Two-Variable Linear Equations

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A two-variable linear equation, like 2x + y = 5, has two variables and makes a straight line when you graph it. Its solutions are ordered pairs (x, y) that make the equation true, such as (1, 3). To graph it, find a few solutions, plot them, and draw a line through them.

Before this lesson: You should know how to plot points on the coordinate plane, solve two-step equations, and solve literal equations for one variable.

What Is a Two-Variable Linear Equation?

An equation like x + 3 = 7 has one variable and one answer. An equation like x + y = 7 has two variables. Now many pairs of numbers work: 3 and 4, 5 and 2, 10 and -3, and so on.

Each pair that works is a solution. We write it as an ordered pair (x, y), with x first. A two-variable linear equation has infinitely many solutions.

When you plot all the solutions, they line up in a straight line. That is why the equation is called linear. Every point on the line is a solution, and every solution is a point on the line.

The line 2x + y = 5 on a grid with x from −3 to 7 and y from −5 to 5. Solutions (0, 5), (1, 3), (2, 1) and (3, −1) are marked with closed dots on the line. The point (3, 3) is marked with an open orange circle off the line and labeled not a solution. A table beside the graph lists x 0, 1, 2, 3 and y 5, 3, 1, −1.
Every solution of 2x + y = 5 is a point on the line. The point (3, 3) is off the line, so it is not a solution.

The Rule: How to Find and Check Solutions

A two-variable linear equation can be written as Ax + By = C, where A, B and C are numbers and A and B are not both 0. The variables have no exponents, are not multiplied together, and are not in a denominator.

To check an ordered pair

  1. Put the first number in for x and the second number in for y.
  2. Simplify both sides.
  3. If the two sides are equal, the pair is a solution. If not, it is not.

To find a solution and graph the line

  1. Pick a value for x. Small numbers like 0, 1 and 2 are easiest.
  2. Put it in the equation and solve for y.
  3. Repeat to get at least three points. Make a table.
  4. Plot the points. Draw a straight line through them, with arrows on both ends.

Use three points, not two. If the three points don’t line up, one of them has a mistake.

Two-Variable Linear Equations Examples

Example 1: Check an ordered pair

Is (3, -1) a solution of 2x + y = 5? Is (3, 3)?

\begin{aligned}
(3, -1): \quad 2(3) + (-1) &= 6 - 1 = 5 \quad \checkmark \\
(3, 3): \quad 2(3) + 3 &= 6 + 3 = 9 \ne 5
\end{aligned}

Answer: (3, -1) is a solution. (3, 3) is not. In the graph above, (3, -1) is on the line and (3, 3) is off it.

Example 2: Find a missing value

Find the solution of 3x - 2y = 6 that has x = 4.

\begin{aligned}
3(4) - 2y &= 6 \\
12 - 2y &= 6 \\
-2y &= -6 \\
y &= 3
\end{aligned}

Check: 3(4) - 2(3) = 12 - 6 = 6. It works.

Answer: (4, 3).

Example 3: Make a table and graph

Make a table of values for 2x + y = 5 and graph it.

First, solve for y: subtract 2x from both sides to get y = -2x + 5. Now pick x-values.

xy = -2x + 5(x, y)
0-2(0) + 5 = 5(0, 5)
1-2(1) + 5 = 3(1, 3)
2-2(2) + 5 = 1(2, 1)
3-2(3) + 5 = -1(3, -1)

Answer: Plot the four points and draw the line through them. It is the blue line in the graph above.

Example 4: Solve for y first

Make a table for 4x + 2y = 8 using x = -1, 0, 1 and 2.

\begin{aligned}
4x + 2y &= 8 \\
2y &= -4x + 8 \\
y &= -2x + 4
\end{aligned}

Now put in each x. Use parentheses for the negative: -2(-1) + 4 = 2 + 4 = 6.

x-1012
y6420

Answer: The solutions are (-1, 6), (0, 4), (1, 2) and (2, 0).

Example 5: Word problem

A school play sells adult tickets for $4 and student tickets for $2. A family spends exactly $20. Write an equation, and list every possible number of adult tickets a and student tickets s.

Each adult ticket adds $4, and each student ticket adds $2: 4a + 2s = 20.

Tickets come in whole numbers, so try a = 0, 1, 2, and so on. For a = 1: 4(1) + 2s = 20, so 2s = 16 and s = 8.

a (adult)012345
s (student)1086420

Answer: The family could buy 0 and 10, 1 and 8, 2 and 6, 3 and 4, 4 and 2, or 5 and 0 (adult and student tickets).

Here only whole-number points make sense, so the graph is a set of dots, not a full line. This is the domain idea from Unit 3.

Common Mistakes

MistakeExample of the errorFix
Swapping x and yChecking (3, -1) with x = -1 and y = 3The first number is always x.
Dropping a negative sign-2(-1) + 5 = 3Use parentheses: -2(-1) + 5 = 2 + 5 = 7.
Thinking there is only one answerStopping after finding (1, 3)A linear equation in two variables has infinitely many solutions.
Graphing from only two pointsTwo points with one mistake still make a line, just the wrong onePlot a third point to check.
Stopping the line at the pointsDrawing a segment from (0, 5) to (3, −1)The solutions go on forever. Draw arrows on both ends.

Linear or Not Linear?

An equation is linear when each variable has an exponent of 1 and no variables are multiplied or divided by each other.

EquationLinear?Why
3x + 2y = 7YesBoth variables have exponent 1
y = \frac{1}{2}x - 3YesFractions as coefficients are fine
x = 4YesA vertical line (it is not a function)
y = x^2Nox is squared
xy = 6NoThe variables are multiplied
y = \frac{4}{x}Nox is in a denominator

Next you will learn faster ways to graph lines, using x-intercepts and y-intercepts and slope-intercept form.

Practice Problems

  1. Is (2, 1) a solution of x + 3y = 5?
  2. Is (-1, 4) a solution of 3x + y = 2?
  3. Find y so that (3, y) is a solution of 2x + y = 9.
  4. Find x so that (x, -2) is a solution of x - 4y = 10.
  5. Find the solution of 5x + 2y = 12 that has x = 4.
  6. Make a table for y = 3x - 1 using x = -1, 0, 1 and 2.
  7. Solve 6x + 3y = 9 for y. Then find the solution with x = 2.
  8. Which are linear? y = 5x - 2, y = x^2 + 1, 2x - 7y = 0, xy = 12
  9. Find k if (k, 4) is a solution of 2x - y = 6.
  10. Pens cost $3 and notebooks cost $5. You spend exactly $30. Write an equation and list every whole-number solution.
Show answers
  1. Yes. 2 + 3(1) = 5.
  2. No. 3(-1) + 4 = 1, not 2.
  3. 2(3) + y = 9, so y = 3. The pair is (3, 3).
  4. x - 4(-2) = 10, so x + 8 = 10 and x = 2. The pair is (2, -2).
  5. 5(4) + 2y = 12, so 2y = -8 and y = -4. The solution is (4, -4).
  6. (-1, -4), (0, -1), (1, 2), (2, 5).
  7. 3y = -6x + 9, so y = -2x + 3. When x = 2: y = -1, so (2, -1).
  8. y = 5x - 2 and 2x - 7y = 0 are linear. y = x^2 + 1 has a square, and xy = 12 multiplies the variables.
  9. 2k - 4 = 6, so 2k = 10 and k = 5.
  10. 3p + 5n = 30. The solutions (p, n) are (10, 0), (5, 3) and (0, 6).

Two-Variable Linear Equations FAQ

What is a linear equation in two variables?

It is an equation like Ax + By = C with two variables, each with exponent 1. Its graph is a straight line.

How many solutions does a two-variable linear equation have?

Infinitely many. Every point on its line is a solution.

How do you check if an ordered pair is a solution?

Put the x-value and y-value into the equation. If both sides come out equal, the pair is a solution.

How do you graph a linear equation in two variables?

Find at least three solutions with a table, plot them, and draw a straight line through them. Solving for y first makes the table easier.

Summary

  • A two-variable linear equation, like 2x + y = 5, has a straight-line graph.
  • A solution is an ordered pair (x, y) that makes the equation true.
  • There are infinitely many solutions, one for every point on the line.
  • To find solutions, pick x-values and solve for y. A table keeps them organized.
  • Plot at least three points and draw the line with arrows.

Previous lesson: Key Features of Functions

Next lesson: X-Intercepts and Y-Intercepts

Back to the Algebra 1 Index

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