NCERT MCQ Solutions for Class 9 Maths Chapter 4 Linear Equations in Two Variables

Simplify the equation
2x+1=x+3
Subtract x from both sides to get all x-terms on one side.
2x−x+1=3
x+1=3
Subtract 1 from both sides to isolate x.
x=3−1
x=2
The solution to the equation is x=2x = 2x=2, so the correct answer is 2.

When we multiply or divide both sides of a linear equation by a non-zero number, the solution of the equation remains the same. This is because multiplying or dividing both sides of an equation by the same non-zero number does not change the equality of the equation.
Here’s why:
Multiplication: If you multiply both sides of the equation by the same non-zero number, the equality is maintained. For example, if the equation is x=3, multiplying both sides by 2 gives 2x=6, which still has the solution x = 3.
Division: Similarly, if you divide both sides of the equation by the same non-zero number, the equality holds. For example, if the equation is 2x=6, dividing both sides by 2 gives x=3.
Conclusion:
The solution remains the same whether we multiply or divide both sides by a non-zero number. The correct answer is remains the same.

To solve the equation x – 2y = 4, we will substitute the value of x and y from the given options and check which one satisfies the equation.
Checking the options:
Option (a): (0, 2) Substitute x = 0 and y = 2 into the equation: 
0 -2(2) = 0 – 4 = -4
This does not satisfy the equation x – 2y = 4. So, (a) is incorrect.
Option (b): (4, 0)
Substitute x = 4 and y = 0 into the equation:
4 – 2(0) = 4 – 0 = 4
This satisfies the equation x – 2y = 4. So , (b) is correct.
Option (c): (2, 0)
Substitute x = 2 and y = 0 into the equation:
2−2(0)=2−0=2
This does not satisfy the equation x−2y=4. So, (c) is incorrect.
Conclusion:
The correct solution is (4, 0).

To determine how many linear equations in x and y can be satisfied by x=1 and y=2, let’s analyze the situation.
A linear equation in x and y typically has the form: Ax+By=C
where A, B and C are constants. For a given pair ( x = 1, y= 2), we can substitute these values into the equation to see if it holds.
Substitute x=1 and y=2 into the general form of the equation Ax + By = C:
A(1) + B(2) = C
A + 2B = C
This equation can be true from many different values of A, B and C. So, there is not just one equation, but many possible equations that can be formed depending on the values of A and B.
General conclusion 
There is no unique solution for the values of A, B and C. This means infinitely many linear equations can be satisfied by the point (x=1,y=2).
Conclusion:
The correct answer is infinitely many.

The equation y=−4 represents a horizontal line where the value of  y is always−4 for any value of x.
Analyzing the options: 
Option (a): parallel to the x-axis Since the value of y remains constant at−4 for all values of x, the line is horizontal, which means it is parallel to the x-axis. This option is correct.
Option (b): parallel to the y-axis
A line parallel to the y-axis would have an equation of the form x=constant, not y=−4. Therefore, this option is incorrect.
Option (c): passes through the origin
The origin is at (0,0), but the equation y=−4 represents a line where y is always −4, so the line does not pass through the origin. This option is incorrect.
Option (d): None of these
Since we have already found the correct option, this is incorrect.
Conclusion:
The correct answer is parallel to the x-axis.

The point of the form (a,−a) represents a point where the x-coordinate is aaa and the y-coordinate is −a.
Step 1: Let’s check each option:
Option (a): x=a
For the point (a,−a) the x-coordinate is a. So, x=a is true. However, the equation x=a does not necessarily describe the relationship between x and y because the y-coordinate  −a, not a. Therefore, (a) is incorrect.
Option (b): y=−a
The point (a, -a) has a y-coordinate of -a. This satisfies y = -a, but it doesn’t describe the relation between x and y in general, because it does not account for both coordinates. (b) is incorrect.
Option (c): y = x
For the point (a, -a), the y – coordinate is -a, not a. Therefore, y = x does not hold true for the point. (c) is incorrect.
Option(d): x + y = 0
For the point (a, -a), if we add the coordinates, we get: x + y = a + (-a) = 0
This is true! The point (a, -a) always satisfies the equation x + y = 0.
Conclusion:
The correct answer is (d) x+y=0x + y = 0x+y=0.

The solution of a linear equation in two variables is an ordered pair (x,y) that satisfies the equation when the values of x and y are substituted into the equation.
A linear equation in two variables typically has the form ax+by=c, where a, b, and c are constants, and x and y are the variables. The solution is the set of values x and y (an ordered pair) that, when substituted into the equation, make the equation true.
Option (a): a number which satisfies the given equation
This is not correct because the solution is not just a single number but an ordered pair of numbers. So, (a) is incorrect.
Option (b): an ordered pair which satisfies the given equation
This is partially correct, but it’s a bit incomplete because it doesn’t clarify that the ordered pair is specific to the equation and needs to satisfy the equation. (b) is incomplete.
Option (c): an ordered pair, whose respective values when substituted for x and y in the given equation, satisfies it
This is the most accurate description. The solution is indeed an ordered pair, and when the values of x and y are substituted into the equation, the equation must be satisfied. (c) is correct.
Option (d): none of these
Since option (c) is correct, this option is incorrect. (d) is incorrect.
Conclusion: The correct answer is (c) an ordered pair, whose respective values when substituted for x and y in the given equation, satisfies it.

To determine the point through which the graph of the line x−y=0 passes, we can solve for y in terms of x or substitute the given options into the equation.
Step 1: Rewrite the equation
The equation is: x−y=0
This simplifies to: x=y
Step 2: Check the options by substituting the coordinates into the equation x=y.
Option (a): (2, 3) Substitute x=2 and y=3 into the equation: 2 = 3
This is false. So, (a) is incorrect.
Option (b): (3, 4)
Substitute x = 3 and y = 4 into the equation: 3 = 4
This is false. So, (b) is incorrect.
Option (c): (5, 6)
Substitute x = 5y and y = 6 into the equation: 5 = 6
This is false. So, (c) is incorrect. 
Option (d): (0, 0)
Substitute x = 0 and y = 0 into the equation: 0 = 0
This is true. So (d) is correct. 
Conclusion:
The graph of the line x−y=0x – y = 0x−y=0 passes through (0, 0), so the correct answer is (d) (0, 0).

To determine which of the following is not a linear equation in two variables, let’s analyze each option:
Option (a): ax+by=c 
This is a standard form of a linear equation in two variables x and y, where a, b, and c are constants. This is a linear equation. (a) is correct.
Option (b):ax² + by = c
This equation has x², which is a quadratic term, making the equation non-linear. Therefore, this is not a linear equation in two variables. (b) is incorrect.
Option (c): 2x+3y=5
This is a linear equation in two variables x and y because both x and y are raised to the first power. (c) is correct.
Option (d): 3x + 2y = 6
This is also a linear equation in two variables, where both x and y are raised to the first power. (d) is correct.
The equation that is not a linear equation in two variables is (b)ax² + by = c

A linear equation in two variables has infinitely many solutions, provided that the equation represents a line in a two-dimensional plane.
A linear equation in two variables typically has the form: ax + by = c
where a, b, and c are constants. This equation represents a straight line on a coordinate plane, and every point on this line is a solution to the equation.
Infinitely many solutions: Since a straight line has infinitely many points, the equation has infinitely many solutions, as each point on the line represents a valid pair of values for x and y.
The correct answer is infinitely many solutions.