Every point on a grid has an address \((x, y)\). Work out a few addresses that fit an equation, plot them, and a ruler does the rest: the graph of a linear equation is always a straight line.
A point is written \((x, y)\): across first, then up or down. To draw a linear graph, choose some \(x\) values, work out \(y\) for each, plot the points and join them with ONE ruled straight line. If a point is off the line, one of your calculations is wrong.
Fig. 1: the table of values for y = -2x + 5 gives five points, and all five lie on one straight line
Section 01
Every point has an address.
In the lab, set the gradient to 2 and the intercept to \(-1\). Pick out three places where the line passes exactly through a grid crossing and write down their coordinates, across first and then up or down. Check each one: is the \(y\) value always \(2 \times x - 1\)?
The axes cross at the origin \((0, 0)\). The \(x\)-axis runs across; the \(y\)-axis runs up.
Order matters: \((3, -2)\) means 3 to the right, then 2 down. \((-2, 3)\) is a different point.
Four quadrants: left of the \(y\)-axis \(x\) is negative; below the \(x\)-axis \(y\) is negative.
On an axis: every point on the \(y\)-axis has \(x = 0\), and every point on the \(x\)-axis has \(y = 0\).
Worked example Finding the fourth corner
\(A(-2, 3)\), \(B(4, 3)\) and \(C(4, -1)\) are three corners of a rectangle \(ABCD\). Find the coordinates of \(D\).
\(A\) and \(B\) have the same \(y\) value, so \(AB\) is horizontal. \(B\) and \(C\) have the same \(x\) value, so \(BC\) is vertical.
\(D\) must be directly below \(A\), so it has the same \(x\) value as \(A\): \(x = -2\).
\(D\) must be level with \(C\), so it has the same \(y\) value as \(C\): \(y = -1\).
Answer: \(D(-2, -1)\)
Examiner trap
Writing the coordinates the wrong way round. \((x, y)\) is always across first, then up: 'along the corridor, then up the stairs'. \((-1, 3)\) and \((3, -1)\) are in different quadrants.
Section 02
From equation to table.
An equation such as \(y = -2x + 5\) is a rule: put in an \(x\) value, get out the matching \(y\) value. A table of values lists a few of these pairs. Work each column out carefully, especially with negative numbers.
Worked example Completing a table
Complete the table of values for \(y = -2x + 5\) for \(x = -1, 0, 1, 2, 3\).
Complete the table of values for \(y = 3x - 1\) for \(x = -2, -1, 0, 1, 2\).
\(3 \times (-2) - 1 = -7\), \(3 \times (-1) - 1 = -4\), then \(-1\), \(2\), \(5\). So \(y\): \(-7, -4, -1, 2, 5\), going up by 3 each time.
Examiner trap
Sign slips with negative \(x\): \(-2 \times (-1)\) is \(+2\), not \(-2\). If one value breaks the steady pattern in your table, recheck that column before you plot.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
A point is 4 units to the left of the origin and 3 units down. What are its coordinates?
Quick check+10 XP
Where is the point \((0, 5)\)?
Quick check+10 XP
For \(y = 4x - 3\), what is \(y\) when \(x = -1\)?
Quick check+10 XP
Which describes the line \(y = -2\)?
Section 03
Plot, rule, check.
Plot every point from the table with a small cross, then draw ONE straight line through them all with a ruler, across the full range of \(x\) asked for. Some lines need no table at all.
Points not in a line? Recheck the table: a linear graph never bends.
\(x = k\) is a vertical line through \(k\) on the \(x\)-axis; \(y = k\) is a horizontal line through \(k\) on the \(y\)-axis.
Equations like \(3x + 2y = 12\) (Extended): put \(x = 0\) to find where it crosses the \(y\)-axis, and \(y = 0\) for the \(x\)-axis.
Is a point on the line? Substitute its \(x\) and \(y\): the equation must balance exactly.
Worked example Using the axis crossings (Extended)
Draw the graph of \(3x + 2y = 12\).
Put \(x = 0\): \(2y = 12\), so \(y = 6\). The line passes through \((0, 6)\).
Put \(y = 0\): \(3x = 12\), so \(x = 4\). The line passes through \((4, 0)\).
Find a third point as a check: \(x = 2\) gives \(6 + 2y = 12\), so \(y = 3\).
Plot \((0, 6)\), \((2, 3)\) and \((4, 0)\) and rule a line through all three.
Answer: A straight line through \((0, 6)\), \((2, 3)\) and \((4, 0)\)
Worked example Vertical and horizontal lines
Write down the coordinates of the point where the lines \(x = 3\) and \(y = -2\) cross.
Every point on \(x = 3\) has \(x\)-coordinate 3 (a vertical line).
Every point on \(y = -2\) has \(y\)-coordinate \(-2\) (a horizontal line).
The crossing point lies on both, so it has both properties.
Answer: \((3, -2)\)
Worked example Testing a point
Does the point \((5, -6)\) lie on the line \(y = 4 - 2x\)?
Mixing up \(x = 3\) and \(y = 3\). The line \(x = 3\) is VERTICAL (every point has \(x = 3\)), even though it crosses the \(x\)-axis. Picture a few points on it before you draw.
Stage 2 · Exam
Exam-style questions
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Exam style+20 XP
A table of values is drawn up for \(y = 3 - 2x\). What is the value of \(y\) when \(x = -2\)?
Exam style+20 XP
(Extended) Which of these points lies on the line \(3x + 2y = 12\)?
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The lines \(x = -2\) and \(y = 3\) are drawn on a grid. Where do they cross?
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\(A(-3, 1)\), \(B(1, 1)\) and \(C(1, -2)\) are three corners of a rectangle \(ABCD\). What is the area of the rectangle?
Skill unlocked
Coordinates and straight-line graphs, plotted.
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