine which ordered pairs given are solutions of the linear inequality in two variables. Inequality Point 1 Point 2 a. y - x < -2 (2,1) ( 5, -1) b. 3x - 5y <= -4 ( -1, 4) ( 4, 0) Graph each inequality 3. 2x _ y > -4 M10005_C4_notes.sxw 5/8 rev. 10/17/05 KSU M10005 4. x + 2y < = 3 5. 3x – 7y >=0 6. 1/2x – 1/3y <= -1 M10005_C4_notes.sxw 6/8 rev. 10/17/05 KSU M10005 4.6 Graphing SYSTEMS of Linear INEQUALITIES Note: • Finding the 'solution' to a system of linear EQUATIONS finds their point of intersection • Finding the 'solution' to a LINEAR INEQUALITY finds the region on one side or the other of the line that contains all points that satisfy the original inequality. • Finding the 'solution' to s SYSTEM OF LINEAR EQUALITIES finds the intersection of the two regions that satisfy each of the two inequalities. To find the 'solution' to a set of inequalities, graph each as in Section 4.5. 1. Graph the line (dashed or solid). 2. Find a test point for each inequality and determine which side of the line is the region containing points that satisfy the original equality 3. Shade that side of the line. 4. Repeat for the other inequality. 5. The 'solution' to the 'system' is the overlapping shaded regions. Examples: Graph the solution to the following systems of inequalities. 1. y ≤ x−4 y ≥ −2 x 5 2. 2 y x−5 3 1 y x−3 4 M10005_C4_notes.sxw 7/8 rev. 10/17/05 KSU M10005 3. y ≥ x−4 1 y x2 2 4. 2x y ≤ 4 xy ≥ 5 5. x 2y ≥ 6 x − 2y ≤ 4 M10005_C4_notes.sxw 8/8 rev. 10/17/05