Pair of linear equations: the graphical method
Draw both lines on one graph. The point where they meet is the solution. The lines can intersect, coincide or be parallel.
What a pair of linear equations is
A linear equation in two variables has the form ax + by + c = 0, where a and b are not both zero. Its graph is a straight line. Two such equations in the same x and y make a pair of linear equations. A solution is a pair of values (x, y) that satisfies both equations at once. On a graph, it is the point that lies on both lines.
Three possible pictures
| The two lines | Number of solutions | The pair is called |
|---|---|---|
| Intersect at one point | Exactly one (unique) | Consistent |
| Coincide (lie on top of each other) | Infinitely many | Consistent (dependent) |
| Are parallel | None | Inconsistent |
How to draw the graph: x + 2y = 8 and 2x − y = 1
- 1Write y in terms of x for each equation: y = (8 − x)/2 and y = 2x − 1.
- 2Make a table of at least 3 points for each line. Line 1: (0, 4), (2, 3), (4, 2). Line 2: (0, −1), (1, 1), (2, 3).
- 3Draw the x-axis and y-axis on graph paper. Choose a scale such as 1 cm = 1 unit and write it in a corner. Label O, x and y.
- 4Plot the points of line 1. Join them with a ruler and extend both ends. Label the line x + 2y = 8.
- 5Plot and join the points of line 2 the same way. Label it 2x − y = 1.
- 6★Mark the point where the lines meet and write its coordinates: (2, 3).
- 7Check in both equations: 2 + 2(3) = 8 and 2(2) − 3 = 1. So x = 2, y = 3.
★ key step
Example 1: a word problem solved by graph
2 pencils and 3 erasers cost ₹ 19. 1 pencil and 1 eraser cost ₹ 8. Find the cost of each by drawing a graph.
- Let one pencil cost ₹ x and one eraser cost ₹ y.
- Equations: 2x + 3y = 19 and x + y = 8.
- Points on x + y = 8: (0, 8), (5, 3), (8, 0).
- Points on 2x + 3y = 19: (2, 5), (5, 3), (8, 1).
- Plot both lines on one graph. They meet at (5, 3).
- Check: 2(5) + 3(3) = 10 + 9 = 19 and 5 + 3 = 8.
A pencil costs ₹ 5 and an eraser costs ₹ 3.
Example 2: the triangle formed with the x-axis
Draw the graphs of x + y = 6 and x − 2y = 0. Find the vertices of the triangle formed by the two lines and the x-axis, and its area.
- Points on x + y = 6: (0, 6), (4, 2), (6, 0).
- Points on x − 2y = 0: (0, 0), (2, 1), (4, 2).
- Plot both lines. They meet at (4, 2).
- x + y = 6 cuts the x-axis at (6, 0). x − 2y = 0 passes through (0, 0).
- Vertices of the triangle: (4, 2), (0, 0) and (6, 0). Shade it.
- Base along the x-axis = 6 − 0 = 6 units. Height = 2 units (the y-coordinate of (4, 2)).
- Area = (1/2) × base × height = (1/2) × 6 × 2 = 6 square units.
Vertices (4, 2), (0, 0) and (6, 0); area 6 square units.
Two auto-rickshaw fares with different fixed charges and different rates per km give two straight lines. The point where the lines meet is the distance at which both fares are the same.
1 marks: “The graphs of x + y = 4 and 2x + 2y = 8 are a pair of which kind of lines?” Coincident lines, because the second equation is 2 times the first.
3 marks: “Solve graphically: x + y = 5 and x − y = 1.” Points on x + y = 5: (0, 5), (5, 0), (3, 2). Points on x − y = 1: (1, 0), (0, −1), (3, 2). The lines meet at (3, 2), so x = 3, y = 2.
5 marks: “Draw the graphs of x − y = 1 and 2x + y = 8. Shade the triangle formed by these lines and the x-axis, and find its area.” The lines meet at (3, 2). They cut the x-axis at (1, 0) and (4, 0). Area = (1/2) × 3 × 2 = 3 square units.
Meet = one answer, same line = endless answers, parallel = no answer.