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Math Practice

Standard Form of a Line Practice Problems

Thirteen hand-built problems on standard form, each fully worked. They cover the conventions that make a form standard, converting in both directions, clearing fractions and common factors, the one-step intercept method, and the vertical line that only this form can express.

  1. 1.

    Find the \(x\)-intercept of \(5x + 2y = 20\). Give the \(x\)-value.

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    Set \(y = 0\): \(5x = 20\), so \(x = 4\).

  2. 2.

    Find the \(y\)-intercept of \(5x + 2y = 20\). Give the \(y\)-value.

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    Set \(x = 0\): \(2y = 20\), so \(y = 10\).

  3. 3.

    State the slope of \(7x - 3y = 21\).

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    \(m = -\dfrac{A}{B} = -\dfrac{7}{-3} = \dfrac{7}{3}\).

  4. 4.

    Convert \(y = -2x + 6\) to standard form written as \(2x + y = C\). Give \(C\).

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    Move the \(x\)-term across: \(2x + y = 6\).

  5. 5.

    Convert \(3x - 4y = 24\) to slope-intercept form. Give only the right-hand side.

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    \(-4y = -3x + 24\), so \(y = \dfrac{3}{4}x - 6\).

  6. 6.

    Convert \(y = \dfrac{2}{5}x - 3\) to standard form written as \(2x - 5y = C\). Give \(C\).

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    Multiply by 5: \(5y = 2x - 15\). Rearranged, \(-2x + 5y = -15\), then multiply by \(-1\) so \(A\) is positive: \(2x - 5y = 15\).

  7. 7.

    Write \(6x + 15y = 9\) in proper standard form as \(2x + 5y = C\). Give \(C\).

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    All three coefficients share a factor of 3, so divide through: \(2x + 5y = 3\). Standard form should carry no common factor.

  8. 8.

    Convert \(y = \dfrac{3}{4}x - \dfrac{1}{2}\) to standard form written as \(3x - 4y = C\). Give \(C\).

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    Multiply every term by 4: \(4y = 3x - 2\). Then \(-3x + 4y = -2\), and multiplying by \(-1\) gives \(3x - 4y = 2\).

  9. 9.

    Find the \(x\)-intercept of \(2x - 5y = 10\). Give the \(x\)-value.

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

  10. 10.

    Find the \(y\)-intercept of \(2x - 5y = 10\). Give the \(y\)-value.

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

  11. 11.

    A line has intercepts \((6,\ 0)\) and \((0,\ -4)\). Written as \(2x - 3y = C\), what is \(C\)?

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    The slope is \(\dfrac{-4 - 0}{0 - 6} = \dfrac{2}{3}\), so \(y = \dfrac{2}{3}x - 4\). Multiplying by 3 and rearranging gives \(2x - 3y = 12\). Check at \((6,\ 0)\): \(12 - 0 = 12\).

  12. 12.

    Concert tickets are \(\$12\) in advance and \(\$18\) at the door, and takings were \(\$1{,}080\). Written in reduced standard form \(2a + 3d = C\), what is \(C\)?

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    \(12a + 18d = 1080\). Every coefficient divides by 6, giving \(2a + 3d = 180\).

  13. 13.

    Convert \(y - 3 = \dfrac{1}{2}(x + 4)\) to standard form written as \(x - 2y = C\). Give \(C\).

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    Multiply by 2: \(2y - 6 = x + 4\). Rearranged, \(-x + 2y = 10\), and multiplying by \(-1\) gives \(x - 2y = -10\).
M
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Standard Form Practice Problems

The Standard Form lesson covers the conventions and both conversions; this set is thirteen problems to practise on, each fully solved.

How to use this set

Several problems give you the target shape — “written as , what is ?” — so you only need to type one number. That is deliberate: it keeps the focus on the conversion arithmetic rather than on which of several equivalent arrangements you happened to write.

The conversion routine is always the same three steps: clear fractions, move the -term left, make positive.

Common mistakes to watch for

Leaving a fraction in. Standard form wants integers — multiply by the LCD first.

Leaving negative. Multiply the whole equation by , and remember that changes too.

Forgetting to reduce. should become .

Reading the slope as . It is .

Multiplying only the left side. The constant gets multiplied as well.

Where to go next

x- and y-Intercepts leans on the method this form makes fastest, and Slope-Intercept Form and Point-Slope Form are the other two ways to write the same line.

Frequently Asked Questions

What is standard form?+

with , and integers, not negative, and no common factor across the three. Both variables sit on the left and the constant on the right.

Why is standard form good for intercepts?+

The two variables sit in separate terms, so setting one to zero deletes its term entirely and leaves a single division. Each intercept costs one step.

What is the slope in standard form?+

, provided . For the slope is . The minus sign is part of the formula.

How do I clear fractions?+

Multiply every term — including the constant on the right — by the least common denominator. Then move the -term across and, if came out negative, multiply the whole equation by .

Why can standard form write a vertical line?+

Because it never divides by anything. is , which fits the template. Slope-intercept form needs a slope, and a vertical line has none.

Does 2x + 4y = 6 count as standard form?+

Not strictly — the coefficients share a factor of 2. Reduce it to . It is the same line either way, but the reduced version is the conventional answer.

More practice problems

x- and y-Intercepts Practice Problems Slope-Intercept Form Practice Problems Point-Slope Form Practice Problems