Step-by-step solutions for Class 9 Mathematics Ganita Manjari Part 1I Chapter 13 Exercise Set 13.1 (Questions 1–3). Covers expressing linear equations in standard form ax + by + c = 0, identifying coefficients of x and y, determining the constant term, handling equations with missing variables, and translating word problems into linear equations.
Any equation that can be put in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero (a² + b² ≠ 0), is called a linear equation in two variables x and y.
In the standard form ax + by + c = 0, the real multiplier of x is the coefficient a, the multiplier of y is the coefficient b, and the term independent of variables is the constant term c. Always retain the positive or negative sign (+ or −) with each coefficient.
To express an equation in standard form, transpose all terms to the left-hand side (LHS) so that the right-hand side (RHS) becomes 0. Then arrange the terms in the standard sequence: x-term first, followed by the y-term, and then the constant term.
An equation in one variable can be represented as a linear equation in two variables by setting the coefficient of the missing variable to 0. For example, x = 8 is written as 1x + 0y − 8 = 0, and 3y = 1 is written as 0x + 3y − 1 = 0.
To formulate a linear equation from a real-world statement, identify the two related quantities, assign a distinct variable to each, and represent their stated condition algebraically.
Statements indicating that one quantity is 'twice' another translate to multiplication by 2 (e.g. t = 2p). Statements where two quantities combine to form a sum translate to addition equal to that total (e.g. x + y = 176).
Write a linear equation in two variables in which a = 3, b = 0 and c = −1/5.
The standard form of a linear equation in two variables is:
a = 3, b = 0, c = −15
Since 0y = 0, this can also be written as:
Complete the following table after expressing the given linear equations in standard form.
| Linear equation in two variables |
Standard Form |
Coefficient of x |
Coefficient of y |
Constant term |
|---|---|---|---|---|
| y − 15 = √2x | ||||
| 3y − 2x = 0 | ||||
| 5x = 3y | ||||
| x = 8 | ||||
| 3y = 1 |
The standard form of a linear equation in two variables is:
The completed table after expressing each equation in standard form is shown below:
| Linear equation | Standard form (ax + by + c = 0) |
Coefficient of x (a) | Coefficient of y (b) | Constant term (c) |
|---|---|---|---|---|
| y − 15 = √2x | −√2x + y − 15 = 0 | −√2 | 1 | −15 |
| 3y − 2x = 0 | −2x + 3y + 0 = 0 | −2 | 3 | 0 |
| 5x = 3y | 5x − 3y + 0 = 0 | 5 | −3 | 0 |
| x = 8 | x + 0y − 8 = 0 | 1 | 0 | −8 |
| 3y = 1 | 0x + 3y − 1 = 0 | 0 | 3 | −1 |
(i) The cost of a notebook is twice the cost of a pen. Consider the cost of a notebook to be ₹t and that of a pen to be ₹p. Charlie wrote t = 2p, whereas Meera wrote p = 2t. Which of these two representations is correct? (ii) In a one-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Manisha expressed this situation as x + y = 176, where the number of runs scored by one batsman is x, and the number of runs scored by the other is y. Is this a correct representation?
According to the given condition:
The cost of a notebook is twice the cost of a pen.
Cost of notebook = 2 × Cost of pen
In standard form, Charlie's equation is written as:
Number of runs scored by the first batsman = x
Number of runs scored by the second batsman = y
According to the given situation:
Runs scored by first batsman + Runs scored by second batsman = Total runs
Manisha expressed this situation as x + y = 176, which correctly equates the sum of the runs scored by the two batsmen to 176.
In standard form, this equation is written as:
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