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Graph Each System of Linear Equations to Find the Solution

5.1: Solve Systems of Equations by Graphing

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    15151

    • Learning Objectives

    By the end of this section, you will be able to:

    • Determine whether an ordered pair is a solution of a system of equations
    • Solve a system of linear equations by graphing
    • Determine the number of solutions of linear system
    • Solve applications of systems of equations by graphing

    Note

    Before you get started, take this readiness quiz.

    1. For the equation \(y=\frac{2}{3}x−4\)
      ⓐ is (6,0) a solution? ⓑ is (−3,−2) a solution?
      If you missed this problem, review Exercise 2.1.1.
    2. Find the slope and y-intercept of the line 3x−y=12.
      If you missed this problem, review Exercise 4.5.7.
    3. Find the x- and y-intercepts of the line 2x−3y=12.
      If you missed this problem, review Exercise 4.3.7.

    Determine Whether an Ordered Pair is a Solution of a System of Equations

    In the section on Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation. Now we will work with systems of linear equations, two or more linear equations grouped together.

    Definition: SYSTEM OF LINEAR EQUATIONS

    When two or more linear equations are grouped together, they form a system of linear equations.

    We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.

    An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

    \[\begin{cases}{2 x+y=7} \\ {x-2 y=6}\end{cases}\]

    A linear equation in two variables, like 2x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.

    To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (x, y) that make both equations true. These are called the solutions to a system of equations.

    Definition: SolutionS OF A SYSTEM OF EQUATIONS

    Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x, y).

    To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

    Let's consider the system below:

    \[\begin{cases}{3x−y=7} \\ {x−2y=4}\end{cases}\]

    Is the ordered pair (2,−1) a solution?

    This figure begins with a sentence,

    The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.

    Let's try another ordered pair. Is the ordered pair (3, 2) a solution?

    This figure begins with the sentence,

    The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

    Exercise \(\PageIndex{1}\)

    Determine whether the ordered pair is a solution to the system: \(\begin{cases}{x−y=−1} \\ {2x−y=−5}\end{cases}\)

    1. (−2,−1)
    2. (−4,−3)
    Answer

    1.
    This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence,
    (−2, −1) does not make both equations true. (−2, −1) is not a solution.

    2.
    This figure begins with the sentence,
    (−4, −3) does not make both equations true. (−4, −3) is a solution.

    Exercise \(\PageIndex{2}\)

    Determine whether the ordered pair is a solution to the system: \(\begin{cases}{3x+y=0} \\ {x+2y=−5}\end{cases}\)

    1. (1,−3)
    2. (0,0)
    Answer
    1. yes
    2. no

    Exercise \(\PageIndex{3}\)

    Determine whether the ordered pair is a solution to the system: \(\begin{cases}{x−3y=−8} \\ {−3x−y=4}\end{cases}\)

    1. (2,−2)
    2. (−2,2)
    Answer
    1. no
    2. yes

    Solve a System of Linear Equations by Graphing

    In this chapter we will use three methods to solve a system of linear equations. The first method we'll use is graphing. The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we'll find the solution to the system.

    Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions. Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in Figure \(\PageIndex{1}\):

    This figure shows three x y-coordinate planes. The first plane shows two lines which intersect at one point. Under the graph it says,
    Figure \(\PageIndex{1}\)

    For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we'll use a different method in each section. After seeing the third method, you'll decide which method was the most convenient way to solve this system.

    Exercise \(\PageIndex{4}\): How to Solve a System of Linear Equations by Graphing

    Solve the system by graphing: \(\begin{cases}{2x+y=7} \\ {x−2y=6}\end{cases}\)

    Answer

    This table has four rows and three columns. The first column acts as the header column. The first row reads, The second row reads, The third row reads, The fourth row reads,

    Exercise \(\PageIndex{5}\)

    Solve each system by graphing: \(\begin{cases}{x−3y=−3} \\ {x+y=5}\end{cases}\)

    Answer

    (3,2)

    Exercise \(\PageIndex{6}\)

    Solve each system by graphing: \(\begin{cases}{−x+y=1} \\ {3x+2y=12}\end{cases}\)

    Answer

    (2,3)

    The steps to use to solve a system of linear equations by graphing are shown below.

    TO SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING.

    1. Graph the first equation.
    2. Graph the second equation on the same rectangular coordinate system.
    3. Determine whether the lines intersect, are parallel, or are the same line.
    4. Identify the solution to the system.
      • If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
      • If the lines are parallel, the system has no solution.
      • If the lines are the same, the system has an infinite number of solutions.

    Exercise \(\PageIndex{7}\)

    Solve the system by graphing: \(\begin{cases}{y=2x+1} \\ {y=4x−1}\end{cases}\)

    Answer

    Both of the equations in this system are in slope-intercept form, so we will use their slopes and y-intercepts to graph them. \(\begin{cases}{y=2x+1} \\ {y=4x−1}\end{cases}\)

    Find the slope and y-intercept of the
    first equation.    		 .
    Find the slope and y-intercept of the
    first equation.    		 .
    Graph the two lines.
    Determine the point of intersection. The lines intersect at (1, 3).
    .
    Check the solution in both equations. \(\begin{array}{l}{y=2 x+1} & {y = 4x - 1}\\{3\stackrel{?}{=}2 \cdot 1+1} &{3\stackrel{?}{=}4 \cdot 1-1} \\ {3=3 \checkmark}&{3=3 \checkmark} \end{array}\)
    The solution is (1, 3).

    Exercise \(\PageIndex{8}\)

    Solve the system by graphing: \(\begin{cases}{y=2x+2} \\ {y=-x−4}\end{cases}\)

    Answer

    (−2,−2)

    Exercise \(\PageIndex{9}\)

    Solve the system by graphing: \(\begin{cases}{y=3x+3} \\ {y=-x+7}\end{cases}\)

    Answer

    (1,6)

    Both equations in Exercise \(\PageIndex{7}\) were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we'll first re-write the equations into slope–intercept form.

    Exercise \(\PageIndex{10}\)

    Solve the system by graphing: \(\begin{cases}{3x+y=−1} \\ {2x+y=0}\end{cases}\)

    Answer

    We'll solve both of these equations for yy so that we can easily graph them using their slopes and y-intercepts. \(\begin{cases}{3x+y=−1} \\ {2x+y=0}\end{cases}\)

    Solve the first equation for y.

    Find the slope and y-intercept.

    Solve the second equation for y.

    Find the slope and y-intercept.

    		 \(\begin{aligned} 3 x+y &=-1 \\ y &=-3 x-1 \\ m &=-3 \\ b &=-1 \\ 2 x+y &=0 \\ y &=-2 x \\ b &=0 \end{aligned}\)
    Graph the lines. .
    Determine the point of intersection. The lines intersect at (−1, 2).
    Check the solution in both equations. \(\begin{array}{rllrll}{3x+y}&{=}&{-1} & {2x +y}&{=}&{0}\\{3(-1)+ 2}&{\stackrel{?}{=}}&{-1} &{2(-1)+2}&{\stackrel{?}{=}}&{0} \\ {-1}&{=}&{-1 \checkmark}&{0}&{=}&{0 \checkmark} \end{array}\)
    The solution is (−1, 2).

    Exercise \(\PageIndex{11}\)

    Solve each system by graphing: \(\begin{cases}{−x+y=1} \\ {2x+y=10}\end{cases}\)

    Answer

    (3,4)

    Exercise \(\PageIndex{12}\)

    Solve each system by graphing: \(\begin{cases}{ 2x+y=6} \\ {x+y=1}\end{cases}\)

    Answer

    (5,−4)

    Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We'll do this in Exercise \(\PageIndex{13}\).

    Exercise \(\PageIndex{13}\)

    Solve the system by graphing: \(\begin{cases}{x+y=2} \\ {x−y=4}\end{cases}\)

    Answer

    We will find the x- and y-intercepts of both equations and use them to graph the lines.

    .
    To find the intercepts, let x = 0 and solve
    for y, then let y = 0 and solve for x.    		 \(\begin{aligned} x+y &=2 \quad x+y=2 \\ 0+y &=2 \quad x+0=2 \\ y &=2 \quad x=2 \end{aligned}\) .
    .
    To find the intercepts, let
    x = 0 then let y = 0.    		 \begin{array}{rlr}{x-y} & {=4} &{x-y} &{= 4} \\ {0-y} & {=4} & {x-0} & {=4} \\{-y} & {=4} & {x}&{=4}\\ {y} & {=-4}\end{array}
    .
    Graph the line. This graph shows two lines intersection at point (3, -1) on an x y-coordinate plane.
    Determine the point of intersection. The lines intersect at (3, −1).
    Check the solution in both equations.

    \(\begin{array}{rllrll}{x+y}&{=}&{2} & {x-y}&{=}&{4}\\{3+(-1)}&{\stackrel{?}{=}}&{2} &{3 - (-1)}&{\stackrel{?}{=}}&{4} \\ {2}&{=}&{2 \checkmark}&{4}&{=}&{4 \checkmark} \end{array}\)

    The solution is (3, −1).

    Exercise \(\PageIndex{14}\)

    Solve each system by graphing: \(\begin{cases}{x+y=6} \\ {x−y=2}\end{cases}\)

    Answer

    (4,2)

    Exercise \(\PageIndex{15}\)

    Solve each system by graphing: \(\begin{cases}{x+y=2} \\ {x−y=-8}\end{cases}\)

    Answer

    (5,−3)

    Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.

    Exercise \(\PageIndex{17}\)

    Solve each system by graphing: \(\begin{cases}{y=−1} \\ {x+3y=6}\end{cases}\)

    Answer

    (9,−1)

    Exercise \(\PageIndex{18}\)

    Solve each system by graphing: \(\begin{cases}{x=4} \\ {3x−2y=24}\end{cases}\)

    Answer

    (4,−6)

    In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we'll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.

    Exercise \(\PageIndex{20}\)

    Solve each system by graphing: \(\begin{cases}{y=-\frac{1}{4}x+2} \\ {x+4y=-8}\end{cases}\)

    Answer

    no solution

    Exercise \(\PageIndex{21}\)

    Solve each system by graphing: \(\begin{cases}{y=3x−1} \\ {6x−2y=6}\end{cases}\)

    Answer

    no solution

    Exercise \(\PageIndex{23}\)

    Solve each system by graphing: \(\begin{cases}{y=−3x−6} \\ {6x+2y=−12}\end{cases}\)

    Answer

    infinitely many solutions

    Exercise \(\PageIndex{24}\)

    Solve each system by graphing: \(\begin{cases}{y=\frac{1}{2}x−4} \\ {2x−4y=16}\end{cases}\)

    Answer

    infinitely many solutions

    If you write the second equation in Exercise \(\PageIndex{22}\) in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept.

    When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept.

    COINCIDENT LINES

    Coincident lines have the same slope and same y-intercept.

    Determine the Number of Solutions of a Linear System

    There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.

    We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Exercise \(\PageIndex{4}\) through Exercise \(\PageIndex{16}\) all had two intersecting lines. Each system had one solution.

    A system with parallel lines, like Exercise \(\PageIndex{19}\), has no solution. What happened in Exercise \(\PageIndex{22}\)? The equations have coincident lines, and so the system had infinitely many solutions.

    We'll organize these results in Figure \(\PageIndex{2}\) below:

    This table has two columns and four rows. The first row labels each column
    Figure \(\PageIndex{2}\)

    Parallel lines have the same slope but different y-intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Exercise \(\PageIndex{19}\).

    \(\begin{array} {cc} & \begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\\ \text{The first line is in slope–intercept form.} &\text { If we solve the second equation for } y, \text { we get } \\ &x-2 y =4 \\ y = \frac{1}{2}x -3& x-2 y =-x+4 \\ &y =\frac{1}{2} x-2 \\ m=\frac{1}{2}, b=-3&m=\frac{1}{2}, b=-2 \end{array}\)

    The two lines have the same slope but different y-intercepts. They are parallel lines.

    Figure \(\PageIndex{3}\) shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.

    This table is entitled
    Figure \(\PageIndex{3}\)

    Let's take one more look at our equations in Exercise \(\PageIndex{19}\) that gave us parallel lines.

    \[\begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\)]

    When both lines were in slope-intercept form we had:

    \[y=\frac{1}{2} x-3 \quad y=\frac{1}{2} x-2\]

    Do you recognize that it is impossible to have a single ordered pair (x,y) that is a solution to both of those equations?

    We call a system of equations like this an inconsistent system. It has no solution.

    A system of equations that has at least one solution is called a consistent system.

    CONSISTENT AND INCONSISTENT SYSTEMS

    A consistent system of equations is a system of equations with at least one solution.

    An inconsistent system of equations is a system of equations with no solution.

    We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent equations, they each have their own set of solutions. Intersecting lines and parallel lines are independent.

    If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.

    INDEPENDENT AND DEPENDENT EQUATIONS

    Two equations are independent if they have different solutions.

    Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

    Let's sum this up by looking at the graphs of the three types of systems. See Figure \(\PageIndex{4}\) and Figure \(\PageIndex{5}\).

    This figure shows three x y coordinate planes in a horizontal row. The first shows two lines intersecting. The second shows two parallel lines. The third shows two coincident lines.
    Figure \(\PageIndex{4}\)
    This table has four columns and four rows. The columns are labeled,
    Figure \(\PageIndex{5}\)

    Exercise \(\PageIndex{25}\)

    Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{y=3x−1} \\ {6x−2y=12}\end{cases}\)

    Answer

    \(\begin{array}{lrrl} \text{We will compare the slopes and intercepts} & \begin{cases}{y=3x−1} \\ {6x−2y=12}\end{cases} \\ \text{of the two lines.} \\ \text{The first equation is already in} \\ \text{slope-intercept form.} \\ & {y = 3x - 1}\\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ & 6x-2y &=&12 \\ & -2y &=& -6x - 12 \\ &\frac{-2y}{-2} &=& \frac{-6x + 12}{-2}\\ &y&=&3x-6\\\\ \text{Find the slope and intercept of each line.} & y = 3x-1 & y=3x-6 \\ &m = 3 & m = 3 \\&b=-1 &b=-6 \\ \text{Since the slopes are the same andy-intercepts} \\ \text{are different, the lines are parallel.}\end{array}\)

    A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.

    Exercise \(\PageIndex{26}\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    \(\begin{cases}{y=−2x−4} \\ {4x+2y=9}\end{cases}\)

    Answer

    no solution, inconsistent, independent

    Exercise \(\PageIndex{27}\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    \(\begin{cases}{y=\frac{1}{3}x−5} \\ {x-3y=6}\end{cases}\)

    Answer

    no solution, inconsistent, independent

    Exercise \(\PageIndex{28}\)

    Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{2x+y=−3} \\ {x−5y=5}\end{cases}\)

    Answer

    \(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts} & \begin{cases}{2x+y=-3} \\ {x−5y=5}\end{cases} \\ \text{of the two lines.} \\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ &2x+y&=&-3 & x−5y&=&5\\ & y &=& -2x -3 & -5y &=&-x+5 \\ &&&&\frac{-5y}{-5} &=& \frac{-x + 5}{-5}\\ &&&&y&=&\frac{1}{5}x-1\\\\ \text{Find the slope and intercept of each line.} & y &=& -2x-3 & y&=&\frac{1}{5}x-1 \\ &m &=& -2 & m &=& \frac{1}{5} \\&b&=&-3 &b&=&-1 \\ \text{Since the slopes are the same andy-intercepts} \\ \text{are different, the lines are parallel.}\end{array}\)

    A system of equations whose graphs are intersect has 1 solution and is consistent and independent.

    Exercise \(\PageIndex{29}\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    \(\begin{cases}{3x+2y=2} \\ {2x+y=1}\end{cases}\)

    Answer

    one solution, consistent, independent

    Exercise \(\PageIndex{30}\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    \(\begin{cases}{x+4y=12} \\ {−x+y=3}\end{cases}\)

    Answer

    one solution, consistent, independent

    Exercise \(\PageIndex{31}\)

    Without graphing, determine the number of solutions and then classify the system of equations. \(\begin{cases}{3x−2y=4} \\ {y=\frac{3}{2}x−2}\end{cases}\)

    Answer

    \(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts of the two lines.}& \begin{cases}{3x−2y} &=&{4} \\ {y}&=&{\frac{3}{2}x−2}\end{cases} \\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ &3x-2y&=&4 \\ & -2y &=& -3x +4 \\ &\frac{-2y}{-2} &=& \frac{-3x + 4}{-2}\\ &y&=&\frac{3}{2}x-2\\\\ \text{Find the slope and intercept of each line.} &y&=&\frac{3}{2}x-2\\ \text{Since the equations are the same, they have the same slope} \\ \text{and samey-intercept and so the lines are coincident.}\end{array}\)

    A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.

    Exercise \(\PageIndex{32}\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    \(\begin{cases}{4x−5y=20} \\ {y=\frac{4}{5}x−4}\end{cases}\)

    Answer

    infinitely many solutions, consistent, dependent

    Exercise \(\PageIndex{33}\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    \(\begin{cases}{ −2x−4y=8} \\ {y=−\frac{1}{2}x−2}\end{cases}\)

    Answer

    infinitely many solutions, consistent, dependent

    Solve Applications of Systems of Equations by Graphing

    We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We'll modify the strategy slightly here to make it appropriate for systems of equations.

    USE A PROBLEM SOLVING STRATEGY FOR SYSTEMS OF LINEAR EQUATIONS.

    1. Read the problem. Make sure all the words and ideas are understood.
    2. Identify what we are looking for.
    3. Name what we are looking for. Choose variables to represent those quantities.
    4. Translate into a system of equations.
    5. Solve the system of equations using good algebra techniques.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

    Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

    Exercise \(\PageIndex{34}\)

    Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?

    Answer

    Step 1. Read the problem.

    Step 2. Identify what we are looking for.

    We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.

    Step 3. Name what we are looking for. Choose variables to represent those quantities.

      Let f= number of quarts of fruit juice.
    c= number of quarts of club soda

    Step 4. Translate into a system of equations.

    This figure shows sentences converted into equations. The first sentence reads,

    We now have the system. \(\begin{cases}{ f+c=10} \\ {f=4c}\end{cases}\)

    Step 5. Solve the system of equations using good algebra techniques.

    This figure shows two equations and their graph. The first equation is f = 4c where b = 4 and b = 0. The second equation is f + c = 10. f = negative c +10 where b = negative 1 and b = 10. The x y coordinate plane shows a graph of these two lines which intersect at (2, 8).

    The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.

    Step 6. Check the answer in the problem and make sure it makes sense.

    Does this make sense in the problem?

    Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.

    Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.

    Step 7. Answer the question with a complete sentence.

    Sondra needs 8 quarts of fruit juice and 2 quarts of soda.

    Exercise \(\PageIndex{35}\)

    Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

    Answer

    Manny needs 3 quarts juice concentrate and 9 quarts water.

    Exercise \(\PageIndex{36}\)

    Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

    Answer

    Alisha needs 15 ounces of coffee and 3 ounces of milk.

    Key Concepts

    Glossary

    coincident lines
    Coincident lines are lines that have the same slope and same y-intercept.
    consistent system
    A consistent system of equations is a system of equations with at least one solution.
    dependent equations
    Two equations are dependent if all the solutions of one equation are also solutions of the other equation.
    inconsistent system
    An inconsistent system of equations is a system of equations with no solution.
    independent equations
    Two equations are independent if they have different solutions.
    solutions of a system of equations
    Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x, y).
    system of linear equations
    When two or more linear equations are grouped together, they form a system of linear equations.

    Graph Each System of Linear Equations to Find the Solution

    Source: https://math.libretexts.org/Bookshelves/Algebra/Book%3A_Elementary_Algebra_(OpenStax)/05%3A_Systems_of_Linear_Equations/5.01%3A_Solve_Systems_of_Equations_by_Graphing