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Lesson 3-1 Solving Systems of Equations by Graphing Lesson 3-2 Solving Systems of Equations Algebraically Lesson 3-3 Solving Systems of Inequalities by Graphing Lesson 3-4 Linear Programming Lesson 3-5 Solving Systems of Equations in Three Variables Contents

Example 1 Solve by Graphing Example 2 Break-Even Point Analysis Example 3 Intersecting Lines Example 4 Same Line Example 5 Parallel Lines Lesson 1 Contents

Solving Systems of Equations by Graphing If you can graph a straight line, you can solve systems of equations graphically! The process is very easy.  To solve a system of equations graphically, graph both equations and see where they intersect.  The intersection point is the solution.

Example Solve by graphing: 4x-6y=12 2x+2y=6 Step 1: Put equations in slope intercept form to find slope (m) and y-intercept (b) and graph. 2x+2y=6 2y=-2x+6 y=(-2/2)x+(6/2) y=-x+3 slope=-1/1 y-intercept = 3 4x-6y=12 4x=6y+12 4x-12=6y 6y=4x-12 y=(4/6)x-(12/6) y=(2/3)x-2 slope = 2/3 y-intercept = -2

The slope intercept method of graphing was used in this example. Example Continued: Graph the equations. The slope intercept method of graphing was used in this example. The point of intersection of the two lines (3,0) is the solution to the system of equations. This means that (3,0), when substituted into either equation, will make them both true.

Solve the system of equations by graphing. Write each equation in slope-intercept form. The graphs appear to intersect at (4, 2). Example 1-1a

Check Substitute the coordinates into each equation. Original equations Replace x with 4 and y with 2. Simplify. Answer: The solution of the system is (4, 2). Example 1-1a

Solve the system of equations by graphing. Answer: (4, 1) Example 1-1b

Fund-raising A service club is selling copies of their holiday cookbook to raise funds for a project. The printer’s set-up charge is $200, and each book costs $2 to print. The cookbooks will sell for $6 each. How many cookbooks must the members sell before they make a profit? Let Cost of books is cost per book plus set-up charge. y = 2x + 200 Example 1-2a

Income from books is price per book times number of books. y = 6 x Answer: The graphs intersect at (50, 300). This is the break-even point. If the group sells less than 50 books, they will lose money. If the group sells more than 50 books, they will make a profit. Example 1-2a

The student government is selling candy bars The student government is selling candy bars. It cost $1 for each candy bar plus a $60 set-up fee. The group will sell the candy bars for $2.50 each. How many do they need to sell to break even? Answer: 40 candy bars Example 1-2b

Consistent vs. Inconsistent Now, keep in mind that you are applying these to a system of linear equations. We say that a point is a "solution" to the system when it makes BOTH equations true, right? This is to say that there exists a point (or set of points) that "work" in one equation and also "work" in the other one. So we say that this point is CONSISTENT from one equation to the next. On the other hand, if there are NO points that work in both, then we say that the equations are INCONSISTENT. NO numbers that work in one are consistent with the other. To sum up, a consistent system has at least one solution. An inconsistent system has NO solution at all.

Dependent vs. Independent When a system is "dependent," it means that ALL points that work in one of them ALSO work in the other one. Graphically, this means that one line is lying entirely on top of the other one, so that if you graphed both, you would really see only one line on the graph, since they are imposed on top of each other. One of them totally DEPENDS on the other one. When a system is "independent," it means that they are not lying on top of each other. There is EXACTLY ONE solution, and it is the point of intersection of the two lines. It's as if that one point is "independent" of the others. To sum up, a dependent system has INFINITELY MANY solutions. An independent system has EXACTLY ONE solution.

Write each equation in slope-intercept form. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Write each equation in slope-intercept form. Example 1-3a

Answer: The graphs of the equations intersect at (2, –3). Since there is one solution to this system, this system is consistent and independent. Example 1-3a

consistent and independent Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Answer: consistent and independent Example 1-3b

Since the equations are equivalent, their graphs are the same line. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Since the equations are equivalent, their graphs are the same line. Example 1-4a

Answer: Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent. Example 1-4a

consistent and dependent Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Answer: consistent and dependent Example 1-4b

Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Answer: The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent. Example 1-5a

Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Answer: inconsistent Example 1-5b

Assignment: Page 113 #14, 18, 22, 26

End of Lesson 1

Example 1 Solve by Using Substitution Example 2 Compare Values Example 3 Solve by Using Elimination Example 4 Multiply, Then Use Elimination Example 5 Inconsistent System Lesson 2 Contents

The Substitution Method The substitution method is used to eliminate one of the variables by replacement when solving a system of equations.   Think of it as "grabbing" what one variable equals from one equation and "plugging" it into the other equation.

Example of Substitution Solve this system of equations and check:  3y - 2x = 11 y + 2x = 9 1.  Solve one of the equations for either "x" or "y". This example solves the second equation for "y ".   3y - 2x = 11 y = 9 - 2x

Example of Substitution 2.  Replace the "y" value in the first equation by what "y" now equals.  Grab the "y" value and plug it into the other equation.       3y - 2x = 11 y = 9 - 2x 3(9 - 2x) - 2x = 11

Example of Substitution 3.  Solve this new equation for "x".       (27 - 6x) - 2x = 11 27 - 6x - 2x = 11 27 - 8x = 11 -8x = -16 x = 2

Example of Substitution 4.  Place this new "x" value into either of the ORIGINAL equations in order to solve for "y".  Pick the easier one to work with!  y = 9 - 2(2) y = 9 - 4 y = 5

Example of Substitution 5.  Check:  substitute x = 2 and y = 5 into BOTH ORIGINAL equations.  If these answers are correct, BOTH equations will be TRUE! 3y - 2x = 11 3(5) - 2(2) = 11 15 - 4 = 11 11 = 11 (check!) y + 2x = 9 5 + 2(2) = 9 5 + 4 = 9 9 = 9 (check!)  

Use substitution to solve the system of equations. Solve the first equation for x in terms of y. First equation Subtract 4y from each side. Example 2-1a

Substitute 26 – 4y for x in the second equation and solve for y. Subtract 26 from each side. Divide each side by –9. Now substitute the value for y in either of the original equations and solve for x. First equation Replace y with 4. Simplify. Subtract 16 from each side. Answer: The solution of the system is (10, 4). Example 2-1a

Use substitution to solve the system of equations. Answer: (5, 1) Example 2-1b

Try These Solve each system of equations by substitution. 2x + y = 11 6x – 2y = -2 2. 5a – b = 17 3a + 2b = 5

Simultaneous equations got you baffled? Relax! You can do it! Elimination Method Simultaneous equations got you baffled?   Relax!  You can do it!  Think of the adding or subtracting method as simply "eliminating" one of the variables to make your life easier.

Example of Elimination 1.  Solve this system of equations and check:  x - 2y = 14 x + 3y = 9 

Example of Elimination a.  First, be sure that the variables are "lined up" under one another.  In this problem, they are already "lined up". x - 2y = 14 x + 3y = 9

Example of Elimination b.  Decide which variable ("x" or "y") will be easier to eliminate.  In order to eliminate a variable, the numbers in front of them (the coefficients) must be the same or negatives of one another.  Looks like "x" is the easier variable to eliminate in this problem. x - 2y = 14 x + 3y = 9

Example of Elimination c.  Now, subtract to eliminate the "x" variable.   (Remember:  when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.) x - 2y = 14 -x - 3y = - 9         - 5y = 5  

Example of Elimination d.  Solve this simple equation. -5y = 5    y = -1

Example of Elimination e.  Plug "y = -1" into either of the ORIGINAL equations to get the value for "x". x - 2y = 14 x - 2(-1) = 14 x + 2 = 14 x = 12

Example of Elimination f.  Check:  substitute x = 12 and y = -1 into BOTH ORIGINAL equations.  If these answers are correct, BOTH equations will be TRUE! x - 2y = 14 12 - 2(-1) = 14 12 + 2 = 14 14 = 14  (check!) x + 3y = 9 12 + 3(-1) = 9 12 - 3 = 9 9 = 9  (check!)

Use the elimination method to solve the system of equations. In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated. Subtract the equations. Example 2-3a

Now find x by substituting 4 for y in either original equation. Second equation Replace y with 4. Subtract 4 from each side. Answer: The solution is (2, 4). Example 2-3a

Use the elimination method to solve the system of equations. Answer: (17, –4) Example 2-3b

Use the elimination method to solve the system of equations. Multiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable. Multiply by 2. Multiply by 3. Example 2-4a

Replace x with 3 and solve for y. First equation Replace x with 3. Multiply. Subtract 6 from each side. Divide each side by 3. Answer: The solution is (3, 2). Example 2-4a

Use the elimination method to solve the system of equations. Answer: (–5, 4) Example 2-4b

Use the elimination method to solve the system of equations. Use multiplication to eliminate x. Multiply by 2. Answer: Since there are no values of x and y that will make the equation true, there are no solutions for the system of equations. Example 2-5a

Use the elimination method to solve the system of equations. Answer: There are no solutions for this system of equations. Example 2-5b

Try These Solve each system of equations by elimination. x + y = 7 2x + y = 11 2. 6g – 8h = 50 4g + 6h = 22

Assignment: Page 120 #18, 24, 26, 30

End of Lesson 2

Example 1 Intersecting Regions Example 2 Separate Regions Example 3 Write and Use a System of Inequalities Example 4 Find Vertices Lesson 3 Contents

System of Inequalities If you can graph an inequality, you can graph a system of inequalities! Simply graph each inequality separately on the same set of axes.  The area where the shadings overlap is the solution to the system of inequalities. 

Solve the system of inequalities by graphing. solution of Regions 1 and 2 solution of Regions 2 and 3 Example 3-1a

Answer: The intersection of these regions is Region 2, which is the solution of the system of inequalities. Notice that the solution is a region containing an infinite number of ordered pairs. Example 3-1a

Solve the system of inequalities by graphing. The inequality can be written as and Graph all of the inequalities on the same coordinate plane and shade the region or regions that are common to all. Example 3-1a

Answer: Example 3-1a

Solve each system of inequalities by graphing. Answer: Example 3-1b

Solve each system of inequalities by graphing. Answer: Example 3-1b

Solve the system of inequalities by graphing. Graph both inequalities. The graphs do not overlap, so the solutions have no points in common. Answer: The solution set is . Example 3-2a

Solve the system of inequalities by graphing. Answer:  Example 3-2b

Medicine Medical professionals recommend that patients have a cholesterol level below 200 milligrams per deciliter (mg/dL) of blood and a triglyceride level below 150 mg/dL. Write and graph a system of inequalities that represents the range of cholesterol levels and trigyceride levels for patients. Let c represent the cholesterol levels in mg/dL. It must be less than 200 mg/dL. Since cholesterol levels cannot be negative, we can write this as Let t represent the triglyceride levels in mg/dL. It must be less than 150 mg/dL. Since triglyceride levels also cannot be negative, we can write this as Example 3-3a

Graph all of the inequalities Graph all of the inequalities. Any ordered pair in the intersection of the graphs is a solution of the system. Answer: Example 3-3a

Safety The speed limits while driving on the highway are different for trucks and cars. Cars must drive between 45 and 65 miles per hour, inclusive. Trucks are required to drive between 40 and 55 miles per hour, inclusive. Let c represent the speed range of speed for cars and t represent the range of speeds for trucks. Write and graph a system on inequalities to represent this situation. Example 3-3b

Answer: Example 3-3b

Find the coordinates of the vertices of the figure formed by and Graph each inequality. The intersection of the graphs forms a triangle. Answer: The vertices of the triangle are at (0, 1), (4, 0), and (1, 3). Example 3-4a

Find the coordinates of the vertices of the figure formed by and Answer: (–1, 1), (0, 3), and (5, –2) Example 3-4b

Assignment: Page 126 #12, 14, 16, 18

End of Lesson 3

Example 1 Bounded Region Example 2 Unbounded Region Example 3 Linear Programming Lesson 4 Contents

Step 1 Find the vertices of the region. Graph the inequalities. Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Step 1 Find the vertices of the region. Graph the inequalities. The polygon formed is a triangle with vertices at (–2, 4), (5, –3), and (5,4). Example 4-1a

(x, y) 3x – 2y f(x, y) (–2, 4) 3(– 2) – 2(4) (5, –3) 3(5) – 2(–3) 21 Step 2 Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function. (x, y) 3x – 2y f(x, y) (–2, 4) 3(– 2) – 2(4) – 14 (5, –3) 3(5) – 2(–3) 21 (5, 4) 3(5) – 2(4) 7 Answer: The vertices of the feasible region are (–2, 4), (5, –3), and (5, 4). The maximum value is 21 at (5, –3). The minimum value is –14 at (–2, 4). Example 4-1a

Graph the following system of inequalities Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Answer: vertices: (1, 5), (4, 5) (4, 2); maximum: f(4, 2) = 10, minimum: f(1, 5) = –11 Example 4-1b

Graph the following system of inequalities Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Graph the system of inequalities. There are only two points of intersection, (–2, 0) and (0, –2). Example 4-2a

(x, y) 2x + 3y f(x, y) (–2, 0) 2(–2) + 3(0) –4 (0, –2) 2(0) + 3(–2) –6 The minimum value is –6 at (0, –2). Although f(–2, 0) is –4, it is not the maximum value since there are other points that produce greater values. For example, f(2,1) is 7 and f(3, 1) is 10. It appears that because the region is unbounded, f(x, y) has no maximum value. Answer: The vertices are at (–2, 0) and (0, –2). There is no maximum value. The minimum value is –6 at (0, –2). Example 4-2a

Answer: vertices: (0, –3), (6, 0); maximum: f(6, 0) = 6; no minimum Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Answer: vertices: (0, –3), (6, 0); maximum: f(6, 0) = 6; no minimum Example 4-2b

m = the number of mowing jobs p = the number of pruning jobs Landscaping A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day from one of its crews. Step 1 Define the variables. m = the number of mowing jobs p = the number of pruning jobs Example 4-3a

m  0, p  0 p  2 Step 2 Write a system of inequalities. Since the number of jobs cannot be negative, m and p must be nonnegative numbers. m  0, p  0 Mowing jobs take 1 hour. Pruning jobs take 3 hours. There are 9 hours to do the jobs. There are no more than 2 pruning jobs a day. p  2 Example 4-3a

Step 3 Graph the system of inequalities. Example 4-3a

Step 4 Find the coordinates of the vertices of the feasible region. From the graph, the vertices are at (0, 2), (3, 2), (9, 0), and (0, 0). Step 5 Write the function to be maximized. The function that describes the income is We want to find the maximum value for this function. Example 4-3a

(m, p) 40m + 120p f(m, p) (0, 2) 40(0) + 120(2) 240 (3, 2) Step 6 Substitute the coordinates of the vertices into the function. (m, p) 40m + 120p f(m, p) (0, 2) 40(0) + 120(2) 240 (3, 2) 40(3) + 120(2) 360 (9, 0) 40(9) + 120(0) (0, 0) 40(0) + 120(0) Step 7 Select the greatest amount. Example 4-3a

Answer: The maximum values are 360 at (3, 2) and 360. at (9, 0) Answer: The maximum values are 360 at (3, 2) and 360 at (9, 0). This means that the company receives the most money with 3 mows and 2 prunings or 9 mows and 0 prunings. Example 4-3a

Landscaping A landscaping company has crews who rake leaves and mulch Landscaping A landscaping company has crews who rake leaves and mulch. The company schedules 2 hours for mulching jobs and 4 hours for raking jobs. Each crew is scheduled for no more than 2 raking jobs per day. Each crew’s schedule is set up for a maximum of 8 hours per day. On the average, the charge for raking a lawn is $50 and the charge for mulching is $30. Find a combination of raking leaves and mulching that will maximize the income the company receives per day from one of its crews. Example 4-3b

0 raking jobs and 4 mulching jobs Answer: 0 raking jobs and 4 mulching jobs Example 4-3b

End of Lesson 4

Example 2 Infinite Solutions Example 3 No Solution Example 1 One Solution Example 2 Infinite Solutions Example 3 No Solution Example 4 Write and Solve a System of Equations Lesson 5 Contents

Solve the system of equations. Step 1 Use elimination to make a system of two equations in two variables. First equation Second equation Multiply by 2. Add to eliminate z. Example 5-1a

Subtract to eliminate z. First equation Third equation Subtract to eliminate z. Notice that the z terms in each equation have been eliminated. The result is two equations with the two same variables x and y. Example 5-1a

Step 2 Solve the system of two equations. Multiply by 5. Add to eliminate y. Divide by 29. Substitute –2 for x in one of the two equations with two variables and solve for y. Equation with two variables Replace x with –2. Multiply. Simplify. Example 5-1a

Equation with three variables Step 3 Substitute –2 for x and 6 for y in one of the original equations with three variables. Equation with three variables Replace x with –2 and y with 6. Multiply. Simplify. Answer: The solution is (–2, 6, –3). You can check this solution in the other two original equations. Example 5-1a

Solve the system of equations. Answer: (–1, 2, –4) Example 5-1b

Solve the system of equations. Eliminate y in the first and third equations. Multiply by 3. Example 5-2a

The equation. is always true The equation is always true. This indicates that the first and third equations represent the same plane. Check to see if this plane intersects the second plane. Multiply by 6. Divide by the GCF, 3. Answer: The planes intersect in a line. So, there are an infinite number of solutions. Example 5-2a

Solve the system of equations. Answer: There are an infinite number of solutions. Example 5-2b

Solve the system of equations. Eliminate x in the second two equations. Multiply by 3. Multiply by 2. Answer: The equation is never true. So, there is no solution of this system. Example 5-3a

Solve the system of equations. Answer: There is no solution of this system. Example 5-3b

Explore Read the problem and define the variables. Sports There are 49,000 seats in a sports stadium. Tickets for the seats in the upper level sell for $25, the ones in the middle level cost $30, and the ones in the bottom level are $35 each. The number of seats in the middle and bottom levels together equals the number of seats in the upper level. When all of the seats are sold for an event, the total revenue is $1,419,500. How many seats are there in each level? Explore Read the problem and define the variables. Example 5-4a

Plan There are 49,000 seats. When all the seats are sold, the revenue is 1,419,500. Seats cost $25, $30, and $35. The number of seats in the middle and bottom levels together equal the number of seats in the upper level. Example 5-4a

Solve Substitute in each of the first two equations. Replace u with m + b. Simplify. Divide by 2. Replace u with m + b. Distributive Property Simplify. Example 5-4a

Now, solve the system of two equations in two variables. Multiply by 55. Substitute 14,400 for b in one of the equations with two variables and solve for m. Equation with two variables Subtract 14,400 from each side. Example 5-4a

Equation with three variables Substitute 14,400 for b and 10,100 for m in one of the original equations with three variables. Equation with three variables Add. Answer: There are 24,500 upper level, 10,100 middle level, and 14,400 bottom level seats. Example 5-4a

Examine Check to see if all the criteria are met. There are 49,000 seats in the stadium. The number of seats in the middle and bottom levels equals the number of seats in the upper level. When all of the seats are sold, the revenue is $1,419,500. 24,500($25) + 10,100($30) + 14,400($35) = $1,419,500 Example 5-4a

Answer: 10 pens, 7 pencils, 8 packs of paper Business The school store sells pens, pencils, and paper. The pens are $1.25 each, the pencils are $0.50 each, and the paper is $2 per pack. Yesterday the store sold 25 items and earned $32. The number of pens sold equaled the number of pencils sold plus the number of packs of paper sold minus 5. How many of each item did the store sell? Answer: 10 pens, 7 pencils, 8 packs of paper Example 5-4b

End of Lesson 5

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