ax+by

Chapter 3 · Class X Mathematics · MCQ Practice

MCQ Practice Arena

Pair of Linear Equations in Two Variables

Consistent, Inconsistent or Dependent — Decide in 5 Seconds and Score

📋 50 MCQs ⭐ 32 PYQs ⏱ 80 sec/Q

🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs

16Easy

22Medium

12Hard

32PYQs

80 secAvg Time/Q

8Topics

Easy 32% Medium 44% Hard 24%

Why Practise These MCQs?

CBSE Class XNTSEState BoardsOlympiad

This is the highest-MCQ-density chapter in Class X Boards — 10–12 marks over multiple question types. The consistency ratio test (a₁/a₂ vs b₁/b₂ vs c₁/c₂) is a guaranteed MCQ. Word problems form 30% of this bank. NTSE includes elegant disguised simultaneous equation problems. Cross-multiplication is tested for speed in BITSAT-style papers.

Topic-wise MCQ Breakdown

Graphical Method & Geometry of Lines6 Q

Consistency Conditions (Ratio Test)10 Q

Substitution Method8 Q

Elimination Method8 Q

Cross-Multiplication Method5 Q

Reducible Equations (1/x, 1/y type)5 Q

Word Problems6 Q

k-value Problems (Find k for consistency)2 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\text{Unique solution: } a_1/a_2 \ne b_1/b_2$

$\text{Inconsistent: } a_1/a_2 = b_1/b_2 \ne c_1/c_2$

$\text{Infinitely many: } a_1/a_2 = b_1/b_2 = c_1/c_2$

$x = (b_1c_2-b_2c_1)/(a_1b_2-a_2b_1)\ \text{(Cross-mult.)}$

$y = (c_1a_2-c_2a_1)/(a_1b_2-a_2b_1)$

MCQ Solving Strategy

For consistency MCQs, immediately write the three ratios a₁/a₂, b₁/b₂, c₁/c₂ and compare — this takes under 15 seconds. For word problems, define variables in the first line, write two equations, then choose the fastest algebraic method. Elimination method is fastest when coefficients of one variable are already equal or easily made equal. For reducible equations, substitute u = 1/x, v = 1/y.

⚠ Common Traps & Errors

⚠c₁/c₂ ratio uses the constant term — check sign (ax+by+c=0 form)
⚠Infinitely many solutions ≠ unique solution — ratios must ALL be equal
⚠Word problem: "sum" gives addition, "difference" gives subtraction — read carefully
⚠Cross-multiplication denominator a₁b₂−a₂b₁ = 0 means no unique solution
⚠Reducible: substitute back u,v to get x,y at the end

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Identify consistent/inconsistent from ratio test, solve by substitution

② Medium

Elimination and cross-multiplication, k-value for infinite solutions

③ Hard

Reducible equations, complex word problems (ages, fractions, boats)

★ PYQ

CBSE — graphical + algebraic combo; NTSE — elegant word problems

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 📘✔️ Exercises Step-by-step NCERT solutions ✔️❌ True / False Concept-based True & False ➡ Next Chapter Quadratic Equations

🎯 Knowledge Check

Maths — PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

50 Questions Class 10 MCQs

Which of the following is a linear equation in two variables?

a a. $2x^2 + 3 = 0$ b b. $x + y = 7$ c c. $3x^2 + y = 1$ d d. $4y^2 - x = 9$

The graph of a linear equation in two variables is always a:

a a. Curve b b. Circle c c. Straight line d d. Parabola

A pair of linear equations represents lines that intersect at one point. The system is:

a a. Inconsistent b b. Consistent with a unique solution c c. Dependent d d. Inconsistent with infinite solutions

For parallel lines, the system of equations has:

a a. One solution b b. Two solutions c c. No solution d d. Infinite solutions

For coincident lines, the system has:

a a. Exactly one solution b b. Two solutions c c. Infinite solutions d d. No solution

Condition for unique solution is:

a a. a1a2=b1b2\frac{a_1}{a_2} = \frac{b_1}{b_2}a2?a1??=b2?b1?? b b. $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$?? c c. $\frac{b_1}{b_2} = \frac{c_1}{c_2}$ d d. $\frac{a_1}{a_2} = \frac{c_1}{c_2}$

The equation $2x + 3y - 6 = 0$ represents:

a a. A line b b. A point c c. A curve d d. A parabola

Which method eliminates one variable directly?

a a. Substitution b b. Elimination c c. Graphical d d. Trial and error

In substitution method, we:

a a. Subtract equations b b. Add equations c c. Express one variable in terms of another d d. Multiply equations

Cross multiplication method applies only when:

a a. Lines coincide b b. Lines intersect c c. $a_1b_2 - a_2b_1 \neq 0$ d d. $a_1b_2 - a_2b_1 = 0$

The graph of $3x=3$ is:

a a. Horizontal line b b. Vertical line c c. Slant line d d. Curve

The graph of $y = 5$ is:

a a. Vertical line b b. Horizontal line c c. Curve d d. Circle

If two equations represent the same line, they are:

a a. Independent b b. Inconsistent c c. Dependent d d. Parallel

Two equations with no common solution are called:

a a. Consistent b b. Inconsistent c c. Dependent d d. One solution

Which is the standard form of a linear equation?

a a. $ax^2 + by + c = 0$ b b. $ax + by^2 + c = 0$ c c. $ax + by + c = 0$ d d. $x^2 + y^2 = 0$

A pair of equations represents intersecting lines if:

a a. Ratios of coefficients are equal b b. Lines coincide c c. Ratios of $a_1/a_2$and $b_1/b_2$? are different d d. Lines are parallel

The solution of a pair of linear equations is:

a a. A point b b. A line c c. A curve d d. A circle

Which is an example of parallel lines?

a a. $x + y = 5$ and $x - y = 1$ b b. $x + 3y = 6$ and $4x + 6y = 12$ c c. $x + y = 7$ and $x - y = 2$ d d. None

Equation $3x - 9 = 0$ in two variable form is:

a a. $3x - 9 + 0y = 0$ b b. $3x + 9y = 0$ c c. $3x - 9y = 0$ d d. $x + y = 0$

Number of solutions of coincident lines is:

a a. 1 b b. 0 c c. 8 d d. 2

If $\frac{a_1}{a_2} = \frac{b_1}{b_2}$?? but $\frac{a_1}{a_2} \neq \frac{c_1}{c_2}$?, lines are:

a a. Intersecting b b. Parallel c c. Coincident d d. Vertical

To find the graph of a linear equation, we need:

a a. Only one point b b. At least two points c c. Three points compulsory d d. No point

A linear equation in two variables has:

a a. No solutions b b. Only one solution c c. Many solutions d d. Depends on equation

Which method is most suitable when one variable is already isolated?

a a. Graphical b b. Elimination c c. Substitution d d. Cross-multiplication

The determinant $D = a_1b_2 - a_2b_1$? equals 0 when:

a a. Lines parallel or coincident b b. Lines intersect c c. Unique solution exists d d. Both variables eliminated

Pair of equations always represents:

a a. Two points b b. Two lines c c. Two curves d d. Two circles

Solve: $x + y = 6$, $x - y = 2$. The value of $x$ is:

a a. 1 b b. 2 c c. 4 d d. 3

In the equation $5x + 0y = 10$, the graph is:

a a. Horizontal b b. Vertical c c. Curve d d. Circle

In elimination, to remove $x$, we:

a a. Multiply equations to equalize coefficients of $x$ b b. Change signs c c. Add or subtract after equalizing d d. Both a and c

Lines$x = 2$ and $x = 3$ are:

a a. Intersecting b b. Parallel c c. Coincident d d. Perpendicular

Lines $y = 3$ and $y = -2$ are:

a a. Parallel b b. Coincident c c. Intersecting d d. Same

In the equation $ax + by + c = 0$, the slope is:

a a. $-a/b$ b b. $a/b$ c c. $-b/a$ d d. $c/a$

Graphical solution is:

a a. Exact b b. Approximate c c. Incorrect d d. Never used

If a pair has one solution, the lines:

a a. Overlap b b. Are parallel c c. Intersect d d. Do not meet

Solving equations means finding:

a a. Variables b b. Values satisfying both equations c c. One equation d d. Only graphs

A solution that satisfies both equations is called:

a a. Common root b b. Intersection c c. Common solution d d. None

Linear equations represent:

a a. First-degree equations b b. Second-degree equations c c. Third-degree equations d d. No degree

Solve by elimination: $2x + 2y = 10,\ x - y = 1$. Value of $y$:

a a. 3 b b. 2 c c. 5 d d. 1

Equation of a line parallel to $x = 5$ is:

a a. $y = 5$ b b. $x = 3$ c c. $x + y = 0$ d d. $x = y$

Equation of a line parallel to$y = 7$ is:

a a. $y = 2$ b b. $x = 7$ c c. $x + y = 7$ and $x - y = 2$ d d. $y = x$

If two lines intersect, their slopes are:

a a. Equal b b. Unequal c c. Zero d d. Infinite

If lines coincide, slopes are:

a a. Different b b. Equal c c. Zero d d. Infinite

If equations have no solution, they are:

a a. Consistent b b. Inconsistent c c. Dependent d d. Coincident

Solve: $3x + y = 7,\ 3x - y = 5$. Value of $y$:

a a. 1 b b. 2 c c. 3 d d. 4

Solve: $x + 2y = 8,\ x - y = 2$. Value of $x$:

a a. 4 b b. 3 c c. 2 d d. 6

Determine nature: $2x + 3y = 6,\ 4x + 6y = 12$.

a a. Intersecting b b. Parallel c c. Coincident d d. Perpendicular

Which of the following is not linear?

a a. $x + y = 5$ b b. $2x - 3 = 0$ c c. $3y = 9$ d d. $xy = 5$

In pair of equations, number of variables is:

a a. One b b. Two c c. Three d d. Four

If elimination results in $0 = 0$, then:

a a. No solution b b. Unique solution c c. Infinite solutions d d. Contradiction

If elimination results in $0 = 5$, then:

a a. Infinite solutions b b. Unique solution c c. No solution d d. Many solutions

📚

ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025

Sharing this chapter

Pair Of Linear Equations | Maths Class 10

Pair Of Linear Equations | Maths Class 10 — Complete Notes & Solutions · academia-aeternum.com

🎓 Class 10 📐 Maths 📖 NCERT ✅ Free Access 🏆 CBSE · JEE

Share on

X / Twitter

Facebook

Instagram

Threads

Quora

Discord

academia-aeternum.com/class-10/mathematics/pair-of-linear-equations/mcqs/ Copy link

💡

Exam tip: Sharing chapter notes with your study group creates a reinforcement loop. Teaching a concept is the fastest path to mastering it.

Frequently Asked Questions

An equation that can be written in the form $ax + by + c = 0$, where $a, b, c$ are real numbers and $a$ and $b$ are not both zero.

Two linear equations involving the same variables $x$ and $y$ that are solved together to find common solutions.

$a x + b y + c = 0$, where $a$, $b$, $c$ are constants.

A pair of values $(x, y)$ that satisfies both equations simultaneously.

Two straight lines on a coordinate plane.

(i) One solution, (ii) No solution, (iii) Infinitely many solutions.

When the two lines intersect at exactly one point.

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

When the lines are parallel and never intersect.

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

When both equations represent the same line (coincident lines).

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Plotting both equations as lines and finding their point of intersection.

The common solution of both equations.

A pair of equations with at least one solution (unique or infinite).

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES – Learning Resources

Detailed Notes

True / False

NCERT Solutions

ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025

Get in Touch

Let's Connect

Questions, feedback, or suggestions?
We'd love to hear from you.

Pair of Linear Equations in Two Variables

MCQ Bank Snapshot

Why Practise These MCQs?

Topic-wise MCQ Breakdown

Must-Know Formulae Before You Start

MCQ Solving Strategy

⚠ Common Traps & Errors

Difficulty Ladder

Continue Your Preparation

Maths — PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Frequently Asked Questions

Q 01 What is a linear equation in two variables?

Q 02 What is a pair of linear equations in two variables?

Q 03 What is the standard form of a linear equation?

Q 04 What is meant by a solution of a pair of linear equations?

Q 05 How many lines does a pair of linear equations represent?

Q 06 What are the three possible cases for the solution of a pair of linear equations?

Q 07 When does a pair have a unique solution?

Q 08 What condition represents intersecting lines?

Q 09 When does a pair have no solution?

Q 10 What condition represents parallel lines?

Q 11 When does a pair have infinitely many solutions?

Q 12 Condition for coincident lines?

Q 13 What is the graphical method of solving equations?

Q 14 What does the point of intersection represent?

Q 15 Define consistent equations.

Recent Posts

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES – Learning Resources

Detailed Notes

True / False

NCERT Solutions

Let's Connect