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Math Practice

Slope Practice Problems

Fifteen hand-built slope problems, each fully worked. They cover the slope formula from two points, reading slope off an equation in either form, the horizontal and vertical special cases, finding a missing coordinate, and slope as a real rate of change.

  1. 1.

    Find the slope of the line through \((1,\ 2)\) and \((3,\ 8)\).

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    \(m = \dfrac{8 - 2}{3 - 1} = \dfrac{6}{2} = 3\)

  2. 2.

    Find the slope of the line through \((0,\ 0)\) and \((4,\ -8)\).

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    \(m = \dfrac{-8 - 0}{4 - 0} = -2\). Negative, so the line falls to the right.

  3. 3.

    Find the slope of the line through \((-2,\ 5)\) and \((4,\ 5)\).

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    \(m = \dfrac{5 - 5}{4 - (-2)} = \dfrac{0}{6} = 0\). Both points sit at the same height, so the line is horizontal.

  4. 4.

    Find the slope of the line through \((3,\ 1)\) and \((3,\ 7)\).

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    The run is \(3 - 3 = 0\), and dividing by zero is undefined. The line is vertical, \(x = 3\).

  5. 5.

    Find the slope of the line through \((-1,\ -4)\) and \((2,\ 5)\).

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    \(m = \dfrac{5 - (-4)}{2 - (-1)} = \dfrac{9}{3} = 3\). Subtracting the negatives is where sign errors creep in.

  6. 6.

    State the slope of \(y = -3x + 7\).

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    In \(y = mx + b\) the slope is the coefficient of \(x\), so \(m = -3\). The \(7\) is the \(y\)-intercept.

  7. 7.

    Find the slope of the line through \((1,\ 2)\) and \((5,\ 4)\).

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    \(m = \dfrac{4 - 2}{5 - 1} = \dfrac{2}{4} = \dfrac{1}{2}\). Always reduce the fraction.

  8. 8.

    State the slope of \(2x + 5y = 10\).

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    Either rearrange to \(y = -\dfrac{2}{5}x + 2\), or use the shortcut \(m = -\dfrac{A}{B} = -\dfrac{2}{5}\).

  9. 9.

    State the slope of \(4x - y = 9\).

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    \(-y = -4x + 9\), so \(y = 4x - 9\) and \(m = 4\). Reading \(4\) straight off the original would have been right here only by luck — check the sign of the \(y\)-term first.

  10. 10.

    A line has slope \(2\) and passes through \((2,\ 3)\) and \((6,\ k)\). Find \(k\).

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    \(\dfrac{k - 3}{6 - 2} = 2\), so \(k - 3 = 8\) and \(k = 11\).

  11. 11.

    What slope is parallel to \(y = \dfrac{3}{4}x - 1\)?

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    Parallel lines have equal slopes, so \(m = \dfrac{3}{4}\).

  12. 12.

    What slope is perpendicular to \(y = \dfrac{3}{4}x - 1\)?

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    The negative reciprocal: flip \(\dfrac{3}{4}\) to \(\dfrac{4}{3}\) and change the sign, giving \(-\dfrac{4}{3}\). Check: \(\dfrac{3}{4} \times -\dfrac{4}{3} = -1\).

  13. 13.

    A wheelchair ramp rises \(18\) inches over a horizontal run of \(240\) inches. What is its slope, as a decimal?

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    \(m = \dfrac{18}{240} = \dfrac{3}{40} = 0.075\). That is a 1-in-13.3 gradient, just inside the 1-in-12 maximum most accessibility codes allow.

  14. 14.

    Find the slope of the line through \((-3,\ 7)\) and \((5,\ -9)\).

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    \(m = \dfrac{-9 - 7}{5 - (-3)} = \dfrac{-16}{8} = -2\).

  15. 15.

    Find the slope of the line through \((a,\ 2a)\) and \((3a,\ 6a)\), where \(a \neq 0\).

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    \(m = \dfrac{6a - 2a}{3a - a} = \dfrac{4a}{2a} = 2\). The \(a\) cancels, so every line in this family has the same slope.
M
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Mathovia Team
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Slope Practice Problems

The Slope lesson explains what the number means and where the formula comes from; this set is fifteen problems to practise it on, each fully solved.

How to use this set

Work each one on paper before typing. Give the slope as a number — a fraction such as 3/4, a decimal, or a negative value. For a vertical line, type undefined.

The problems build in three stages: the slope formula from two points, reading slope off an equation, and then the cases that need a bit more thought — a missing coordinate, a real ramp gradient, and a line whose coordinates contain a letter.

Common mistakes to watch for

Run over rise. The formula is rise over run — the -difference goes on top.

Mixed subtraction order. If you do on top, do underneath. Reversing one and not the other flips the sign.

Zero and undefined swapped. Horizontal is zero; vertical is undefined.

Reading the slope before solving for . has slope , not — rearrange first.

Where to go next

Slope is the input to every line equation: Point-Slope Form builds a line from a slope and a point, Slope-Intercept Form reads it off a graph, and Parallel and Perpendicular Lines uses the two slope relationships.

Frequently Asked Questions

What is the slope formula?+

For two points and , the slope is — the change in divided by the change in .

Does it matter which point I call the first one?+

No, as long as you are consistent. Subtracting in the other order flips the sign of both the numerator and the denominator, and the two changes cancel.

What is the difference between zero slope and undefined slope?+

A horizontal line has slope zero — the rise is zero. A vertical line has undefined slope — the run is zero, and dividing by zero is not defined. Mixing these up is the most common error on this topic.

How do I get the slope from an equation in standard form?+

For the slope is . You can also rearrange into and read the coefficient of .

How should I type a fractional slope?+

Either form works — `1/2` or `0.5` are both accepted, and so is the LaTeX `\frac{1}{2}`. For an undefined slope, type `undefined`.

What does slope mean in a real problem?+

It is a rate of change: how much the output moves for each one-unit increase in the input. On a ramp it is steepness, on a cost graph it is price per unit, on a distance graph it is speed.

More practice problems

Point-Slope Form Practice Problems Slope-Intercept Form Practice Problems Parallel and Perpendicular Lines Practice Problems