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Slope-Intercept Form

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

A nonvertical line has the form y = mx + b. Its slope m is the change in output divided by the corresponding change in input; its intercept b is the output when x = 0. To construct a rule from data, first find the rate, then substitute a known point to find the starting value. Keep subtraction order consistent in numerator and denominator. A slope of -3 liters per minute means the volume decreases by 3 liters whenever time increases by one minute; it does not mean the volume is always -3 liters. A graph is a collection of points satisfying the rule. To draw it, label axes and units, plot the intercept if it is in the relevant domain, and use the slope to locate a second point. Two distinct points determine a line. To interpret a graph, distinguish its height (a value) from its steepness (a rate). Context can restrict the line to a segment: an emptying tank does not continue to a negative physical volume. An intercept outside the observed time interval is an extrapolation that needs justification.

What you'll learn

Why it matters: Linear models can compare rates, predict costs, or describe a tank emptying at a constant rate. The units explain what the slope measures.

Worked example

Problem. A tank contains 34 liters at minute 2 and 22 liters at minute 6. It drains at a constant rate. Find V(t), its initial volume, and the time when it becomes empty under this model.

  1. The rate is (22 - 34)/(6 - 2) = -12/4 = -3 liters per minute.
  2. Substitute (2, 34) into V = -3t + b: 34 = -6 + b, so b = 40.
  3. The rule is V(t) = 40 - 3t. Its intercept means 40 liters at t = 0 under the constant-rate assumption.
  4. To find when the tank is empty, set V(t) = 0: 40 - 3t = 0 gives t = 40/3 minutes.
  5. The physical model is restricted to 0 <= t <= 40/3. Its graph runs from (0, 40) to (40/3, 0), with time horizontal and volume vertical.

Answer: V(t) = 40 - 3t; initially 40 liters; empty at t = 40/3 minutes.

Lesson preview: this page shows 5 practice items. The interactive activity contains 32 practice items and 17 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. A linear function passes through (2, 11) and (5, 20). What is its y-intercept?

Show solution
  1. Slope is (20 - 11)/(5 - 2) = 3.
  2. Using (2, 11), solve 11 = 3(2) + b, so b = 5.

Answer: 5

2. A tank starts with 50 liters and loses 4 liters each minute. After how many minutes will 18 liters remain?

Show solution
  1. The model is V = 50 - 4t.
  2. Set the output to 18: 18 = 50 - 4t.
  3. Then 4t = 32, so t = 8 minutes.

Answer: 8

3. A student sees the point (3, 14) on a line and concludes its slope is 14/3. What additional fact would justify that calculation?

Choices: The line also passes through (0, 0). · The y-coordinate is positive. · The line has a nonzero y-intercept. · The x-coordinate is a whole number.

Show solution
  1. Slope compares two points: change in y divided by change in x.
  2. The ratio 14/3 equals (14 - 0)/(3 - 0), which uses (0, 0) as the other point.
  3. Without that point or another known point, one point does not determine a slope.

Answer: The line also passes through (0, 0).

4. For 0 <= t <= 10, A(t) = 20 + 3t and B(t) = 35 + t. Which statement is true?

Choices: A starts lower, grows faster, and equals B at t = 7.5. · A starts higher and grows slower. · B grows faster because 35 is larger than 20. · A and B never have equal outputs.

Show solution
  1. The initial outputs are 20 and 35, so A starts lower.
  2. The rates are 3 and 1, so A grows faster.
  3. Solve 20 + 3t = 35 + t: 2t = 15, so t = 7.5, inside the given interval.

Answer: A starts lower, grows faster, and equals B at t = 7.5.

5. In y = -3x + 4, what is the slope?

Choices: -3 · 3 · 4 · -4

Show solution
  1. In y = mx + b, m is the slope.
  2. Here m = -3.

Answer: -3

Explain and apply

Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.

Build and critique a graph

A taxi charges $4 to start and $2 per mile. Write C(d), draw and label its graph for trips from 0 to 8 miles, and mark a 5-mile trip. A student connects the graph onward to negative miles. Explain which part describes the taxi model and why.

Review rubric and solution
  • Write C(d) = 4 + 2d with 0 <= d <= 8.
  • Label miles horizontally and dollars vertically.
  • Plot the segment from (0, 4) to (8, 20) and mark (5, 14).
  • Explain that negative trip distances are outside the model's domain.
  1. C(d) = 4 + 2d for 0 <= d <= 8.
  2. The graph is a line segment with endpoints (0, 4) and (8, 20); the point (5, 14) represents a 5-mile, $14 trip.
  3. The entire algebraic line includes negative inputs, but only the stated nonnegative-distance segment describes these trips.

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