Math Practice

Point-Slope Form Practice Problems

Q: What is point-slope form?

y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is any known point on the line. It is the slope formula with the fraction cleared.

Q: When should I use it instead of y = mx + b?

Whenever you know a slope and a point that is not the y-intercept. Slope-intercept form needs b, which you would have to solve for first; point-slope skips that step.

Q: Why do the signs flip with a negative coordinate?

The formula always subtracts, so a negative x₁ produces a double negative. With the point (-3, 4) you write x - (-3), which simplifies to x + 3.

Q: Can point-slope form describe a vertical line?

No. A vertical line has undefined slope, so there is no m to substitute. Write it as x = k instead.

Q: How should I type these answers?

Give only the right-hand side of y = … — for example `3x + 2`. Fractional coefficients can be typed as `(2/3)x` or `0.6667x`; the checker compares the expressions by value, not by how they are written.

Fourteen hand-built problems on point-slope form, each fully worked. They start from a slope and a point, move on to two points, and finish with the sign traps and conversions that this form is really tested on — negative coordinates, fractional slopes, and turning the result into slope-intercept or standard form.

Questions

1.
A line has slope \(3\) and passes through \((1,\ 5)\). Write it in slope-intercept form, giving only the right-hand side.
Show full solution
\(y - 5 = 3(x - 1)\), so \(y - 5 = 3x - 3\) and \(y = 3x + 2\).
2.
A line has slope \(-2\) and passes through \((0,\ 7)\). Write it in slope-intercept form, giving only the right-hand side.
Show full solution
The point is already the \(y\)-intercept, so \(y = -2x + 7\) with no work needed.
3.
A line has slope \(4\) and passes through \((2,\ -3)\). Write it in slope-intercept form, giving only the right-hand side.
Show full solution
\(y - (-3) = 4(x - 2)\) becomes \(y + 3 = 4x - 8\), so \(y = 4x - 11\).
4.
A line has slope \(5\) and passes through \((-3,\ 4)\). Write it in slope-intercept form, giving only the right-hand side.
Show full solution
\(y - 4 = 5(x - (-3)) = 5(x + 3)\), so \(y - 4 = 5x + 15\) and \(y = 5x + 19\). Subtracting the negative coordinate turns into addition.
5.
Convert \(y - 2 = 5(x - 3)\) to slope-intercept form, giving only the right-hand side.
Show full solution
Expand: \(y - 2 = 5x - 15\), then \(y = 5x - 13\).
6.
Write the line through \((2,\ 1)\) and \((5,\ 10)\) in slope-intercept form, giving only the right-hand side.
Show full solution
\(m = \dfrac{10 - 1}{5 - 2} = 3\). Then \(y - 1 = 3(x - 2)\), so \(y = 3x - 5\).
7.
Write the line through \((-1,\ 4)\) and \((3,\ -4)\) in slope-intercept form, giving only the right-hand side.
Show full solution
\(m = \dfrac{-4 - 4}{3 - (-1)} = \dfrac{-8}{4} = -2\). Then \(y - 4 = -2(x + 1)\), so \(y = -2x + 2\).
8.
A line has slope \(-\dfrac{2}{5}\) and passes through \((10,\ 1)\). Write it in slope-intercept form, giving only the right-hand side.
Show full solution
\(y - 1 = -\dfrac{2}{5}(x - 10)\) gives \(y - 1 = -\dfrac{2}{5}x + 4\), so \(y = -\dfrac{2}{5}x + 5\). Choosing the point whose \(x\) is divisible by 5 kept the arithmetic clean.
9.
Write the line through \((-4,\ -1)\) and \((2,\ -7)\) in slope-intercept form, giving only the right-hand side.
Show full solution
\(m = \dfrac{-7 - (-1)}{2 - (-4)} = \dfrac{-6}{6} = -1\). Then \(y + 1 = -(x + 4)\), so \(y = -x - 5\). Check at \((2,\ -7)\): \(-2 - 5 = -7\).
10.
A line has slope \(\dfrac{1}{2}\) and passes through \((-6,\ 0)\). What is its \(y\)-intercept?
Show full solution
\(y - 0 = \dfrac{1}{2}(x + 6)\) gives \(y = \dfrac{1}{2}x + 3\), so the intercept is \((0,\ 3)\).
11.
A line has slope \(0\) and passes through \((6,\ -2)\). Write it in slope-intercept form, giving only the right-hand side.
Show full solution
\(y + 2 = 0(x - 6)\) collapses to \(y + 2 = 0\), so \(y = -2\). A slope of zero kills the \(x\)-term and leaves a horizontal line.
12.
A line passes through \((-2,\ 3)\) and falls \(3\) units for every \(4\) units to the right. Write it in slope-intercept form, giving only the right-hand side.
Show full solution
Falling 3 over a run of 4 means \(m = -\dfrac{3}{4}\). Then \(y - 3 = -\dfrac{3}{4}(x + 2)\), so \(y - 3 = -\dfrac{3}{4}x - \dfrac{3}{2}\) and \(y = -\dfrac{3}{4}x + \dfrac{3}{2}\).
13.
Convert \(y + 1 = -\dfrac{3}{4}(x - 8)\) to standard form \(Ax + By = C\). Give the value of \(C\) when the equation is written as \(3x + 4y = C\).
Show full solution
Multiply by 4: \(4y + 4 = -3(x - 8) = -3x + 24\). Then \(3x + 4y = 20\).
14.
A line passes through \((0,\ -3)\) and \((4,\ 5)\). Using the point \((4,\ 5)\) in point-slope form and simplifying, write the slope-intercept form, giving only the right-hand side.
Show full solution
\(m = \dfrac{5 - (-3)}{4 - 0} = 2\). Then \(y - 5 = 2(x - 4)\) gives \(y = 2x - 3\) — the same answer the other point would give.

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Point-Slope Form Practice Problems

The Point-Slope Form lesson covers where the formula comes from and why it works; this set is fourteen problems to build the habit, each fully solved.

How to use this set

Substitute into with the brackets in place first, then simplify. Writing before tidying it costs three seconds and removes almost every sign error on this topic.

Most problems ask for slope-intercept form as the final answer, so give just the right-hand side: 3x + 2.

Common mistakes to watch for

Dropping the double negative. With the point the bracket is .

Swapping the coordinates. The -value goes on the left; the -value goes inside the bracket.

Distributing to only one term. is .

Mismatched subtraction order in the slope. Keep over .

Where to go next

Slope is the number this form needs, Slope-Intercept Form is usually where the answer ends up, and Standard Form is the third way to write the same line.

Frequently Asked Questions

What is point-slope form?+

, where is the slope and is any known point on the line. It is the slope formula with the fraction cleared.

When should I use it instead of y = mx + b?+

Whenever you know a slope and a point that is not the -intercept. Slope-intercept form needs , which you would have to solve for first; point-slope skips that step.

Why do the signs flip with a negative coordinate?+

The formula always subtracts, so a negative produces a double negative. With the point you write , which simplifies to .

Does it matter which of two points I use?+

No. The two point-slope equations look different but simplify to the same line. That is worth checking once by hand to convince yourself.

Can point-slope form describe a vertical line?+

No. A vertical line has undefined slope, so there is no to substitute. Write it as instead.

How should I type these answers?+

Give only the right-hand side of — for example `3x + 2`. Fractional coefficients can be typed as `(2/3)x` or `0.6667x`; the checker compares the expressions by value, not by how they are written.

y−y₁Point-Slope Form Practice Problems

Point-Slope Form Practice Problems

How to use this set

Common mistakes to watch for

Where to go next

Frequently Asked Questions

More practice problems

Point-Slope Form Practice Problems