Test: Linear Equations In Two Variables - 1 - Class 9 MCQ

To find where the line (2x - 3y = 6) cuts the y-axis, we set (x = 0) and solve for (y).
- Substituting (x = 0):
(2(0) - 3y = 6)
(-3y = 6)
(y = -2
- Thus, the line cuts the y-axis at (0, -2).

So, the correct answer is C:(0, -2)

The linear equation (2x + 3y = 6) can be rewritten in the slope-intercept form (y = mx + b).

- Rearrange to (3y = -2x + 6) and then (y = -\frac{2}{3}x + 2\).
- This is a straight-line equation.
- A straight line has infinitely many points.

Thus, the equation has:

- Infinitely many solutions (C)

Solution: From equation (i) x = 14 - 4y ....(iii)

Substitute the value of x in equation (ii)

⇒ 7 (14 - 4y) - 3y = 5

⇒ 98 - 28y - 3y = 5

⇒ 98 - 31y = 5

⇒ 93 = 31y

⇒ y = 93/31

⇒ y = 3

Now substitute value of y in equation (iii)

⇒ 7x - 3 (3) = 5

⇒ 7x - 3 (3) = 5

⇒ 7x = 14

⇒ x = 14/7 = 2

So the solution is x = 2 and y = 3

A linear equation in two variables has maximum:
C: Infinite solutions
- A linear equation in two variables can be written as \( ax + by = c \).
- This equation represents a straight line on the coordinate plane.
- Every point on this line is a solution to the equation.
- Since there are infinitely many points on a line, there are infinitely many solutions to the equation.

Hence, the correct answer is C: Infinite solutions.

Solution:

Substitution Method:

- From the first equation, x + y = 3
- Rearranging, we get y = 3 - x
- Substitute y = 3 - x into the second equation 3x - 2(3 - x) = 4
- Simplify the equation: 3x - 6 + 2x = 4
- Combine like terms: 5x - 6 = 4
- Add 6 to both sides: 5x = 10
- Divide by 5: x = 2
- Substitute x = 2 back into y = 3 - x
- y = 3 - 2 = 1

Therefore, the solution is x = 2, y = 1, which corresponds to option A.

To find if ( x = 2, y = -1 ) is a solution for a given equation, substitute these values into each equation:

- A: ( 2(2) + 3(-1) =1 not  5 ) ❌
- B: ( 2 + (-1) = 1 not 5 ) ❌
- C: ( 2 + (-1) = 1 ) ✓
- D: ( 2 - (-1) = 3 not 9 ) ❌

Therefore, option C is the correct answer.

Solution : 

The correct option is Option B.

x² + p² = (q + x)²

x² + p² = q² - 2qx + x²

Eliminating x² from both sides,

p² = q² - 2qx

x = q² - p² / 2q

\

The point (2a, -a) lies on the line 2y + x = 0 because:
- For a point to lie on a line, it must satisfy the equation of the line.
- Substituting the coordinates of the point (2a, -a) into the equation 2y + x = 0 gives 2(-a) + 2a = 0.
- Simplifying the equation further, we get -2a + 2a = 0, which is true.
- Therefore, the point (2a, -a) lies on the line 2y + x = 0.

Answer: c
Explanation: Let the two digit number be 10x+y
The number obtained after reversing the digits will be 10y+x
10x+y+10y+x=187
11x+11y=187
x+y=17     (1)
Also, x-y=1     (2)
Adding (1) and (2)
2x=18
x=9
Substituting in equation (1), 9+y=17
y=8
The number is 89 or 98.

- The point of the form (a,0) lies on the x-axis because the y-coordinate is always 0 on the x-axis.
- The x-axis is the horizontal line on the Cartesian plane where the y-coordinate is always zero.
- Therefore, the correct answer is A: the x-axis.

We have to check which straight line given in options, passes through the points (0,0),(−1,1) and (1,−1).
Any point lying on the line satisfies its equation.
Let's take option A,2−x=3y
For point (0,0), LHS =2−0=2, RHS =3(0)=0
LHS ≠ RHS
Hence, the line 2−x=3y does not passes through (0,0). So, no need to check whether other points lies on the line or not.

To determine where the graph of the equation (3x - 2y = 6) cuts the x-axis, follow these steps:
- Identify the x-intercept: The x-intercept occurs where (y = 0).
- Tequation becomes 3x - 2(0) = 6.
- Solve for (x): (3x = 6), hence (x = 2).
Therefore, the graph cuts the x-axis at the point (2, 0)

Answer: C: (2, 0)

Let the ones's digit is y

and ten's digit is x

so, two digit number = 10x + y

and the number formed by interchanged the digits = 10y + x

10x + y + 10y + x = 110

⇒ 11x + 11y = 110

⇒ x + y = 10

⇒ x = 10 - y      ---- (1)

10x + y - 10 = 5(x + y) + 4

⇒ 10x + y - 10 = 5x + 5y + 4

⇒ 10x - 5x + y - 5y = 4 + 10

⇒ 5x - 4y = 14      ---- (2)

5(10 - y) - 4y = 14 (From 1)

⇒ 50 - 5y - 4y = 14

⇒ 50 - 9y = 14

⇒ -9y = 14 - 50

⇒ -9y = - 36

⇒ y = 4

so, x =10 - 4

⇒ x = 6

Two digit number = 10 × 6 + 4

⇒ 64

∴ Two digit number is 64.

In the graph of y = x², the point (0, 0) is called the vertex. The vertex is the minimum point in a parabola that opens upward. In a parabola that opens downward, the vertex is the maximum point. We can graph a parabola with a different vertex.

x + y = 7 
x – y = 3
adding this two equation
2x = 10
x = 5
so y  = 2 
(5,2)

Parallel to the y-axis:.
- Therefore, the graph of the equation 2x - 3 = 3x - 5 solving this we would get
x=2
 is parallel to the y-axis, which is option B.

2. Coincident

• Coincident Lines: Two lines are coincident if they are exactly the same line, meaning they have the same slope and y-intercept. In other words, they overlap completely.
• For the given equations:

•  Because the y-intercepts are different, the lines are not coincident.

3. Intersecting

• Intersecting Lines: Two lines are intersecting if they cross each other at exactly one point. This happens when the lines have different slopes.
• 

4. Perpendicular to Each Other

• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is −1-1−1. In other words, if the slopes are negative reciprocals of each other.