Point-Slope Form Word Problems, Solved

Word problems look harder than they are. Almost every one gives you a starting value and a steady rate, which are just a point and a slope in disguise. Once you spot them, point-slope form writes the equation in one step.

The Two Things to Hunt For

In any point-slope word problem, look for these two pieces:

• The point: one known snapshot, an input paired with its output. Often the starting value at time zero.
• The slope: the constant rate of change, the per-unit amount something rises or falls.

Then drop them into y − y₁ = m(x − x₁).

Example 1: A Gym Membership

A gym charges a $30 signup fee plus $20 per month. At month 0 the cost is $30, so the point is (0, 30). The rate is $20 per month, so the slope is 20. That gives y − 30 = 20(x − 0), which tidies to y = 20x + 30. After 6 months the cost is 20(6) + 30 = $150.

Example 2: A Draining Tank

A tank holds 500 litres and drains 25 litres per minute. At minute 0 it holds 500, so the point is (0, 500). It falls 25 per minute, so the slope is −25. The model is y − 500 = −25(x − 0), or y = −25x + 500. The tank is empty when y = 0, at minute 20.

Example 3: A Known Reading Mid-Way

A plant is 12 cm tall after 3 weeks and grows 2 cm per week. Here the point isn’t at time zero, it’s (3, 12). The slope is 2. So y − 12 = 2(x − 3). This is exactly where point-slope shines, because you don’t need the starting height to write it.

Turn It Into a Prediction

Once you have the equation, you can answer any “how much after N units” question by substituting. Convert to slope-intercept form if you want to read the starting value directly, or check your model with the calculator. For more practice, try the worked examples.