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Find the equation of a line through two points

Start a set of 8
1 OF 6
First point
\(\displaystyle (-1,6)\)
Second point
\(\displaystyle (3,-2)\)

Which equation passes through both points?

Year 11, VCE Units 1 & 2, Chapter 5, 4 difficulty levels, Calculator allowedBuild a worksheet

Equation through two points

Use the two-point gradient, then recover the intercept from either point

\[y=mx+c\]
  1. Check the points This method requires different horizontal coordinates. Coincident points do not determine a unique line
  2. Gradient Take both differences in the same point order \(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)
  3. Intercept Substitute either point with the exact gradient \(\displaystyle c=y_1-mx_1\)
  4. Check The other point must also satisfy the completed equation

Worked example

Find the equation through the two points

\[P(-4,9),\quad Q(2,0)\]
  1. \(\displaystyle m=\frac{0-9}{2-(-4)}\)Keep the point order consistent in both differences
  2. \(\displaystyle m=\frac{-9}{6}\)Evaluate the coordinate changes
  3. \(\displaystyle m=-\frac{3}{2}\)Keep the gradient exact
  4. \(\displaystyle 0=-\frac{3}{2}(2)+c\)Use the second point because its vertical coordinate is zero
  5. \(\displaystyle 0=-3+c\)Simplify the product and retain the equation
  6. \(\displaystyle 0+3=c\)Remove the negative contribution from both sides
  7. \(\displaystyle 3=c\)Evaluate the remaining sum
  8. \(\displaystyle y=-\frac{3}{2}x+3\)Place both coefficients into the rule
  9. \(\displaystyle \begin{aligned}-\frac{3}{2}(-4)+3&=6+3\\&=9\end{aligned}\)Substitution reproduces the first point's vertical coordinate

Common mistakes

Mixing point order
✕\(\displaystyle \begin{aligned}m&=\frac{0-9}{-4-2}\\m&=\frac{3}{2}\end{aligned}\)✓\(\displaystyle \begin{aligned}m&=\frac{0-9}{2-(-4)}\\m&=-\frac{3}{2}\end{aligned}\)
Reversing only one difference negates the gradient. A zero gradient remains zero
Assuming an unseen intercept is zero
✕\(\displaystyle y=-\frac{3}{2}x\)✓\(\displaystyle y=-\frac{3}{2}x+3\)
The intercept must be recovered even when it is not one of the supplied points

Difficulty levels

introRecover both coefficients from simple integer point pairs
standardRecover both coefficients from signed points, including horizontal lines
challengeRecover both coefficients when the graph does not show the intercept
examRecognise a complete equation from supplied points or a graph with an unseen intercept

Questions about this topic

Does it matter which point I substitute?+

Either point gives the same intercept when the gradient is correct. Choose the easier arithmetic

What if the points have the same horizontal coordinate?+

Distinct points then form a vertical line. Use the vertical-line form from the earlier special-lines topic

What if their vertical coordinates agree?+

The line is horizontal. Its gradient is zero and its intercept is the common vertical coordinate

Why keep the gradient exact?+

Rounding it can change the recovered intercept and produce a line that misses the supplied points

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 5: Linear functions and models

In this chapter

Substitute values into a formulaConstruct a table of values from a formulaRecognise linear equationsSolve a linear equationSet up and solve a practical linear equationBuild and use a formula in two unknownsRead coordinates and points from a straight-line graphClassify and compare gradients from graphs5D Straight-line graphs and slope: Recognise a line or find points for its graph5D Straight-line graphs and slope: Find the gradient of a line through two points5E Straight-line equations: Recognise a line or find points for its graph5E Straight-line equations: Find the gradient of a line through two pointsRead the vertical-axis intercept and gradient from an equation5E Straight-line equations: Find the equation of a line from its graph or from its intercept and slope5F Equations from graphs and points: Find the equation of a line from its graph or from its intercept and slopeFind the equation of a line through two pointsConstruct a linear model from a descriptionInterpret the graph of a linear modelRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSet up a pair of simultaneous equations from a descriptionSolve a practical problem with simultaneous equationsUse a piecewise linear modelRead and construct step graphs