Chapter 3 · Class 10 CBSE · MCQ Test
Pair of Linear Equations in Two Variables MCQ Test — Class 10 CBSE
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Pair of Linear Equations in Two Variables — MCQ Questions
1For a pair of linear equations in two variables, (x, y) = (p, q) is a solution if:
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Answer: It satisfies both equations simultaneously.
Hint: A solution to a system of equations must work for *all* equations in the system.
Solution:
A solution to a pair of linear equations must satisfy both equations when the values of x and y are substituted into them.
Graphically, this solution represents the point where the two lines corresponding to the equations intersect.
2If the graph of a pair of linear equations in two variables shows two lines intersecting at a single point, then the system of equations has:
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Answer: Exactly one solution.
Hint: The solution to a system of equations is represented by the point(s) where their graphs meet.
Solution:
When two lines intersect at a single point, that point is the unique common point to both lines.
Therefore, the system of equations has exactly one solution, also known as a unique solution.
3For what value of 'k' will the pair of linear equations x + ky = 3 and 3x + 2y = 1 NOT have a unique solution?
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Answer: 2/3
Hint: For a system to NOT have a unique solution, the ratio a1/a2 must be equal to b1/b2.
Solution:
For a system of linear equations `a1x + b1y + c1 = 0` and `a2x + b2y + c2 = 0` to not have a unique solution, we must have a1/a2 = b1/b2.
From the given equations, `a1=1`, `b1=k`, `a2=3`, `b2=2`.
So, we set the ratios equal: `1/3 = k/2`.
Cross-multiplying gives `2 = 3k`, which means `k = 2/3`.
4Ravi is trying to solve the system: x + y = 7 and 2x - 3y = 4 using the substitution method. He correctly expresses x from the first equation as x = 7 - y. Which of the following is the correct next step to substitute into the second equation?
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Answer: 2(7 - y) - 3y = 4
Hint: The expression for 'x' should replace 'x' in the *other* equation, while the 'y' term in that equation remains as is.
Solution:
Ravi has derived `x = 7 - y` from the first equation.
To use the substitution method, this expression for `x` must be substituted into the second equation, `2x - 3y = 4`.
Replacing `x` with `(7 - y)` in the second equation gives `2(7 - y) - 3y = 4`.
5If x and y are two numbers such that their sum is 8 and twice the first number minus the second number is 7, which pair of linear equations represents this?
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Answer: x + y = 8, 2x - y = 7
Hint: 'Sum' implies addition, 'twice the first number' means multiplication by 2, and 'minus the second number' implies subtraction.
Solution:
The phrase 'their sum is 8' translates directly to the equation `x + y = 8`.
The phrase 'twice the first number minus the second number is 7' translates to `2x - y = 7`.
Combining these, the correct pair of linear equations is `x + y = 8` and `2x - y = 7`.
6A pair of equations is called 'linear' because their graphical representation always forms:
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Answer: Straight lines.
Hint: Recall the definition of a linear equation – the variables have a maximum power of one.
Solution:
An equation is classified as linear if the highest power of its variables is 1.
When such an equation is plotted on a coordinate plane, its graph always produces a straight line.
7Which of the following pairs of equations represents parallel lines (no solution)?
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Answer: x + 2y = 4, 3x + 6y = 10
Hint: For parallel lines, the ratios a1/a2 and b1/b2 must be equal, but c1/c2 must be different.
Solution:
For two linear equations `a1x + b1y + c1 = 0` and `a2x + b2y + c2 = 0` to represent parallel lines, the condition is `a1/a2 = b1/b2 ≠ c1/c2`.
Let's check the ratios for option C: `x + 2y - 4 = 0` and `3x + 6y - 10 = 0`.
`a1/a2 = 1/3`.
`b1/b2 = 2/6 = 1/3`.
`c1/c2 = -4/-10 = 2/5`.
Since `1/3 = 1/3 ≠ 2/5`, option C represents parallel lines.
8If the graphs of two linear equations are the same line, how many solutions does the system have?
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Answer: Infinitely many solutions.
Hint: If the lines are the same, every point on one line is also on the other.
Solution:
When two lines are coincident (meaning they are the same line), they overlap completely.
Every point that lies on one line also lies on the other line.
Therefore, there are infinitely many common points, leading to infinitely many solutions.
9If (x, y) = (2, -1) is a solution to the equation 3x + ky = 8, what is the value of k?
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Answer: -2
Hint: Substitute the given values of x and y into the equation and then solve for k.
Solution:
Substitute `x = 2` and `y = -1` into the equation `3x + ky = 8`.
`3(2) + k(-1) = 8`
`6 - k = 8`
Subtract 6 from both sides: `-k = 8 - 6` which simplifies to `-k = 2`.
Multiply by -1 to find k: `k = -2`.
10A pair of linear equations that has no solution is called an _______ system.
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Answer: inconsistent
Hint: Think about what it means for a system to 'not have a solution' in terms of consistency.
Solution:
A system of linear equations is classified based on the number of solutions it has.
If a system has no solution (graphically, this means parallel lines), it is called an inconsistent system.
If it has at least one solution (intersecting or coincident lines), it is called a consistent system.
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Tips for Pair of Linear Equations in Two Variables MCQs
- 1Read each question carefully and identify what is being asked before looking at the options.
- 2Try to solve the problem mentally or on paper first, then match your answer with the options.
- 3Use elimination — rule out clearly wrong options to improve your chances even when unsure.
- 4Check units, signs, and edge cases — these are common traps in Pair of Linear Equations in Two Variables MCQs.
- 5Review your mistakes after completing the test to build lasting understanding.
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