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Linear Equations Quiz - CBSE Class 10 Mathematics

Free practice quiz: Linear Equations (CBSE Class 10 Mathematics). 12 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 10 Mathematics practice quiz has 12 multiple-choice questions on Linear Equations, and 12 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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About this topic

A pair of linear equations in two variables has the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Its solution is the pair (x, y) satisfying both equations. Solve by substitution, elimination, or graph. The ratios of coefficients tell you in advance how many solutions exist.

General form: ax + by + c = 0, where a and b are not both zero.
Graph of a linear equation in two variables is always a straight line.
A solution must satisfy both equations at the same time.
Substitution: express one variable in terms of the other, then put it in the second equation.
Elimination: make coefficients of one variable equal, then add or subtract equations.
Consistent pair has at least one solution; inconsistent pair has none.
Always verify the answer by putting x and y back in both equations.

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What a Solution Means

A pair of linear equations in x and y has the general form shown below.
A solution is an ordered pair (x, y) that makes BOTH equations true at the same time.
Graphically, the solution is the point where the two lines meet.

General pair

\[a_1x + b_1y = c_1,\quad a_2x + b_2y = c_2\]

Solution as a point

\[(x, y) \text{ satisfies both equations}\]

Common mistake: Checking a found pair in only one equation. It must satisfy both equations, otherwise it is not the solution.

Substitution Method

Make one variable the subject of one equation, for example y in terms of x.
Put this expression into the other equation, solve for one variable, then substitute back.

Step 1: express y

\[y = \frac{c_1 - a_1x}{b_1},\quad b_1 \neq 0\]

Step 2: substitute

\[a_2x + b_2\left(\frac{c_1 - a_1x}{b_1}\right) = c_2\]

Common mistake: Substituting the expression back into the same equation it came from. This gives a true statement but no new information.

Elimination Method

Multiply the equations so that one variable has equal or opposite coefficients in both.
Add or subtract the equations to remove that variable, solve, then substitute back to find the other.

Opposite coefficients: add

\[(ax + by) + (cx - by) = (a + c)x\]

Equal coefficients: subtract

\[(ax + by) - (cx + by) = (a - c)x\]

Common mistake: Multiplying only the left side of an equation by a number and forgetting to multiply the constant on the right side too.

Number of Solutions by Ratios

Compare the coefficient ratios to find whether the lines intersect, coincide or are parallel.
Coincident lines give infinitely many solutions. Parallel lines give no solution.

Unique solution (intersecting)

\[\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\]

Infinitely many (coincident)

\[\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\]

No solution (parallel)

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

Common mistake: Trying to find one intersection point for coincident lines. When one equation is a multiple of the other, every point on the line is a solution.

Exam tips

If a variable has coefficient 1 or -1, substitution is usually quickest.
If one variable already has equal or opposite coefficients, use elimination directly.
Always check your answer by putting x and y into both original equations.

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

The solution of 4x - y = 7 and 2x + y = 1 is:
Show answer
Answer: A. x = 2, y = -3
Add the two equations: 4x - y + 2x + y = 7 + 1. This gives 6x = 8, so x = 2. Substitute x into 2x + y = 1 gives y = -3.
Solve for x and y: x - 2y = 3 and 3x + y = 5.
Show answer
Answer: A. x = 1, y = 1
From x - 2y = 3, x = 2y + 3. Substitute in 3x + y = 5 gives y = 1. Then x = 1.
What is the value of x in the equation: 3x + 4y = 10 and x = 2y + 1?
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Answer: A. x = 3
Substitute x = 2y + 1 in 3x + 4y = 10 simplifies to y = 1, then x = 3.
Determine the y-value in: x + y = 5 and 2x + 2y = 10.
Show answer
Answer: D. Infinite solutions
Both equations represent the same line, thus there are infinite solutions.
Solve for x and y: 4x + y = 9 and 3x - y = 3.
Show answer
Answer: A. x = 2, y = 1
Add equations: (4x + y) + (3x - y) = 12 gives 7x = 12 so x = 2. Substitute to find y = 1.

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