How do you convert 28x − 35y = −7 to standard form ax + by + c = 0?

To convert 28x − 35y = −7 to standard form, move all terms to the left side: 28x − 35y + 7 = 0. Now a = 28, b = −35, c = 7. The key step is transposing −7 from the right side to the left, which changes its sign to +7.

What are the values of a, b, c when the equation has only one variable like 2x = 5?

When only x is present in 2x = 5, rewrite it as 2x + 0·y − 5 = 0. This gives a = 2, b = 0, c = −5. The missing variable y still appears in the standard form with a coefficient of 0. Similarly, y − 2 = 0 becomes 0·x + y − 2 = 0 with a = 0, b = 1, c = −2.

How do you form a linear equation in two variables from a word problem?

The method for word problems is: (1) Identify the two unknown quantities and assign them variable names, typically x and y. (2) Translate the stated condition into a mathematical equation using those variables. (3) Rearrange into standard form ax + by + c = 0. For example, 'the sum of two numbers is 34' becomes x + y = 34, then x + y − 34 = 0, with a = 1, b = 1, c = −34.

Exercise 6.1 — Standard Form | Class 9 Maths | Linear Equations in Two Variables

Q: What is the standard form of a linear equation in two variables in Class 9?

The standard form of a linear equation in two variables is ax + by + c = 0, where a is the coefficient of x, b is the coefficient of y, and c is the constant term. The conditions are: a and b cannot both be zero simultaneously, and a, b, c must be real numbers. The highest power of each variable must be 1 (linear means degree 1).

Linear Equations in Two Variables — Standard Form

A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where x and y are the two variables and a, b, c are real numbers with the condition that a and b are not both zero. The word "linear" refers to the fact that the highest power of each variable is 1 — so the equation, when graphed, always produces a straight line.

Exercise 6.1 develops two skills: converting equations from various rearranged forms into the standard form ax + by + c = 0, and translating real-world word problems into linear equations using two variables.

Standard form of a linear equation in two variables:

ax + by + c = 0

where:
◆ a = coefficient of x (abscissa term)
◆ b = coefficient of y (ordinate term)
◆ c = constant term
◆ a and b cannot both be zero at the same time
◆ a, b, c are real numbers

Important conditions to remember:
◆ If one variable is missing (e.g., only x appears), its coefficient is 0 for the other variable.
◆ The equation must have degree 1 — no x², no y², no xy terms.
◆ Moving terms across the equals sign changes their sign: if 3x = 12, then 3x − 12 = 0, so a = 3, b = 0, c = −12.

Exercise 6.1 — Question 1: Express in ax + by + c = 0 Form

Each equation below is in some rearranged form. Rewrite it so that all terms are on the left side with zero on the right, then read off a, b, and c. The key operation is always transposing every term to the left side by reversing its sign.

(i) 8x + 5y − 3 = 0

Already in standard form

a = 8, b = 5, c = −3

(ii) 28x − 35y = −7

28x − 35y + 7 = 0

a = 28, b = −35, c = 7

(iii) 93x = 12 − 15y

93x + 15y − 12 = 0

a = 93, b = 15, c = −12

(iv) 2x = −5y

2x + 5y + 0 = 0

a = 2, b = 5, c = 0

(v) x/3 + y/4 = 7

(1/3)x + (1/4)y − 7 = 0

a = 1/3, b = 1/4, c = −7

(vi) y = (−3/2)x

(3/2)x + y + 0 = 0

a = 3/2, b = 1, c = 0

(vii) 3x + 5y = 12

3x + 5y − 12 = 0

a = 3, b = 5, c = −12

Exercise 6.1 — Question 2: Single-Variable Equations in ax + by + c = 0

When an equation has only one variable, the missing variable still exists in the standard form — its coefficient is simply 0. For example, 2x = 5 can be written as 2x + 0·y − 5 = 0. This is still a linear equation in two variables where b = 0.

(i) 2x = 5

2x + 0·y − 5 = 0

a = 2, b = 0, c = −5

(ii) y − 2 = 0

0·x + y − 2 = 0

a = 0, b = 1, c = −2

(iii) y/7 = 3

0·x + (1/7)y − 3 = 0

a = 0, b = 1/7, c = −3

(iv) x = (−14/13)y

x + (14/13)y + 0 = 0

a = 1, b = 14/13, c = 0

Equation Type	Example	Standard Form	a, b, c
Both variables present	3x + 5y = 12	3x + 5y − 12 = 0	3, 5, −12
Only x (no y)	2x = 5	2x + 0·y − 5 = 0	2, 0, −5
Only y (no x)	y − 2 = 0	0·x + y − 2 = 0	0, 1, −2
No constant term	2x = −5y	2x + 5y + 0 = 0	2, 5, 0
Fractional coefficients	x/3 + y/4 = 7	(1/3)x + (1/4)y − 7 = 0	1/3, 1/4, −7

Exercise 6.1 — Question 3: Word Problems to Linear Equations

Translating a word problem into a linear equation requires identifying the two unknown quantities, assigning them variable names, and building the equation from the condition given. The strategy is always the same: let x and y represent the unknowns, then write the mathematical relationship between them.

(i) The sum of two numbers is 34.

Let first number = x, second number = y Condition: x + y = 34 Standard form: x + y − 34 = 0

(ii) The cost of a ball pen is ₹5 less than half the cost of a fountain pen.

Let cost of ball pen = ₹x, cost of fountain pen = ₹y Condition: x = y/2 − 5 ⇒ 2x = y − 10 Standard form: 2x − y + 10 = 0

(iii) Bhargavi got 10 more marks than double the marks of Sindhu.

Let Bhargavi's marks = x, Sindhu's marks = y Condition: x = 2y + 10 Standard form: x − 2y − 10 = 0

(iv) Pencil costs ₹2, ball pen costs ₹15. Sheela pays ₹100 for pencils and pens.

Let x = number of pencils, y = number of pens Total cost of pencils = 2x, total cost of pens = 15y Condition: 2x + 15y = 100 Standard form: 2x + 15y − 100 = 0

(v) Yamini and Fatima together contributed ₹200 towards the Prime Minister's Relief Fund.

Let Yamini's contribution = ₹x, Fatima's contribution = ₹y Condition: x + y = 200 Standard form: x + y − 200 = 0

(vi) Sum of a two-digit number and its reverse is 121. Unit's digit = x, ten's digit = y.

Original number = 10y + x (ten's digit × 10 + unit's digit) Reversed number = 10x + y Condition: (10y + x) + (10x + y) = 121 ⇒ 11x + 11y = 121 ⇒ 11(x + y) = 121 ⇒ x + y = 11 Standard form: x + y − 11 = 0

How to Write Any Equation in Standard Form — Step-by-Step

General Method

Step 1: Write the equation as given.

Step 2: Move all terms to the left side (change signs when crossing the = sign).

Step 3: Set the right side to 0.

Step 4: Identify a (coefficient of x), b (coefficient of y), c (constant).

Step 5: Check: if a term is absent, its coefficient is 0.

Result: ax + by + c = 0 with a, b, c clearly identified.

The standard form always has 0 on the right side — every term moves to the left.
When a variable is missing from the original equation, its coefficient is 0 in the standard form, not omitted entirely.
Fractional coefficients are perfectly valid — keep them as fractions unless told to simplify.
For word problems: define variables clearly first, then write the mathematical condition stated in the problem.
The equation ax + by + c = 0 represents a straight line on the Cartesian plane — every linear equation in two variables graphs as a line.

Key insight: The standard form ax + by + c = 0 is universal. Any linear relationship between two quantities — prices and quantities purchased, marks scored, money contributed, digit relationships — can always be written in this form. The coefficients a and b tell you the rate of change of each variable, and c is the adjustment constant that balances the equation.