Find any two solutions of 2x + y = 7. Draw the graph of this equation. Use the graph to find 4 more solutions.
Step-by-step solution
Idea: The quickest two points come from putting x = 0 (where the line meets the y-axis) and y = 0 (where it meets the x-axis). Two points fix a straight line; any other point read off the line is another solution.
- Put x = 0: (2 × 0) + y = 7, so y = 7. One solution is (0, 7).½ mark
- Put y = 0: 2x + 0 = 7, so x = 72. Another solution is (72, 0).½ mark
- Table of values:
½ markx 0 72 y 7 0 - Plot A(0, 7) and B(72, 0) and draw the straight line through them. This line is the graph of 2x + y = 7.1 mark
- Read points that lie on the line: (1, 5), (2, 3), (3, 1) and (32, 4). Each one is a solution.1 mark
- Check them: 2(1) + 5 = 7, 2(2) + 3 = 7, 2(3) + 1 = 7, 2(32) + 4 = 3 + 4 = 7 ✓. So the four more solutions are (1, 5), (2, 3), (3, 1), (32, 4).½ mark
Check: Each point read from the graph must satisfy the equation exactly. (2, 3): 4 + 3 = 7 ✓. A wrong reading such as (2, 4) gives 8 ✗.
Answer to write in the exam
x = 0: y = 7 ⇒ (0, 7)
y = 0: 2x = 7 ⇒ x = 72 ⇒ (72, 0)
| x | 0 | 72 |
| y | 7 | 0 |
Plot A(0, 7), B(72, 0) and join AB (graph of 2x + y = 7)
From the graph: (1, 5), (2, 3), (3, 1), (32, 4) lie on the line
∴ Four more solutions: (1, 5), (2, 3), (3, 1), (32, 4)
Common mistakes that cost marks
- Plotting (72, 0) at 7 or at 2 instead of at 3.5 on the x-axis.
- Swapping coordinates and plotting (7, 0) for (0, 7).
- Reading points from the graph without checking them in the equation. A small reading error gives a point that is not a solution.
How this can come in the exam
The graph of 3x + y = 9 meets the x-axis at
- (0, 9)
- (9, 0)
- (3, 0)
- (0, 3)
Show answer
(C) (3, 0)
On the x-axis y = 0, so 3x = 9, x = 3: (3, 0).
Draw the graph of x + 2y = 6. From the graph, find the value of y when x = 2.
Show answer
Points (0, 3) and (6, 0); join them (2 marks). At x = 2 the line is at y = 2, since 2 + 2(2) = 6 (1 mark).Try one yourself
Find two solutions of x + y = 5, draw its graph and read off two more solutions.
Show answer
(0, 5) and (5, 0). From the graph: e.g. (1, 4) and (3, 2); check 1 + 4 = 5, 3 + 2 = 5 ✓.
More questions like this
- 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.
- Find any two solutions for each of the following equations:
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