Two points determine a line
Choose two different points satisfying the equation, plot them accurately and extend the line in both directions
- Integer points For a fractional gradient, a horizontal step equal to the denominator often gives a convenient second point
- Horizontal line Keep y fixed and choose different x-values \(\displaystyle y=c\)
- Vertical line Keep x fixed and choose different y-values \(\displaystyle x=k\)
- Check both points Substitute both coordinate pairs into the original equation. The drawing is valid when both substitutions work and the points are distinct
Worked example
Construct a line using two points with integer coordinates
\[2y+x=6\]
- \(\displaystyle \begin{aligned}x&=0\\2y&=6\\y&=3\end{aligned}\)One point is P=(0,3)
- \(\displaystyle \begin{aligned}x&=4\\2y&=2\\y&=1\end{aligned}\)A different point is Q=(4,1)
- \(\displaystyle 2(3)+0=6\)Check P in the original equation
- \(\displaystyle 2(1)+4=6\)Check Q, then plot P and Q and extend the straight line through them
Common mistakes
Using the same point twice
✕\(\displaystyle (0,3),(0,3)\)✓\(\displaystyle (0,3),(4,1)\)
Two copies of one point do not determine the direction. For the worked equation, these distinct points doDifficulty levels
introConstruct lines with integer gradients
standardChoose convenient points for half-integer gradients
challengeConstruct a line from a non-isolated equation
examConstruct ordinary, horizontal and vertical lines
Questions about this topic
Must I use the intercept?+
No. Any two different points with integer coordinates satisfying the equation are accepted on this grid
What does the grid assess?+
You choose and plot the points and construct their line. The grid and its scale are supplied; choosing a suitable scale is a separate skill
How do I do this on paper?+
Use the printed grid, mark two points and extend their straight line with a ruler. Keep both axes and their scales visible
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)