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Check a candidate point against two lines

Optional extension for this course. It is not required for core topic progress

Start a set of 8
Step 3 of 7

Two separate comparisons

Use the candidate’s x-coordinate in each rule, then compare both outputs with its y-coordinate

  1. 1Evaluate line 1 Keep signs and order of operations correct \(\displaystyle P=(-2,-3),\quad y=2x+1,\quad 2(-2)+1=-3\)
  2. 2Evaluate line 2 Use the same candidate input \(\displaystyle y=-x+4,\quad -(-2)+4=6\)
  3. 3Classify the result The first output agrees with -3 and the second does not. This point fails line 2 only
  4. 4Intersection condition For distinct intersecting lines, a candidate is their common point only if it satisfies both rules
Year 9, Chapter 4, 1 difficulty level, No calculatorBuild a worksheet

Check a candidate against two lines

Evaluate each rule at the supplied x-coordinate and compare its result with the supplied y-coordinate

  1. Do both checks A point on one line need not lie on the other
  2. Read the verdict wording “Fails neither” means both checks succeed. “Fails both” means both checks fail

Worked example

Check whether the candidate lies on both displayed lines

\[\begin{gathered}y=\frac12x+1\\y=-x+4\\P=(2,2)\end{gathered}\]
  1. \(\displaystyle \frac12(2)+1=2\)Line 1 agrees with the candidate’s y-coordinate
  2. \(\displaystyle -(2)+4=2\)Line 2 also agrees
  3. \(\displaystyle P\in\ell_1\text{ and }P\in\ell_2\)The candidate satisfies both lines and fails neither

Common mistakes

Stopping after one check
✕\(\displaystyle \begin{gathered}2(1)+1=3\\\Rightarrow P\text{ is the intersection}\end{gathered}\)✓\(\displaystyle 5-1=4\ne3\)
The second rule fails for this candidate even though the first passes

Difficulty levels

challengeClassify a candidate against two exact line rules, including fractional coefficients

Questions about this topic

Do I need to solve the simultaneous equations?+

No. The candidate is supplied. Substitute it into each equation and compare the outputs

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 4: Linear relations

In this chapter

Construct a table of values from a formulaRead coordinates and points from a straight-line graph4A Points, tables and linear relations: Decide whether a point lies on a lineFind the axis intercepts of a line4C Horizontal, vertical and through-origin lines: Recognise horizontal, vertical and through-origin graphsClassify and compare gradients from graphsFind the gradient of a line through two points4E Direct proportion: Use a direct-proportion model4E Direct proportion: Read and interpret direct-proportion graphsRead the vertical-axis intercept and gradient from an equation4F Gradient-intercept form: Recognise a line or find points for its graphConstruct a line on a coordinate gridFind the equation of a line from its graph or from its intercept and slopeRecover an intercept or recognise a line from a gradient and pointFind the equation of a line through two pointsMidpoint, endpoint and distance of a coordinate segmentForm an equation of a parallel or perpendicular lineDetermine a perpendicular bisector (optional extension)4J Linear modelling: Construct a linear model from a descriptionInterpret the graph of a linear model4K Graphical simultaneous equations: Check a candidate point against two lines (optional extension)Read the point of intersection of two lines (optional extension)