Solve Systems of Equations by Graphing

Determine whether an ordered pair is a solution of a system of equations

A linear equation in two variables, like $2x + y = 7$, has infinitely many solutions — its graph is a line, and every point on that line is a solution. Now we will work with systems of linear equations, two or more linear equations grouped together.

We will focus our work here on systems of two linear equations in two unknowns. An example of a system of two linear equations is shown below. A brace shows that the two equations are grouped together to form a system:

To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations at once — the ordered pairs $(x, y)$ that make both equations true.

To determine whether 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.

Example. Determine whether the ordered pair is a solution to the system $\left\{\begin{array}{l} x - y = -1 \\ 2x - y = -5 \end{array}\right.$: (a) $(-2, -1)$, (b) $(-4, -3)$.

(a) Substitute $x = -2$ and $y = -1$ into both equations.

The ordered pair makes the first equation true but not the second, so $(-2, -1)$ does not make both equations true — it is not a solution to the system.

(b) Substitute $x = -4$ and $y = -3$ into both equations.

The ordered pair makes both equations true, so $(-4, -3)$ is a solution to the system.

Solve a system of linear equations by graphing

The graph of a linear equation is a line, and every point on that line is a solution of the equation. For a system of two equations, we graph two lines on the same coordinate plane. Then we can see all the points that are solutions to each equation — and by finding what the two lines have in common, we find the solution to the system.

Two lines in the same plane must either intersect, be parallel, or be the same line, so there are three possible outcomes when we graph a system. We’ll explore all three later in this section — for now, we’ll focus on systems where the lines intersect in exactly one point.

Example. Solve the system by graphing: $\left\{\begin{array}{l} y = 2x + 1 \\ y = 4x - 1 \end{array}\right.$

Both equations are already in slope–intercept form:

We graph both lines on the same coordinate plane using their slopes and $y$-intercepts.

The lines intersect at $(1, 3)$. Check the solution in both equations:

The solution is $(1, 3)$.

When equations are given in standard form, it is often more convenient to graph them using their $x$- and $y$-intercepts.

Example. Solve the system by graphing: $\left\{\begin{array}{l} x + y = 2 \\ x - y = 4 \end{array}\right.$

We find the $x$- and $y$-intercepts of each line.

The lines intersect at $(3, -1)$. Check the solution in both equations:

The solution is $(3, -1)$.

If one of the equations in a system has only one variable — like $x = 4$ or $y = 6$ — its graph is a vertical or horizontal line. You can still graph it on the same plane and find where it crosses the other line exactly the way we’ve been doing.

Determine the number of solutions of a linear system

We’ve seen that two lines in a plane can intersect, be parallel, or be the same line. This means a system of two linear equations can have exactly one solution, no solution, or infinitely many solutions.

We can tell which case we’re in without graphing, just by comparing the slopes and intercepts of the two lines. Write each equation in slope–intercept form ($y = mx + b$) and compare:

A system of equations that has at least one solution is called a consistent system; a system with no solution is called an inconsistent system.

We also categorize the equations in a system as independent or dependent. If two equations are independent equations, they each have their own set of solutions; intersecting lines and parallel lines are both independent. If two equations are dependent, every solution of one equation is also a solution of the other — graphing two dependent equations gives coincident lines.

Example. Without graphing, determine the number of solutions and then classify the system of equations: $\left\{\begin{array}{l} 2x + y = -3 \\ x - 5y = 5 \end{array}\right.$

We compare the slope and intercept of each line by writing both equations in slope–intercept form. The first equation is already close to that form:

Solve the second equation for $y$:

so $m = \tfrac{1}{5}$, $b = -1$. The slopes are different, so the lines intersect. A system of equations whose graphs intersect has one solution and is consistent and independent.

Solve applications of systems of equations by graphing

We use the same problem-solving strategy for applications of systems of equations that we’ve used before, adapted to set up two equations instead of one.

Example. 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?

Let $f =$ the number of quarts of fruit juice and $c =$ the number of quarts of club soda. The total is $10$ quarts, and the fruit juice amount is $4$ times the club soda amount:

Graphing both equations (with $c$ on the horizontal axis and $f$ on the vertical axis) shows where they intersect:

The point of intersection is $(2, 8)$ — that is, $c = 2$ and $f = 8$. Check: the fruit juice amount, $8$, is indeed $4$ times the club soda amount, $2$; and $8 + 2 = 10$ quarts total, as required. Sondra needs $8$ quarts of fruit juice and $2$ quarts of club soda.

Key terms

system of linear equations — two or more linear equations grouped together. solution of a system of equations — an ordered pair $(x, y)$ that makes all the equations in the system true. consistent system — a system of equations with at least one solution. inconsistent system — a system of equations with no solution. independent equations — two equations with different solutions (intersecting or parallel lines). dependent equations — two equations whose solutions are identical (coincident lines).

This section is adapted from Elementary Algebra 2e, Section 5.1: Solve Systems of Equations by Graphing by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the two-line coordinate-plane graphs (intersecting, parallel, and coincident cases, and the punch-mixture application) as accessible inline SVGs and the slope/intercept comparisons as markdown tables; omitted the Be Prepared quiz, Media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.