A linear equation in two variables can be written as ax + by + c = 0, where a and b are not both zero. A solution is an ordered pair (x, y) that makes the equation true. One equation of this kind generally has infinitely many real solutions, represented by points on a straight line.
Find and check ordered pairs
For x + y = 7, choosing x = 2 gives y = 5. Thus (2, 5) is a solution. Choosing x = 0 gives (0, 7). Coordinates have an order: the first value is x and the second is y.
To test a proposed solution, substitute both values into the original equation. A point that does not satisfy the equation does not lie on its graph.
- The equation x = 3 gives a vertical line.
- The equation y = 3 gives a horizontal line.
Draw the graph
Find at least two distinct solution points, choose a suitable scale and mark the axes. Plot x horizontally and y vertically. A third point is useful as a check that your calculation and plotting agree.
Join the points with a straight line and extend it as appropriate. The graph represents all real solutions, not only the few points in your table. A word problem may restrict solutions to non-negative values or whole numbers, even when the line continues further.
Put it into practice
A worked example
Find two points on 2x + y = 6.
- Set x = 0: y = 6, giving (0, 6).
- Set y = 0: 2x = 6, so x = 3, giving (3, 0).
- Check: 2 × 3 + 0 = 6.
Answer: Two suitable points are (0, 6) and (3, 0).
Try it yourself
Have a go before revealing the answer.
Does (2, 3) satisfy x + 2y = 8?
Show answer
Yes. 2 + 2 × 3 = 8.
Answer: Yes. 2 + 2 × 3 = 8.
Find y when x = −1 in y = 3x + 2.
Show answer
y = −3 + 2 = −1, giving the point (−1, −1).
Answer: y = −3 + 2 = −1, giving the point (−1, −1).
Watch out for these
- Do not reverse x and y when plotting an ordered pair.
- Two variables do not mean an equation has exactly two solutions.
Continue with official resources
Match these concept notes to your current textbook and syllabus. They are independent explanations, not official board publications.
