A system of equations is a group of two or more equations containing the same variables. In a system of equations, there is more than one unknown since the equations contain more than one variable. We can use these systems to solve for all of the variables, or unknown quantities, in the system. You can see some examples appearing here:
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Since a system of equations is a set of equations, we can also represent the system graphically by graphing all of the equations in the system on the same graph. For example, consider the following system.
The image shows this system represented graphically.
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Notice that the intersection point of the equations on the graph is a point that satisfies both of the equations in the system. We call that point a solution to the system. The solution to a system of equations consists of values of the variables that make all of the equations in the system true. We see that a solution to the system in our example is x = 1 and y = 2, which can also be written as the ordered pair (1, 2).
There are multiple ways of finding solutions to systems of equations, but those strategies are for another lesson. In this lesson, we want to look at applications of systems of equations and how to set up a system of equations that we can use to solve a problem.
Let's suppose you want to know the win/loss record of your school's basketball team. You know they played 24 games during the season, and you also know that they won four more games than they lost. We're looking for the number of wins and the number of losses, so we have two unknowns. Hmmm, more than one unknown, that should ring a bell!
Systems of equations are used to solve applications when there is more than one unknown and there's enough information to set up equations in those unknowns. In general, if there are n unknowns, we need enough information to set up n equations in those unknowns. When we have those two things, setting up a system of equations is a good way to begin to tackle the problem.
Finding your team's win/loss record involves finding more than one unknown, and we are given information to set up equations in those unknowns, so let's go ahead and set up a system of equations to represent this problem.
The first thing we want to do is represent our unknowns using variables. Let's let w = number of wins and l = number of losses. We're given that the team played 24 games total. We know that the number of wins plus the number of losses has to equal the total number of games played. Therefore,
We have our first equation.
Since there are two unknowns, we know we want one more equation. We're told that the team won four more games than they lost. This tells us that the number of losses plus four would give the number of wins. Putting that in equation form, we have that
We have our second equation, so we have our system of equations.
The image shows this system represented graphically.
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We see that the intersection point is (14, 10). Therefore, the solution to the system of equations is w = 14 and l = 10. We have the answer to our problem. Your school's basketball team won 14 games and lost 10 games. Not a bad season!
Let's consider one more example. Suppose you want to find three numbers given the following information:
We see there are three unknowns, so we need three equations. We start by naming the unknowns with variables. Let x = the first number, y = the second number, and z = the third number. If we add all the numbers together, we get 50. Therefore, x + y + z = 50. We have one equation.
The next fact says that two times the first number, or 2x, plus the second number, y, is equal to the third number plus 22, or z + 22. Putting this all in equation form, we have 2x + y = z + 22. We have two equations. We just need one more!
The last fact gives that doubling the sum of the first and second number, or 2(x + y), gives three times the third number, or 3y. Putting this in equation form, we have: 2(x + y) = 3y. We have three equations, so we have our system.
x + y + z = 50
2x + y = z + 22
2(x + y) = 3y
The solution to this system is x = 12, y = 18, z = 20, because if we plug these values in for the variables, they make all of the equations in our system true.
All right, let's now take a moment or two to review. As we learned, a system of equations is a group of two or more equations with the same variables. A solution of a system of equations consists of the values of the variables, or unknown quantities, that make all of the equations in the system true. This is also the intersection point of the equations when we graph them all on the same graph. This is, as we also learned, a point that satisfies both of the equations in the system, known as the solution.
Systems of equations can be used in applications that have more than one unknown and enough information to set up equations in those unknowns. Remember that if there are n unknowns, we need enough information to set up n equations. Recognizing when to use systems of equations and being familiar with how to set them up makes solving applications with more than one unknown much easier.
This activity will help you assess your knowledge regarding systems of equations and their applications.
For this activity, carefully read and select the best answer for each of the given questions. To do this, print or copy this page on a blank paper and circle the letter of your answer.
1.) Which of the following statements is TRUE about the two equations x + 3y = 4 and y - 4x = 23?
A. Each equation signifies lines that are parallel with each other.
B. The given are classified as systems of equations.
C. The solutions for x and y are -3 and 5 respectively.
D. Both B and C
2.) Suppose you have a system of four equations with four unknowns. Which of the following statements regarding these equations is FALSE?
A. Any intersection in the graphical representation of the equations is a solution.
B. All unknown variables can be calculated using four equations.
C. The system of four equations can be reduced to three.
D. Not enough information.
3.) What is the solution for the following group of equations?
y-2x=2
x-4y-z=8
4x+6y+3z=-6
A. x = -6, y = -10, z = 26
B. x = 3, y = -5, z = 26
C. x = -2, y = -10, z = -26
D. x = 3, y = 10, z = 26
4.) Which of the following correctly shows the graph and solution of the given equations?
6x-3y=-3
x-y=-2
A.
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B.
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C.
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D.
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5.) If the solution to a system of equations is x = 2, y = -12, and z = 6, then the ordered solution would be __________.
A. (1, 3, -6)
B. (2, -12, 6)
C. (-12, 6, 2)
D. Both A and B
1.) B
2.) C
3.) A
4.) D
5.) B
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