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Introduction to a System of Equations

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to select the solution to a system of equations. Specifically, this lesson will cover:

Table of Contents

1. What Is a System of Equations?

Simply put, a system of equations consists of at least two equations, and each equation in the system contains the same variables. Additionally, the variables must represent the same quantity.

EXAMPLE

If x in one equation represents "cost per person", but represents "number of people" in another equation, they cannot be part of the same system.

This typically only requires consideration for problems in context. Out of context, if the variables are the same, we can consider them part of the same system.

EXAMPLE

Here are two equations that make up a system:

table attributes columnalign left end attributes row cell y equals 2 x plus 4 end cell row cell y equals 3 x minus 2 end cell end table

term to know
System of Equations
Two or more equations with the same variables, considered at the same time.

2. Solutions to a System of Equations

A solution to a system of equations is a set of values for each variable in the system that satisfies all equations in the system. It is important to remember that the solution to one equation must satisfy every other equation in the system. If it doesn't satisfy all equations, it is not a solution to the system.

EXAMPLE

Consider the system of two equations from above. Test a random value for x, plug it into one of the equations, and get a y-value. This will represent a solution to that specific equation:

y equals 2 x plus 4 For the first equation, plug in a random value for x, for instance, 3, and solve
y equals 2 open parentheses 3 close parentheses plus 4 Multiply 2 and 3
y equals 6 plus 4 Add 6 and 4
y equals 10 Our solution for y when x equals 3

One solution for the first equation y equals 2 x plus 4 is (3, 10).

But is (3, 10) also a solution for the second equation y equals 3 x minus 2? Let's test it out by plugging in x equals 3 and y equals 10 into the other equation in our system:

y equals 3 x minus 2 For the second equation, plug in x equals 3 and y equals 10
10 equals 3 left parenthesis 3 right parenthesis minus 2 Multiply 3 and 3
10 equals 9 minus 2 Subtract 2 from 9
10 equals 7 This is a false statement, thus NOT a solution to y equals 3 x minus 2

As we can see, the point (3, 10) was a solution to the first equation in the system, but it wasn't a solution to the second equation in the system. Since (3, 10) does not satisfy all equations in the system, it is not a solution to the system.

big idea
Solutions to systems of equations satisfy all equations in the system. Be careful for solutions that satisfy one, but not all equations. While a solution satisfies an equation, it does not necessarily represent a solution to the entire system.

EXAMPLE

The solution to the above system is actually the point (6, 16). We can test this point by plugging in 6 for x and 16 for y in both equations.

y equals 2 x plus 4 In the first equation in the system, use the point (6, 16); Plug in 6 for x and 16 for y
16 equals 2 open parentheses 6 close parentheses plus 4 Multiply 2 and 6
16 equals 12 plus 4 Add 12 and 4
16 equals 16 This is a true statement, thus a solution to y equals 2 x plus 4

y equals 3 x minus 2 In the second equation in the system, use the point (6, 16); Plug in 6 for x and 16 for y
16 equals 3 open parentheses 6 close parentheses minus 2 Multiply 3 and 6
16 equals 18 minus 2 Subtract 2 from 18
16 equals 16 This is a true statement, thus a solution to y equals 3 x minus 2

3. Solutions to a System on a Graph

We know that solutions to a system of equations must satisfy all equations in the system. What does this look like on a graph? Let's start by examining the graph of a system of equations:

EXAMPLE

Consider the system of equations:

table attributes columnalign left end attributes row cell y equals 2 x plus 2 end cell row cell y equals 5 x minus 4 end cell end table

The graph of this system is show below:



The point of intersection represents a single coordinate point that is a solution to both equations simultaneously. This intersection point is at the point (2, 6). We can confirm algebraically that (2, 6) is a solution to all equations in the system:

y equals 2 x plus 2 In the first equation in the system, use the point (2,6); Plug in 2 for x and 6 for y
6 equals 2 open parentheses 2 close parentheses plus 2 Multiply 2 and 2
6 equals 4 plus 2 Add 4 and 2
6 equals 6 This is a true statement. (2, 6) satisfies y equals 2 x plus 2

y equals 5 x minus 4 In the second equation in the system, use the point (2,6); Plug in 2 for x and 6 for y
6 minus 5 open parentheses 2 close parentheses minus 4 Multiply 5 and 2
6 equals 10 minus 4 Subtract 4 from 10
6 equals 6 This is a true statement. (2, 6) satisfies y equals 5 x minus 4

summary
When thinking about what is a system of equations, we can define this as two or more equations with the same variables, considered at the same time. The solutions to a system of equations is the solution to all equations in the system. The solutions to a system of equations on a graph is the intersection point of the lines on the graph. There is no solution to a system of equations represented by lines that are parallel because the lines will never intersect. However, there are an infinite number of solutions to a system of equations represented by lines that are identical because the lines share all points.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
System of Equations

Two or more equations with the same variable, considered at the same time.

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