Consider all the solutions of a linear equation in two variables, expressed as ordered pairs. What do we get when we plot the points corresponding to these ordered pairs?
Step-by-step solution
Idea: Rewrite the equation as y = mx + d. Each step of 1 in x changes y by the same amount m, so the plotted points climb or fall at a steady rate and line up.
- Take 3x + 2y = 12. Rewrite it as 2y = −3x + 12, i.e. y = −32x + 6.½ mark
- Some solutions: (−1, 7.5), (0, 6), (1, 4.5), (4, 0). Each time x goes up by 1, y goes down by the same 1.5, so the points sit on a straight line (see the graph).½ mark
- This is true of every linear equation in two variables: plotting all its solutions gives a straight line, called the graph of the equation.½ mark
- It works both ways: each solution (x, y) is a point on the line, and the coordinates of any point on the line form a solution.½ mark
Check: Pick a new point on the line, e.g. (2, 3): 3(2) + 2(3) = 12 ✓. Pick a point off the line, e.g. (2, 4): 6 + 8 = 14 ✗.
Answer to write in the exam
3x + 2y = 12 ⇒ y = −32x + 6
Solutions: (−1, 7.5), (0, 6), (1, 4.5), (4, 0)
Plotted, these points lie on one straight line
∴ The solutions of a linear equation in two variables form a straight line (its graph)
Common mistakes that cost marks
- Joining only the plotted points with short segments and stopping. The graph is the whole line, extended both ways.
- Expecting a curve. A linear equation (powers of x and y equal to 1) always gives a straight line.
How this can come in the exam
Assertion (A): The point (2, 3) lies on the graph of 3x + 2y = 12.
Reason (R): Every solution of a linear equation in two variables is a point on its graph.
- Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show answer
(A) Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
3(2) + 2(3) = 12, so (2, 3) is a solution; R explains why it lies on the line.
Try one yourself
Write three solutions of y = 2x − 1 and say what you get when you plot them.
Show answer
(0, −1), (1, 1), (2, 3); plotted, they lie on one straight line, the graph of y = 2x − 1.
More questions like this
- Find any two solutions of 2x + y = 7. Draw the graph of this equation. Use the graph to find 4 more solutions.
- Find any two solutions for each of the following equations and draw their graphs:
- In the equations shown below, a and b are unknown numbers.
3ax + 4y = −2
2x + by = 14.
If (−3, 4) is the solution of both equations, find the values of a and b. - Four friends — Ranju, Meena, Farhan, and Toshi — are solving problems.
Ranju: I noticed something! If c = 0 in the standard form of a line ax + by + c = 0, then the line must pass through the origin. Look, if I substitute x = 0 and y = 0, in equation ax + by = 0, the equation is satisfied. So, the origin lies on the line! We can also say that the line passes through the origin.
Farhan: Let us try for the equation 2x + 3y = 0. Here a = 2, b = 3 but c = 0. If we substitute x = 0, then we get 3y = 0 or y = 0. This means (0, 0) lies on the line. So yes, this line passes through the origin.
Toshi: Suppose our equation has b = c = 0, say, 5x = 0. This becomes x = 0 which is the equation of the y-axis. And the y-axis passes through the origin.
Meena: And if we take a = c = 0, say, 7y = 0, that means y = 0, which is the equation of the x-axis that also passes through the origin.
So, they conclude: Whenever c = 0, the line ax + by + c = 0 will always pass through the origin, irrespective of the values of a or b. Do you agree with them? - Verify if the ordered pair (4, 3) is a solution of 5x − 6y = 2. Explain your reasoning.
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