Solving Systems of Equations Modeling real-world problems
In business, the point at which income equals expenses is called the break-evenpoint.When starting a business, people want to know the point a which their income equals their expenses, that’s the point where they start to make a profit. In the example above the values of y on the blue line represent dollars made and the value of y on the dotted red line represent dollars spent. Profit Loss
A system of equations is a set of two or more equations that have variables in common. The common variables relate to similar quantities. You can think of an equation as a condition imposed on one or more variables, and a system as several conditions imposed simultaneously. Remember, when solving systems of equations, you are looking for a solution that makes each equation true.
In earlier chapters, you learned to solve an equation for a specified variable, graph the equation, and find how pairs of lines represented by linear equations in two variables are related. When solving a system of equations, you look for a solution that makes each equation true. There are several strategies you can use. To begin with we will be using tables and graphs. Let’s look at the following example and work through the problem step-by-step to find a solution.
Edna leaves the trailhead at dawn to hike 12 miles toward the lake, where her friend Maria is camping. At the same time, Maria starts her hike toward the trailhead. Edna is walking uphill so she averages only 1.5 mi/hr, while Maria averages 2.5 mi/hr walking downhill. When and where will they meet?
Here’s what we need to complete to solve this example. • Define variables for time (x) and for distance (y) from the trailhead. • Write a system of two equations to model this situation. • Solve this system by creating a table and finding values for the variables that make both equations true. Then locate this solution on a graph. • Check your solution and explain its real-world meaning.
Let x represent the time in hours. Both women hike the same amount of time. Let y represent the distance in miles from the trailhead. When Edna and Maria meet they will both be the same distance from the trailhead, although they will have hiked different distances.
The system of equations that models this situation is grouped in a brace.Edna starts at the trailhead so she increases her distance from it as she hikes 1.5 mi/hr for x hours. Maria starts 12 miles from the trailhead and reduces her distance from it as she hikes 2.5 mi/hr for x hours.
Create a table from the equations. Fill in the times and calculate each distance. The table shows the x-value that gives equal y-values for both equations. When x = 3, both y-values are 4.5. So the solution is the ordered pair (3, 4.5). We say these values “satisfy” both equations.
Let’s create the data on your graphing calculator. Enter the equations in y = on your calculator and create a table.
Do both tables model this situation?What do you notice about y when x increases in each equation?Why are the values different?
On the graph this solution is the point where the two lines intersect. You can use trace function or calculate function on your calculator to approximate the coordinates of the solution point, though sometimes you’ll get an exact answer as in our example here.
Solving Systems of Equations Graphing Method
A system of linear equations is a set of two or more equations with the same variable. The solution of a system in x and y is any ordered pair (x, y) that satisfies each of the equations in the system.The solution of a system of equations is the intersection of the graphs of the equations.
If you can graph a straight line, you can solve systems of equations graphically!The process is very easy. Simply graph the two lines and look for the point where they intersect (cross).Remember using the graphing method many times only approximates the solution, so sometimes it can be unreliable.
4x – 6y = 12 4x = 6y + 12 4x – 12 = 6y 6y = 4x – 12 6 6 6 y = 2/3x – 2 slope = 2/3 y-intercept = -2 2x + 2y = 6 2y = -2x + 6 2 2 2 y = -x + 3 slope = -1/1 y-intercept = 3 Solving Systems of Equations by Graphing.To solve a system of equations graphically, graph both equations and see where they intersect. The intersection point is the solution.
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.
Use a graph to solve the system of equations below. Graph both equations on the same coordinate plane.Graph x + y = 5 using the intercepts: (5, 0) and (0, 5)Graph y = 2x – 1 using the slope-intercept method.
Locate the point where the lines intersect. From the graph, the solution appears to (2, 3).Check to be sure that (2, 3) is the solution, substitute 2 for x and 3 for y into each equation.x + y = 5 y = 2x – 12 + 3 = 5 3 = 2(2) - 1
Use a graph to solve each system of equations. If the system has no solution, write none.
Summary of Solutions of Systems of Linear Equations The lines intersect so there is one solution. y – 2x = 7 Y = 2x + 3 The lines are parallel so there are no solutions. x + 2y = 7 x = y + 4 The lines are the same so there are infinitely many solutions. -3x = 5 – y 2y = 6x + 10
NoteSome systems of equations may be very difficult to solve using the graphing method. The exact solution would be hard to determine from a graph because the coordinates are not integers. Solving a system algebraically is better than graphing when you need an accurate solution.Take for example the system of equations:3x + 2y = 12x – y = 3x = 3/8 and y = 15/4
Solving Systems of Equations Algebraically 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.
Solve this system of equations using the substitution method. Step 1 3y – 2x = 11 Y + 2x = 9 Solve one of the equations for either “x” or “y”. In this example it is easier to solve the second equation for “y”, since it only involves one step. Y = 9 – 2x
Step 2Replace the “y” value in the first equation by what “y” now equals (y = 9 – 2x).Grab the “y”value and plug it into the other equation.3y – 2x = 113(9 – 2x) – 2x = 11
Step 3Solve this new equation for “x”.3(9 – 2x) – 2x = 1127 – 6x – 2x = 1127 – 6x – 2x = 1127 – 8x = 11-8x = -16x = 2
Step 4Now that we know the “x” value(x = 2), we place it into either of the ORIGINALequations in order to solve for “y”.Pick the easierone to work with!Y + 2x = 9y + 2(2) = 9y + 4 = 9y = 5
Step 5Check:substitute x = 2 and y = 5 into BOTH ORIGINAL equations. If these answers are correct BOTH equations will be true!3y – 2x = 113(5) – 2(2) = 1115 – 4 = 11 11 = 11 True?Y + 2x = 95 + 2(2) = 95 + 4 = 99 = 9 True?
The Substitution Method • Step 1 Solve one equation for x (or y). • Step 2 Substitute the expression from Step 1 into the other equation. • Step 3 Solve for y (or x). • Step 4 Take the value of y (or x) found in Step 3 and substitute it into one of the original equations. Then solve for the other variable. • Step 5 The ordered pair of values from Steps 3 and 4 is the solution. If the system has no solution, a contradictory statement will result in either Step 3 or 4.
Use the substitution method to solve each system of equations. Check your answers.
Solving systems of equations in real-world problems. April sold 75 tickets to a school play and collected a total of $495. If the adult tickets cost $8 each and child tickets cost $5 each, how many adult tickets and how many child tickets did she sell? Solution: Let a represent the adult tickets and c represent the child tickets. Individual tickets sold equaled 75, so a + c = 75 All total April sold $495 in tickets, since adult tickets are $8 and child tickets are $5, so 8a + 5c = 495. System of Equations a + c = 75 8a + 5c == 495
Solutiona + c = 758a + 5c = 495a = 75 – c8(75 – c) + 5c = 495600 – 8c + 5c = 495600 - 3c = 495105 = 3c35 = ca + c = 75a + 35 = 75a = 40There were 40 adult tickets and 35 child tickets sold. 40 + 35 = 758(40) + 5(35) = 495320 + 175 = 495
Write a system of equations and solve.At a baseball game, Jose bought five hot dogs and three sodas for $17. At the same time, Allison bought two hot dogs and four sodas for $11. Find the cost of one hot dog and one soda.
Solving Systems of Equations Elimination method
You can use the Addition and SubtractionProperties of Equality to solve a system by the elimination method. You can add or subtract equations to eliminate (getting rid of) a variable. Step 1 5x – 6y = -32 3x + 6y = 48 Eliminate ybecause the sum of the coefficients of y is zero 5x – 6y = -32 3x + 6y = 48 8x + 0 = 16 x = 2 If you add the two equations together, the +6y and -6y cancel each other out because of the Property of Additive Inverse Addition Property of Equality Solve for x
Step 2Solve for the eliminated variable y using either of the original equations.3x + 6y = 483(2) + 6y = 486 + 6y = 486y = 42y = 7 Choose the 2nd equation Remember x = 2 Substitute 2 for x Simplify. Then solve for y.
Since x = 2 and y = 7, the solution is (2, 7) Check 5x – 6y = -32 3x + 6y = 48 5(2) – 6(7) = -32 3(2) + 6(7) = 48 10 – 42 = -32 6 + 42 = 48 -32 = -32 48 = 48 Remember, the order pair (2, 7) must make both equations true. True True
Suppose your community center sells a total of 292 tickets for a basketball game. An adult ticket cost $3. A student ticket cost $1. The sponsors collected $470 in ticket sales. Write and solve a system to find the number of each type of ticket sold.
Let a = number of adult ticketsLet s = number of student ticketstotal number of ticket total number of salesa + s = 292 3a + 1s = 470Solve by elimination (get rid of s) because the difference of the coefficients of s is zero.a + s = 2923a + s = 470-2a + 0 = -178a = 89 This is the number of adult tickets sold. That means you must subtract the two equations so, -3a – a = -470 is what you must subtract. Next Step
Solve for the eliminated variable using either of the original equations.a + s = 29289 + s = 292s = 203There were 89 adult tickets sold and 203 student tickets sold.Is the solution reasonable? The total number of tickets is 89 + 203 = 292. The total sales is $3(89) + $1(203) = $470. The solution is correct. This is the number of student tickets sold.
If you have noticed in the last few examples that to eliminate a variable its coefficients must have a sum or difference of zero. Sometime you may need to multiply one or both of the equations by a nonzero number first so that you can then add or subtract the equations to eliminate one of the variables. 2x + 5y = 17 7x + 2y = 10 2x + 5y = -22 6x – 5y = -19 -7x + y = -16 10x + 3y = 22 If you notice the systems of equations above, two of them have something in common. The third doesn’t. We can add these two equations together to eliminate the y variable. We can add these two equations together to eliminate the x variable. What are we going to do with these equations, can’t eliminate a variable the way they are written?
Multiplying One Equation Be careful when you subtract. All the signs in the equationthat isbeing subtracted change. -10x – 3y = -22 Solve by Elimination 2x + 5y = -22 10x + 3y = 22 Step 1 2x + 5y = -22 5(2x + 5y = -22) 10x + 25y = -110 10x + 3y = 22 10x + 3y = 22-(10x + 3y = 22) 0 + 22y = -132 y = -6 To prepare for eliminating x, multiply the first equation by 5. Subtract the equations to eliminate x. Start with the given system. Ask: Is one coefficient a factor of the other coefficient for the same variable? NEXT
Step 2Solve for the eliminated variable using either of the original equations. 2x + 5y = -22 Choose the first equation.2x +5(-6) = -22 Substitute -6 for y. 2x – 30 = -22Solve for x. 2x = 8 x = 4The solution is (4, -6).
Solve by elimination.-2x + 5y = -327x – 5y = 173x – 10y = -254x + 40y = 202x – 3y = 612x + y = -7 Ask: Is one coefficient a factor of the other coefficient for the same variable?
Multiplying Both Equations To eliminate a variable, you may need to multiply both equations in a system by a nonzero number. Multiply each equation by values such that when you write equivalent equations, you can then add or subtract to eliminate a variable. 4x + 2y = 14 7x + 3y = -8 In these two equations you cannot use graphing or substitution very easily. However ever if we multiply the first equation by 3 and the second by 2, we can eliminate the y variable. Find the least common multiple LCM of the coefficients of one variable, since working with smaller numbers tends to reduce the likelihood of errors. 4 x 7 = 28 2 x 3 = 6 NEXT
4X + 2Y = 14 3(4X + 2Y = 14) 12X + 6Y = 427X – 3Y = - 8 2(7X – 3Y = -8) 14X – 6Y = -16 26X + 0 = 26 26X = 26X = 1Solve for the eliminated variable y using either of the original equations.4x + 2y = 144(1) + 2y = 14 4 + 2y = 14 2y = 10y = 5 The solution is (1, 5). Add the equations to eliminate y. Start with the given system. To prepare to eliminate y, multiply the first equation by 3 and the second equation by 2.
Practice and Problem Solving Solve by elimination: 1) 2x + 5y = 17 2) 7x + 2y = 10 6x + 5y = -9 -7x + y = -16 3) 2x – 3y = 61 4) 24x + 2y = 52 2x + y = -7 6x – 3y = -36 5) y = 2x 6) 9x + 5y = 34 y = x – 1 8x – 2y = -2 You choose what method you what to use to solve question 5 thru 10. Word Problems
7) The sum of two numbers is 20. Their difference is 4. Write and solve a system of equations.8) Your school sold 456 tickets for a school play. An adult ticket cost $3.50. A student ticket cost $1. Total ticket sales equaled $1131. Let a = adult tickets sold, and s = student tickets sold. How many tickets of each were sold?9) Suppose the band sells cans of popcorn for $5 each and mixed nuts for $8 each. The band sells a total of 240 cans and makes a total of $1614. Find the number of cans of each sold. One more problem, my favorite.
10) A farmer raises chicken and cows. He has a total of 34 animals in his barnyard. His six-year son came in one day all excited saying, “Daddy, daddy, did you know all your animals have a total of 110 legs.”Write a system of equations to represent this situation.How many chickens and how many cows does the farmer have?