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Math101learn.math101.caPoint-Slope Form
Point-slope form $y-y_1=m(x-x_1)$ describes the line with slope $m$ through $(x_1,y_1)$. It comes from the slope relation $(y-y_1)/(x-x_1)=m$ and is especially useful when one point and slope are known.
Point-slope form translates local rate and one known state directly into a line. It supports modelling, tangent lines, and conversions among linear forms.
Intuition and core definition
Point-slope form $y-y_1=m(x-x_1)$ describes the line with slope $m$ through $(x_1,y_1)$. It comes from the slope relation $(y-y_1)/(x-x_1)=m$ and is especially useful when one point and slope are known.
Notation, language, and conditions
The paired coordinates must stay together: both subscripts refer to the same known point. Subtracting a negative coordinate creates addition. A vertical line has undefined slope and cannot be written in point-slope form; through $(a,b)$ it is $x=a$.
Why this idea matters
Point-slope form builds a line directly from one known point and a rate of change without first calculating the vertical intercept.
A dependable method
- Identify the given slope and one complete point.
- Substitute into $y-y_1=m(x-x_1)$ with parentheses around signed coordinates.
- Simplify double negatives but need not solve for $y$ unless another form is requested.
- Test the known point in the equation.
- Use the slope to generate a second point or convert forms for an additional check.
Worked example
Representations and interpretation
Starting at $(x_1,y_1)$ on a coordinate plane, slope $m=\Delta y/\Delta x$ generates a second point. Point-slope form records this displacement relationship relative to the anchor point.
Reasoning about variations
Any point on the same line can produce a different-looking point-slope equation. The forms are equivalent. For two given points, first compute slope, checking whether their $x$-coordinates are equal.
Common mistakes
How to check your work
- Substitute the anchor point and obtain a true equality.
- Convert to slope-intercept form and verify the slope coefficient.
- Generate a second point using rise and run and test it.
Practice
- Write a point-slope equation of slope $4$ through $(2,-3)$.
- Find the slope of the line $y-1=-2(x+5)$.
- What is the line through $(7,2)$ and $(7,-4)$?
Answers and brief solutions
Show answers
- $y+3=4(x-2)$ $y-(-3)=4(x-2)$.
- $-2$ Point-slope form displays the multiplier as slope.
- $x=7$ Equal $x$-coordinates make a vertical line.
Synthesis and transfer
A sensor reading with a known value at a nonzero time can be modelled immediately from its slope; substituting the anchor point should reduce both sides to zero.
A sensor reads $14$ units at time $5$ and increases by $1.8$ units per second. The equation $y-14=1.8(t-5)$ keeps both the rate and the measured anchor visible. Substituting $t=5$ makes the right side zero and returns $y=14$, while expanding gives a slope-intercept form useful for predicting the reading at time zero. Either form represents the same line, but the point-slope form avoids an unnecessary intercept calculation when the known data are a point and a slope. If units are included, both sides measure sensor units because the rate multiplies a time difference.
Related topics
Teaching and accessibility note
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Point-Slope Form.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Write a point-slope equation of slope $4$ through $(2,-3)$.
- $y-(-3)=4(x-2)$.
End of lesson
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