Unit 3 · Lesson 1 of 5

The coordinate plane and graphing lines

About 25 minutes · 5 steps · one idea: the points that make an equation true form its graph

You'll be able to: plot points, check whether a point is on a line, and graph a line from a table or from its two intercepts.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.

Solve \(|x - 2| = 5\).

Two cases: \(x - 2 = 5\) gives \(x = 7\), and \(x - 2 = -5\) gives \(x = -3\).

Answer: \(x = 7\) or \(x = -3\).

Evaluate \(3x - 2\) when \(x = -4\).

Put \(-4\) in parentheses: \(3(-4) - 2 = -12 - 2 = -14\).

Step 2 of 5The idea

The coordinate plane is two number lines that cross at 0. The flat one is the x-axis. The upright one is the y-axis. The crossing point is the origin, \((0, 0)\).

Every point has an address called an ordered pair \((x, y)\). The first number says how far to go across. The second says how far to go up or down.

To plot \((3, 2)\): start at the origin, go right 3, then up 2.

An equation like \(y = 2x - 1\) has many solutions: every ordered pair that makes it true. Plot all of them and you get the graph. For a linear equation the graph is a straight line, so two points are enough to draw it.

Plot \((x, y)\). Start at the origin. Move \(x\) across (right if positive, left if negative). Then move \(y\) up or down.

Is a point on the line? Put its \(x\) and \(y\) into the equation. A true statement means yes.

Intercepts. The x-intercept: set \(y = 0\) and solve. The y-intercept: set \(x = 0\) and solve.

Flat and upright lines. \(y = k\) is a horizontal line through \((0, k)\). \(x = h\) is a vertical line through \((h, 0)\).

The horizontal line \(y = 2\) and the vertical line \(x = -3\).

Why this shows up on the placement test: questions ask you to pick the graph of an equation, find its intercepts, or check a point, and every later graphing skill starts here.

Step 3 of 5Worked examples

Example 1: graph \(y = 2x - 1\) with a table

  1. Pick a few easy x-values, like \(-1, 0, 1, 2\), and evaluate \(2x - 1\) for each.
  2. \(x = -1\): \(2(-1) - 1 = -3\). \(x = 0\): \(-1\). \(x = 1\): \(1\). \(x = 2\): \(3\).
  3. So the points are \((-1, -3)\), \((0, -1)\), \((1, 1)\), \((2, 3)\). Plot them and draw one straight line through all of them. If one point is off the line, recheck that row.
  4. Is \((3, 5)\) on this line? \(2(3) - 1 = 5\). True, so yes.
The table points all sit on one line.

Example 2: graph \(2x + 3y = 6\) using intercepts

  1. x-intercept: set \(y = 0\). Then \(2x = 6\), so \(x = 3\). The point is \((3, 0)\).
  2. y-intercept: set \(x = 0\). Then \(3y = 6\), so \(y = 2\). The point is \((0, 2)\).
  3. Plot both points and draw the line through them.
  4. Check with a third point. Try \(x = -3\): \(2(-3) + 3y = 6\), so \(3y = 12\) and \(y = 4\). Is \((-3, 4)\) on your line? It should be. A third point catches mistakes.
The intercepts \((3, 0)\) and \((0, 2)\), plus the check point \((-3, 4)\).

Step 4 of 5Practice

Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.

Write the ordered pair for the point shown.

Each grid square is 1 unit.

From the origin, the point is 4 units left (x is negative) and 3 units up (y is positive).

Answer: \((-4, 3)\).

Is the point \((4, -1)\) on the line \(x - 2y = 6\)?

Put in \(x = 4\) and \(y = -1\): \(4 - 2(-1) = 4 + 2 = 6\).

\(6 = 6\) is true, so the answer is yes.

You are making a table of values for \(y = -3x + 4\). What y-value goes with \(x = -2\)?

\(y = -3(-2) + 4 = 6 + 4 = 10\).

Answer: \(10\), so the point is \((-2, 10)\).

Find the x-intercept of \(3x - 4y = 12\). Write it as a point.

Set \(y = 0\): \(3x - 0 = 12\), so \(x = 4\).

Answer: \((4, 0)\).

Find the y-intercept of \(5x + 2y = -10\). Write it as a point.

Set \(x = 0\): \(2y = -10\), so \(y = -5\).

Answer: \((0, -5)\).

Write the equation of the horizontal line through \((5, -2)\).

On a horizontal line every point has the same y-value. Here that y-value is \(-2\).

Answer: \(y = -2\).

On paper, graph \(x - 2y = 4\) using its two intercepts. Then open the answer and compare.

The line through \((4, 0)\) and \((0, -2)\).

x-intercept: set \(y = 0\), so \(x = 4\). Point \((4, 0)\).

y-intercept: set \(x = 0\), so \(-2y = 4\) and \(y = -2\). Point \((0, -2)\).

Draw the line through \((4, 0)\) and \((0, -2)\). It rises gently from left to right.

Which point is on the line \(2x - 5y = 10\)?

  1. \((0, 2)\)
  2. \((2, 0)\)
  3. \((5, 2)\)
  4. \((-5, -4)\)

(a) \(2(0) - 5(2) = -10\). Not 10.

(b) \(2(2) - 5(0) = 4\). Not 10.

(c) \(2(5) - 5(2) = 0\). Not 10.

(d) \(2(-5) - 5(-4) = -10 + 20 = 10\). This one.

Step 5 of 5Wrap-up

You can now plot points, test whether a point is on a line by substituting, and graph a line from a table or from its intercepts (set the other letter to 0).

Read or watch next: OpenStax Intermediate Algebra 2e, 3.1 Graph Linear Equations in Two Variables (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Two-variable linear equations intro (short videos). Quick rules: Lines and systems cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "Why is x = 3 a vertical line and not a horizontal one?" It knows this course and can explain things another way.