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Class 10 › Mathematics › Pair of Linear Equations in Two Variables
CBSE2026Class Class 10 · Mathematics1 Marks · Long✅ Verified
<b>Assertion (A) :</b> The value of $p$ for which the system of equations $4x+py+8=0$ and $2x+2y+2=0$ is consistent is 4.
<b>Reason (R) :</b> The system of equations $a_{1}x+b_{1}y=c_{1}$ and $a_{2}x+b_{2}y=c_{2}$ is consistent with infinitely many solutions, if $\dfrac{a_{1}}{a_{2}}=\dfrac{b_{1}}{b_{2}}=\dfrac{c_{1}}{c_{2}}$.
Directions: Select the correct answer from the codes given below —
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.
✅ Answer & Solution
Step 1: Write the given equations in standard form.
$$4x+py+8=0,\qquad 2x+2y+2=0$$
So $a_{1}=4,\;b_{1}=p,\;c_{1}=8$ and $a_{2}=2,\;b_{2}=2,\;c_{2}=2$.
Step 2: Compute the ratios.
$$\frac{a_{1}}{a_{2}}=\frac{4}{2}=2,\qquad \frac{b_{1}}{b_{2}}=\frac{p}{2},\qquad \frac{c_{1}}{c_{2}}=\frac{8}{2}=4$$
Step 3: Put $p=4$ (as claimed in the Assertion). Then $\dfrac{b_{1}}{b_{2}}=2$, so
$$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=2\neq 4=\frac{c_{1}}{c_{2}}$$
Step 4: This is the condition for NO solution (parallel lines), i.e. the system is INCONSISTENT for $p=4$. So Assertion (A) is FALSE. (For consistency we actually need $p\neq 4$.)
Step 5: Reason (R) states the standard condition for infinitely many solutions, which is correct. So (R) is TRUE.
Correct option is (D) — (A) is false, but (R) is true.
