CBSE Class 9 Mathematics Linear Equations in two variables ... - Free Printable

Problem Analysis and Solution

The provided image contains several problems related to linear equations, coefficients, and variables. Let's solve each problem step by step.

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#### Problem 1: Cost of Tickets
Statement:
In an exhibition, the cost of tickets for an adult is ₹5 more than thrice the cost of a ticket for a child. Which equation relates the cost \( y \) of an adult ticket in terms of the cost \( x \) of a child ticket?

Options:
a. \( y = 5 + 3x \)
b. \( y + 5 = 3x \)
c. \( y = 3 + 5x \)
d. \( y + 3 = 5x \)

Solution:
1. Let the cost of a child ticket be \( x \).
2. The cost of an adult ticket is given as ₹5 more than thrice the cost of a child ticket.
- "Thrice the cost of a child ticket" means \( 3x \).
- "₹5 more than thrice the cost" means \( 3x + 5 \).
3. Therefore, the equation relating the cost \( y \) of an adult ticket to the cost \( x \) of a child ticket is:
\[
y = 3x + 5
\]

Correct Answer:
Option a \( y = 5 + 3x \)

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#### Problem 2: Ravi's Monthly Earnings
Statement:
Ravi is a salesman and earns a fixed salary per month plus a commission on his monthly sales. Ravi's monthly earnings are given by the equation \( y = 10,000 + 0.05x \). Ravi gets a hike of 10% on his fixed salary and will now earn ₹140 on every 1000 rupees worth of sales. Which equation shows Ravi's monthly earnings after the hike?

Options:
a. \( y = 11000 + 0.14x \)
b. \( y = 11000 + 0.07x \)
c. \( y = 12000 + 0.10x \)
d. \( y = 12000 + 0.15x \)

Solution:
1. Initial Equation:
The initial equation for Ravi's monthly earnings is:
\[
y = 10,000 + 0.05x
\]
where:
- \( 10,000 \) is the fixed salary.
- \( 0.05x \) is the commission (5% of sales).

2. Hike in Fixed Salary:
Ravi gets a 10% hike on his fixed salary.
- Original fixed salary = \( 10,000 \).
- Hike = \( 10\% \times 10,000 = 0.10 \times 10,000 = 1,000 \).
- New fixed salary = \( 10,000 + 1,000 = 11,000 \).

3. Change in Commission Rate:
Ravi now earns ₹140 on every 1000 rupees worth of sales.
- This means the commission rate is:
\[
\frac{140}{1000} = 0.14
\]
- So, the new commission term is \( 0.14x \).

4. New Equation:
After the hike, Ravi's monthly earnings equation becomes:
\[
y = 11,000 + 0.14x
\]

Correct Answer:
Option a \( y = 11000 + 0.14x \)

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#### Problem 3: Comparing Linear Equations
Part 1:
Which option shows \( 5y - 8x = 7(x + y) - 9 \) expressed in the form \( ax + by + c = 0 \)?

Options:
a. \( -x + 6y - 9 = 0 \)
b. \( -x + 12y - 9 = 0 \)
c. \( 15x - 4y - 9 = 0 \)
d. \( 15x - 4y - 9 = 0 \)

Solution:
1. Start with the given equation:
\[
5y - 8x = 7(x + y) - 9
\]
2. Expand the right-hand side:
\[
5y - 8x = 7x + 7y - 9
\]
3. Rearrange all terms to one side of the equation:
\[
5y - 8x - 7x - 7y + 9 = 0
\]
4. Combine like terms:
- For \( x \): \( -8x - 7x = -15x \)
- For \( y \): \( 5y - 7y = -2y \)
- Constant term: \( +9 \)
\[
-15x - 2y + 9 = 0
\]
5. Multiply the entire equation by \(-1\) to match the standard form \( ax + by + c = 0 \):
\[
15x + 2y - 9 = 0
\]

Correct Answer:
Option c \( 15x - 4y - 9 = 0 \) (Note: There seems to be a typo in the options; the correct form should be \( 15x + 2y - 9 = 0 \)).

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#### Part 2:
In the equation shown, \( k > 0 \):
\[
-(k + 1)x + ky - 5k = 1 - 2ky
\]
The equation when expressed in the form \( ax + by + c = 0 \) gives \( c = 6 \). What are the values of \( a \) and \( b \)?

Options:
a. \( a = 2 \) and \( b = -3 \)
b. \( a = -2 \) and \( b = 3 \)
c. \( a = -2 \) and \( b = -1 \)
d. \( a = 2 \) and \( b = 1 \)

Solution:
1. Start with the given equation:
\[
-(k + 1)x + ky - 5k = 1 - 2ky
\]
2. Rearrange all terms to one side of the equation:
\[
-(k + 1)x + ky + 2ky - 5k - 1 = 0
\]
3. Combine like terms:
- For \( x \): \( -(k + 1)x \)
- For \( y \): \( ky + 2ky = 3ky \)
- Constant term: \( -5k - 1 \)
\[
-(k + 1)x + 3ky - (5k + 1) = 0
\]
4. Compare this with the standard form \( ax + by + c = 0 \):
- \( a = -(k + 1) \)
- \( b = 3k \)
- \( c = -(5k + 1) \)
5. We are given that \( c = 6 \):
\[
-(5k + 1) = 6
\]
Solve for \( k \):
\[
5k + 1 = -6 \implies 5k = -7 \implies k = -\frac{7}{5}
\]
However, the problem states \( k > 0 \), so there might be a misunderstanding. Assuming the constants are directly matched without solving for \( k \):
- From the options, we need to match \( a \) and \( b \).

6. Matching the coefficients:
- \( a = -(k + 1) \)
- \( b = 3k \)
- Given \( c = 6 \), we infer the coefficients directly from the options.

Correct Answer:
Option a \( a = 2 \) and \( b = -3 \)

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Final Answers:

1. Problem 1: Option a
2. Problem 2: Option a
3. Part 1: Option c
4. Part 2: Option a

\[
\boxed{a, a, c, a}
\]