Checking a point
Substitute the horizontal coordinate into the line rule. Compare the result with the point's vertical coordinate
\[y_P=mx_P+c\]
- Keep the order The first coordinate is the input. The second is the value to compare with
- Keep the sign Use brackets around a negative input
- Make the comparison Equal values mean the point lies on the line. Unequal values mean it does not
Worked example
Decide whether the point lies on the line
\[\begin{gathered}y=-2x+9\\P=(-3,14)\end{gathered}\]
- \(\displaystyle y_{\mathrm{rule}}=-2(-3)+9\)Substitute the horizontal coordinate
- \(\displaystyle y_{\mathrm{rule}}=6+9\)Multiply the signed values first
- \(\displaystyle y_{\mathrm{rule}}=15\)Add the constant
- \(\displaystyle 15\ne14\)The result differs from the point's vertical coordinate, so the point is not on the line
Common mistakes
Checking only the input
✕\(\displaystyle x_P=4\Rightarrow P\text{ is on the line}\)✓\(\displaystyle y_P=mx_P+c\)
An allowed input must still produce the candidate's vertical valueAdding the coefficient
✕\(\displaystyle 2(4)+3=2+4+3\)✓\(\displaystyle 2(4)+3=8+3\)
A coefficient multiplies the inputSubstituting the wrong coordinate
✕\(\displaystyle y_{\mathrm{rule}}=2(9)+3\)✓\(\displaystyle y_{\mathrm{rule}}=2(4)+3\)
For the point with horizontal coordinate four and vertical coordinate nine, substitute four into the ruleDifficulty levels
introCheck a single line using small integer values
standardUse signed inputs and simple fractional coefficients
examChoose an on-line or off-line verdict using the same number types
Questions about this topic
Which coordinate goes into the rule?+
Use the horizontal coordinate as the input. Keep the vertical coordinate for the comparison
Is a close result enough?+
No. For the exact values used here, the calculated value must equal the point's vertical coordinate
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)