State whether the following sentences are True or False. Justify your answer.
- (i) A linear equation in two variables has only one solution.
- (ii) The graph of a linear equation in two variables always passes through the origin.
- (iii) A linear equation in two variables can never have rational solutions.
- (iv) x = 3 is a valid linear equation in two variables.
- (v) The equation 2x + 3y = 7 has infinitely many solutions.
- (vi) The point (1, 2) is a solution of the equation 2x + 3y = 7.
Step-by-step solution
Idea: A linear equation in two variables has infinitely many solutions forming a straight line. Justify each “False” with a counter-example and each “True” with a reason.
(i) A linear equation in two variables has only one solution.
- False. x + y = 4 has (0, 4), (1, 3), (2, 2), … Every value of x gives a solution, so there are infinitely many.½ mark
(ii) The graph of a linear equation in two variables always passes through the origin.
- False. x + y = 4: (0, 0) gives 0 ≠ 4, so its graph misses the origin. Only lines with c = 0 pass through (0, 0).½ mark
(iii) A linear equation in two variables can never have rational solutions.
- False. 2x + 3y = 5 has the rational (in fact whole-number) solution (1, 1).½ mark
(iv) x = 3 is a valid linear equation in two variables.
- True. It can be written 1×x + 0×y − 3 = 0, with a = 1, b = 0 (not both zero). Its graph is the vertical line through (3, 0).½ mark
(v) The equation 2x + 3y = 7 has infinitely many solutions.
- True. For any x, y = 7 − 2x3 is a matching value, e.g. (2, 1), (5, −1), (−1, 3).½ mark
(vi) The point (1, 2) is a solution of the equation 2x + 3y = 7.
- False. 2(1) + 3(2) = 2 + 6 = 8 ≠ 7.½ mark
Check: (v) Check (5, −1): 10 − 3 = 7 ✓. (iv) Points like (3, 0), (3, 5) all satisfy x = 3 ✓.
Answer to write in the exam
(i)
False. e.g. x + y = 4 has (0, 4), (1, 3), (2, 2), …: infinitely many solutions
(ii)
False. For x + y = 4, (0, 0) gives 0 ≠ 4
Only ax + by = 0 (i.e. c = 0) passes through the origin
(iii)
False. (1, 1) is a solution of 2x + 3y = 5 and 1 is rational
(iv)
True. x = 3 ⇒ 1×x + 0×y − 3 = 0, with a = 1, b = 0
(v)
True. For every x, y = 7 − 2x3; e.g. (2, 1), (5, −1), (−1, 3)
(vi)
False. 2(1) + 3(2) = 8 ≠ 7
Common mistakes that cost marks
- Marking (iv) False because y is missing. A missing variable just has coefficient 0.
- Writing True/False without a reason or example. The question asks you to justify each one.
How this can come in the exam
Which statement is true?
- y = −2 is not a linear equation in two variables
- The graph of x − y = 0 passes through the origin
- x + y = 1 has exactly two solutions
- (2, 2) is a solution of x + y = 5
Show answer
(B) The graph of x − y = 0 passes through the origin
(0, 0) satisfies x − y = 0. The others are false.
Try one yourself
True or False: The point (0, 0) lies on 5x − 2y = 3. Justify.
Show answer
False: 0 − 0 = 0 ≠ 3.
More questions like this
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