Linear Models, Tables, and Predictions

A linear model describes a constant change in the output per unit input. Distinguish its slope from its starting value and distinguish a model prediction from a measured observation.

2 worked examples and 3 practice questions, with explained answers. No sign-in required.

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A reliable method

  1. Identify the input and output units.
  2. Compute change in output divided by change in input.
  3. Use one data point to determine the intercept.
  4. Interpret predictions only in a sensible range; real data may not follow the model exactly.

Example 1

Write an equation for the line through \(\left( 0, \ 8\right)\) and \(\left( 4, \ 20\right)\).

  1. Compute the slope: \(m=3\).
  2. Use point-slope form: \(y-(8)=3(x-(0))\).
  3. Expand and isolate y: \(y=3 x + 8\).
  4. Substitute both given points to verify the equation.

Answer: \(y=3 x + 8\)

Example 2

Find the slope between \(\left( 2, \ 14\right)\) and \(\left( 6, \ 30\right)\).

  1. Subtract coordinates in a consistent order: \(\Delta y=16\), \(\Delta x=4\).
  2. Use slope = rise/run. \(m=\frac{16}{4}=4\).
  3. A vertical line has undefined slope; a horizontal line has slope 0.

Answer: \(4\)

Common mistakes

  • A positive association alone does not prove causation.
  • An intercept can be a mathematical parameter without a sensible real-world interpretation at input zero.

Practice without the answer first

  1. A linear cost model is \(C=5+3*x\) dollars for x items. Predict the cost for 12 items.
  2. Evaluate \(4 x + 18\) when \(x\)=\(7\).
  3. Write an equation for the line through \(\left( 2, \ 11\right)\) and \(\left( 8, \ 29\right)\).
Answers and step-by-step explanations

Practice 1

  1. The fixed cost is 5 dollars.
  2. The variable cost is 3 times 12, or 36 dollars.
  3. Add fixed and variable costs: 5+36=41 dollars.

Answer: \(41\) dollars

Practice 2

  1. Substitute each value for every occurrence of its variable. Use parentheses around negative values.
  2. After substitution: \(18+4\times 7\).
  3. Complete multiplication/division, then addition/subtraction: \(46\).

Answer: \(46\)

Practice 3

  1. Compute the slope: \(m=3\).
  2. Use point-slope form: \(y-(11)=3(x-(2))\).
  3. Expand and isolate y: \(y=3 x + 5\).
  4. Substitute both given points to verify the equation.

Answer: \(y=3 x + 5\)

These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.

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Original Testinar instructional content. Updated November 11, 2026. Editorial standards ยท Report a correction

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