KS3 · Maths

Drawing linear graphs efficiently

Pin one point and a ruler swings any way it likes. Add a second point and it snaps into place. That snap is the secret.

Maths · Linear graphs

Pin one point. Then lock the line.
0.0°0.0°xyPRQS

Spread pair: P–Q is off the true line by 0.0°. Close pair: R–S is off the true line by 0.0°. Relationship: One point leaves the line free to swing; a second point locks it. Nudge a close pair and the line tilts far more than the same nudge to a spread pair.

Spread pair: P–Q is off the true line by0.0°Close pair: R–S is off the true line by0.0°Drag Q and S

One point leaves the line free to swing; a second point locks it. Nudge a close pair and the line tilts far more than the same nudge to a spread pair.

P, Q, R and S all sit on the line y = x + 2: P (−5, −3), Q (5, 7), R (1, 3), S (2, 4). First, drag Q anywhere and watch the line through P swing. Then put Q back near (5, 7) and give Q and S the same small nudge. Which line goes further wrong?

Same graph, two routes

Full table of valuesvsTwo coordinates

Take y = 5x − 7. Which route would you rather do?

Focus

Why it works

Full table of values

Work out lots of points, one after another: x = −3 gives −22, x = −2 gives −17, and on.

Two coordinates

A linear equation forms a straight line, so only two points are needed: (−3, −22) and (3, 8).

The insight

The straight-line shape is the whole reason two points are enough. A line has no surprises hiding between the points.

How much work

Full table of values

A lot of work, and not always needed.

Two coordinates

Two substitutions, then a ruler.

What it shows

Full table of values

The gradient (every +1 in x gives +5 in y) and the y-intercept (−7, when x = 0) jump out of the table.

Two coordinates

The two points you chose, ready to plot.

Maths · ax + by = c

Draw 5x + 3y = 15: you fill in the gaps

Draw the graph of 5x + 3y = 15 by finding two coordinates.

  1. 5x + 3y = 15
  2. missing step
Which line is step 2?

Predict, then check

You need two plottable coordinates for y = 2.5 − 0.25x.

Which two x values would you choose?

Maths · Choosing your method

Which way finds two good coordinates fastest?

Sort each equation by the quickest way to find two coordinates for it.

Still to sort

Any two well-spaced x values (0)

y = mx + c with whole numbers: every x gives a whole-number y.

Where the line is: Whole-number m and c, so you can just pick x values that are far apart.

Choose x to make y a whole number (0)

y = mx + c with decimals in it.

Where the line is: The decimals mean most x values give a y you cannot plot accurately, so you choose x on purpose.

Put x = 0, then y = 0 (0)

ax + by = c, with x and y both on the left.

Where the line is: x and y sit together on the left of the equals sign, so zero kills one of them.

6 of 6 still to sort.

Look at the equation, pick a method, then read why.

Maths · Checking

One coordinate is wrong. Which line?

A student draws y = 2 + x/4 using two coordinates. Where does the working go wrong?

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

You don’t need a big table. You need two good points.

What you need to know

  • Linear means the graph is a straight line when you plot the two variables on a pair of axes.
  • Choose pairs of values that satisfy the equation and you can draw the graph accurately.
  • A table of values works and shows the gradient and the y-intercept, but it is a lot of work.
  • Because a linear equation forms a straight line, only two points (two coordinates) are needed to draw it.
  • One coordinate is never enough: infinitely many lines pass through it, and a second coordinate fixes one unique line.
  • Have a goPriya pushes one drawing pin through a point on her wall and balances a ruler on it. Can she say how the ruler will lie?

    No. The ruler can swing round the pin to any angle until a second point holds it.

    One coordinate leaves infinitely many lines; the second coordinate is what fixes one unique line.

  • Choose two points well apart. Close neighbours lose accuracy as the line is extended towards the edges.
  • Have a goLeo says: “For y = x + 2, any two points on it draw the line equally well, so I'll use (1, 3) and (2, 4).” Is he right?

    Not quite. Both points are on the line, but they are too close together.

    Neighbouring points lose accuracy as the line is extended, so a pair that is far apart draws it more accurately.

  • To test a coordinate, substitute it into the equation and see whether it lies on the line.
  • Linear equations often come as y = mx + c (m is the gradient, c helps define the y-intercept) or as ax + by = c.
  • For ax + by = c, substitute x = 0 and y = 0 to find two coordinates, then plot and join them.
  • Have a goFor 2x + 6y = 12, put x = 0. What coordinate do you get?

    (0, 2)

    2 × 0 + 6y = 12 gives 6y = 12, so y = 2. The 12 is 6y, not y, so you must divide by 6.

  • For y = mx + c with decimals, choose x values that make y a whole number so you can plot accurately.

The big picture

A straight-line graph is fixed by two points, so you don't need a full table of values. Choose two coordinates that satisfy the equation, keep them well apart, and check each by substitution. For ax + by = c use x = 0 and y = 0; for decimals, pick x to make y a whole number.

Key points

1A linear equation forms a straight line, so two coordinates that satisfy it are enough to draw the graph.
2One coordinate is never enough, and two points that are too close lose accuracy towards the edges, so choose them well apart.
3ax + by = c: substitute x = 0 and y = 0. y = mx + c with decimals: choose x so y is a whole number. Then check each coordinate by substitution.

Worked example

Problem

Draw the graph of 4x + 2y = 16. Find two coordinates, check them, then draw the line.

⚠ Watch out

Thinking any two points will do. Two neighbouring points give a line that drifts further off the true line towards the edges of the grid, and with only two points a wrong coordinate goes unnoticed unless you substitute it into the equation.

🧠

Memory hook

Two points lock it. Far apart keeps it true. Substitute to check.

✓

Check yourself

Cold try: draw 6x + 2y = 12. Which two substitutions? Which two coordinates (you should get (0, 6) and (2, 0))? How will you know each is right?

Flashcards

(12)
How many coordinates do you need to draw a linear graph?
Two. A linear equation forms a straight line, so two points are enough.
Why can't one coordinate draw a linear graph?
Infinitely many lines pass through a single coordinate. A second coordinate fixes one unique line.
What makes a relationship between two variables linear?
When plotted on a pair of axes, it forms a straight line.
Why should your two points be well apart?
Two closely neighbouring points lose accuracy as the line is extended to its extremities.
What does a table of values show for y = 5x − 7?
The gradient (+5 in y for each +1 in x) and the y-intercept (−7, when x = 0). It is a lot of work, though.
Which two forms of linear equation does this lesson use?
y = mx + c (m is the gradient, c helps define the y-intercept) and ax + by = c.
How do you find two coordinates quickly for ax + by = c?
Substitute x = 0 and y = 0. Plot both coordinates and draw the line through them.
y = mx + c has decimals in it. How do you pick coordinates?
Choose x values that give whole-number y values, so the points can be plotted accurately.
How do you test whether a coordinate lies on a line?
Substitute it into the equation. If the equation does not balance, the point is not on the line.
Why must you check calculations when you use only two points?
No third point will expose a slip, so one wrong coordinate gives a wrong line.
For y = 2.5 − 0.25x, why does every change of 4 in x give another whole-number y?
Because 0.25 × 4 = 1, so y changes by exactly 1.
Why is rearranging 5x + 3y = 15 into y = mx + c a slower choice?
It gives y = 5 − (5/3)x, whose gradient −5/3 is not easy to sketch.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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