Consider the graph of the equation 3x − 7y = 21 shown below. Does the point C (2, 3) lie on the line? Does it satisfy the equation? Can points that do not lie on the line satisfy the equation?
Step-by-step solution
Idea: The graph of an equation is exactly the set of its solutions. So “lies on the line” and “satisfies the equation” always go together.
- From the graph, C(2, 3) is above the x-axis while the line at x = 2 is below it, so C does not lie on the line.½ mark
- Substitute: 3(2) − 7(3) = 6 − 21 = −15 ≠ 21. So C does not satisfy the equation.1 mark
- On the line at x = 2: 6 − 7y = 21 ⇒ y = −157 ≈ −2.1, which agrees with the picture.½ mark
- No point off the line can satisfy the equation: every solution of a linear equation is a point on its graph, and every point on the graph is a solution. A point that satisfied the equation would have to be on the line.1 mark
Check: The marked points A(7, 0) and B(0, −3) do satisfy it: 21 − 0 = 21 ✓, 0 + 21 = 21 ✓.
Answer to write in the exam
At C(2, 3): LHS = 3(2) − 7(3) = 6 − 21 = −15
RHS = 21; LHS ≠ RHS
∴ C does not satisfy the equation and does not lie on the line
Every solution lies on the graph and every point of the graph is a solution
∴ Points not on the line cannot satisfy the equation
Common mistakes that cost marks
- Working out 3(2) − 7(3) as 6 − 21 = 15 and missing the minus sign.
- Thinking a point can satisfy an equation without being on its graph. The graph is, by definition, all the solutions.
How this can come in the exam
Assertion (A): The point (1, 1) does not lie on the graph of 2x − 5y = 4.
Reason (R): 2(1) − 5(1) = −3 ≠ 4.
- 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.
Both true, and failing the equation is exactly why the point is off the line.
For what value of k does the point (k, 2) lie on the line 3x − 7y = 21?
Show answer
3k − 14 = 21 (1 mark) ⇒ 3k = 35 ⇒ k = 353 (1 mark).Try one yourself
Does the point (14, 3) lie on the line 3x − 7y = 21?
Show answer
42 − 21 = 21 ✓. Yes, it lies on the line.
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