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Find the axis intercepts of a line

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\(\displaystyle 3x+7y=42\)

Which point is the y-intercept?

Year 9, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Finding axis intercepts

An axis intercept lies on both the line and an axis. Substitute the coordinate that is zero on that axis, then solve for the other coordinate

  1. Horizontal axis The vertical coordinate is zero \(\displaystyle y=0\)
  2. Vertical axis The horizontal coordinate is zero \(\displaystyle x=0\)
  3. Write a point Give the horizontal coordinate first, including the zero coordinate \(\displaystyle \begin{gathered}I_x=(x_0,0)\\I_y=(0,y_0)\end{gathered}\)
  4. Use different points Two different intercept points determine the line. When both intercepts are the origin, find another point before drawing the line

Worked example

Find both axis intercepts

\[4x-5y=30\]
  1. \(\displaystyle 4x-5(0)=30\)Use zero for the vertical coordinate at the horizontal-axis crossing
  2. \(\displaystyle 4x=30\)Simplify the zero term
  3. \(\displaystyle x=\frac{15}{2}\)Divide both sides by the horizontal variable's coefficient and simplify the fraction
  4. \(\displaystyle I_x=\left(\frac{15}{2},0\right)\)Include the zero vertical coordinate
  5. \(\displaystyle 4(0)-5y=30\)Use zero for the horizontal coordinate at the vertical-axis crossing
  6. \(\displaystyle -5y=30\)Simplify the zero term
  7. \(\displaystyle y=-6\)Divide both sides by the negative coefficient
  8. \(\displaystyle I_y=(0,-6)\)Include the zero horizontal coordinate. Plot the two distinct crossings and extend the line through them

Common mistakes

Setting the wrong coordinate to zero
✕\(\displaystyle \text{At the x-intercept: }x=0\)✓\(\displaystyle \text{At the x-intercept: }y=0\)
Every point on the horizontal axis has zero vertical coordinate
Losing a negative sign
✕\(\displaystyle \begin{aligned}-3y&=33\\y&=11\end{aligned}\)✓\(\displaystyle \begin{aligned}-3y&=33\\y&=-11\end{aligned}\)
Divide by the signed coefficient. A positive number divided by a negative number is negative
Swapping the coordinates
✕\(\displaystyle \text{A y-intercept: }(-11,0)\)✓\(\displaystyle \text{A y-intercept: }(0,-11)\)
Write the horizontal coordinate first. It must be zero for a point on the vertical axis

Difficulty levels

introCalculate integer intercept coordinates from a rule giving the vertical coordinate
standardCalculate integer intercepts from general form or recognise the matching graph
challengeCalculate intercepts or recognise a graph in either equation form, with at least one non-integer half coordinate
examChoose an intercept pair in either equation form, using integer and half-unit coordinates

Questions about this topic

Why use the other coordinate?+

At a crossing, the coordinate along the axis can vary. The coordinate perpendicular to that axis is zero

What if both intercepts are the origin?+

You have found the same point twice. Choose another horizontal coordinate and use the rule to find its vertical coordinate before sketching the line

Can a line miss an axis?+

Yes. A horizontal line away from the origin misses the horizontal axis. A vertical line away from the origin misses the vertical axis. These special lines are studied next

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)