Math Core

A single linear equation like $y = 2x - 1$ has infinitely many solutions: every point on its line. Real problems often come with two conditions that must hold at the same time, such as a budget and a head count. This unit is about finding the values that satisfy both, and the most visual way to start is to graph.

Systems and their solutions

A system of linear equations is a set of two or more linear equations in the same variables. You write the equations stacked, often with a brace:

\begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases}

Definition

Solution of a system

A solution of a system of two equations in $x$ and $y$ is an ordered pair $(x, y)$ that makes both equations true at once. On a graph, it is a point that lies on both lines: a point of intersection.

Every point on the first line satisfies the first equation, and every point on the second line satisfies the second. Only a point on both lines satisfies both, which is why the intersection is the answer.

Checking a possible solution

You can test an ordered pair without drawing anything. Substitute it into each equation. It is a solution only if every equation comes out true.

Worked example: Testing ordered pairs

Decide whether $(2, 3)$ and $(4, 1)$ are solutions of the system $y = 2x - 1$ and $x + y = 5$ .

For $(2, 3)$ :

$y = 2x - 1$ : $2(2) - 1 = 3$ , and $y = 3$ . True.
$x + y = 5$ : $2 + 3 = 5$ . True.

Both equations hold, so $(2, 3)$ is a solution.

For $(4, 1)$ : the second equation works, since $4 + 1 = 5$ . But in the first, $2(4) - 1 = 7$ , not $1$ . One false equation is enough: $(4, 1)$ is not a solution. It lies on the second line but not the first.

Solving a system by graphing

Solving by graphing

Write each equation in slope-intercept form, $y = mx + b$ , if it isn't already.
Graph both lines on the same coordinate plane.
Locate the point where the lines cross and read its coordinates.
Check the point in both original equations.

Step 4 matters more here than with any other method. A graph is only as precise as your drawing, so checking is how you know you read the point correctly.

Worked example: Two lines in slope-intercept form

Solve by graphing.

\begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases}

The first line has $y$ -intercept $-1$ and slope $2$ : start at $(0, -1)$ , then go up $2$ and right $1$ . The second has $y$ -intercept $5$ and slope $-1$ : start at $(0, 5)$ , then go down $1$ and right $1$ .

The lines y = 2x − 1 and y = −x + 5 intersect at (2, 3).Open in grapher →

The lines cross at $(2, 3)$ .

Check: $2(2) - 1 = 3$ and $-2 + 5 = 3$ . Both give $y = 3$ , so the solution is $(2, 3)$ .

When an equation is in standard form, $Ax + By = C$ , solve it for $y$ first. (You could also plot its two intercepts, as you did in the linear functions unit.)

Worked example: Starting from standard form

Solve by graphing.

\begin{cases} 2x + y = 1 \\ x - y = 5 \end{cases}

Rewrite each equation:

\begin{aligned} 2x + y = 1 \quad &\Longrightarrow \quad y = -2x + 1 \\ x - y = 5 \quad &\Longrightarrow \quad -y = -x + 5 \quad \Longrightarrow \quad y = x - 5 \end{aligned}

The lines y = −2x + 1 and y = x − 5 intersect at (2, −3).Open in grapher →

The lines cross at $(2, -3)$ .

Check in the original equations: $2(2) + (-3) = 1$ and $2 - (-3) = 5$ . The solution is $(2, -3)$ .

One, none, or infinitely many

Two lines in a plane can relate in only three ways, so a linear system has exactly one of three kinds of solution sets.

the lines	slopes and intercepts	number of solutions	name
intersect once	different slopes	exactly one	consistent, independent
are parallel	same slope, different $y$ -intercepts	none	inconsistent
are the same line	same slope, same $y$ -intercept	infinitely many	consistent, dependent

A system with at least one solution is called consistent; one with no solution is inconsistent.

y = 2x + 3 and y = 2x − 2 have the same slope and different intercepts. They never meet, so the system has no solution.Open in grapher →

Parallel lines rise at the same rate, so the vertical gap between them (here, $5$ units) never closes. For the same-line case, the two equations may look different but describe identical lines; every point on that line solves both.

Worked example: Classifying without graphing

How many solutions does each system have?

(a) $y = 3x - 2$ and $6x - 2y = 4$

(b) $y = 3x - 2$ and $-9x + 3y = 6$

(c) $y = 3x - 2$ and $y = -3x - 2$

Put every equation in slope-intercept form and compare.

(a) $6x - 2y = 4$ gives $-2y = -6x + 4$ , so $y = 3x - 2$ . That's the same line as the first equation: infinitely many solutions.

(b) $-9x + 3y = 6$ gives $3y = 9x + 6$ , so $y = 3x + 2$ . Same slope $3$ , different intercepts ( $-2$ and $2$ ): parallel lines, no solution.

(c) The slopes $3$ and $-3$ are different, so the lines cross exactly once: one solution. (They share the $y$ -intercept, so that solution is $(0, -2)$ .)

Common mistake

Don't decide "no solution" just because two equations look different, or "one solution" just because they look alike. Always rewrite both in $y = mx + b$ form. In part (a) above, $6x - 2y = 4$ looked nothing like $y = 3x - 2$ , yet it is the same line.

The limits of graphing

Graphing shows you the whole picture: whether there is a solution and roughly where. But it struggles when the intersection isn't at whole-number coordinates. The lines $y = x + 1$ and $y = -2x + 3$ cross at $\left(\tfrac{2}{3}, \tfrac{5}{3}\right)$ . On a graph, you would see a point near $(0.7, 1.7)$ and have no way to be sure of its exact value.

The intersection is not at grid-line coordinates, so a graph gives only an estimate.Open in grapher →

That's why the next two lessons develop algebraic methods, substitution and elimination, which give exact answers every time. Graphing remains the best way to see what a system means and to catch an answer that doesn't make sense.

Tip

Before solving any system, compare slopes. Different slopes guarantee exactly one solution, so if an algebraic method later tells you "no solution," you know you made an arithmetic slip.

Practice

Practice 1

Is $(3, -1)$ a solution of the system $x + y = 2$ and $2x - y = 7$ ?

Practice 2

Solve by graphing: $y = x + 2$ and $y = -2x + 8$ .

Enter a point like (2, -3)

Practice 3

Solve by graphing: $y = -x + 1$ and $y = \dfrac{1}{2}x - 5$ .

Enter a point like (2, -3)

Practice 4

Solve by graphing: $x - y = 4$ and $x + 2y = -2$ .

Enter a point like (2, -3)

Practice 5

How many solutions does the system $y = -4x + 3$ and $8x + 2y = 6$ have?

Practice 6

How many solutions does the system $2x - 3y = 6$ and $y = \dfrac{2}{3}x + 1$ have?

Practice 7

For what value of $k$ does the system $y = kx + 5$ and $y = 3x - 1$ have no solution?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Phone plan A costs $20 per month plus $5 per gigabyte of data. Plan B costs $10 per gigabyte with no monthly fee. Let $x$ be the gigabytes used and $y$ the monthly cost. Graph $y = 5x + 20$ and $y = 10x$ and find the point where the plans cost the same. Give your answer as $(x, y)$ .

Enter a point like (2, -3)