ढाल-Intercept Form Calculator

Convert between y=mx+b, standard form, two points & point-ढाल — with interactive graph

Enter coefficients for Ax + By = C

Conversion Formula Quick Reference

Standard → ढाल-Int
Ax + By = C
m = −A/B | b = C/B

ढाल-Int → Standard
y = mx + b
Multiply by LCD, keep A > 0

Two Points → ढाल
m = (y₂−y₁) / (x₂−x₁)
b = y₁ − m·x₁

Parallel & Perpendicular
Parallel: m₂ = m₁
Perpendicular: m₂ = −1/m₁

What is ढाल-Intercept Form?

ढाल-intercept form is the equation y = mx + b, where m is the ढाल of the line and b is the Y-अंतःखंड — the point where the line crosses the vertical axis. It is by far the most widely used way to express a linear equation in algebra, pre-calculus, and beyond, because both key properties of the line are immediately readable from the equation itself without any additional algebra.

The ढाल m tells you how steep the line is and in which direction it travels. A positive ढाल means the line rises from left to right; a negative ढाल means it falls. The absolute value of m describes exactly how many units the line rises (or falls) for every one unit you move to the right. A ढाल of 2, for example, means the line goes up 2 units for each 1 unit of horizontal movement — a "rise over run" ratio of 2:1.

The Y-अंतःखंड b anchors the line on the coordinate plane. It is the y-value of the line when x = 0, so you can instantly plot the point (0, b) without substituting any value. From that anchor point, you apply the ढाल to trace the rest of the line. This makes y = mx + b the most graphing-friendly form of a linear equation, which is why it dominates secondary-school mathematics curricula worldwide and is the preferred form for cost functions, speed-distance relationships, and any other linear model.

Forms of Linear Equations Compared

Form	Equation	Best Used When
ढाल-Intercept	y = mx + b	Graphing, reading ढाल and intercept directly
Standard Form	Ax + By = C	Integer coefficients, finding both intercepts quickly
Point-ढाल	y − y₁ = m(x − x₁)	Writing an equation from one point and a known ढाल
Intercept Form	x/a + y/b = 1	When both X-अंतःखंड (a) and Y-अंतःखंड (b) are known

How to Use This Calculator

Choose the tab matching the information you have: Standard Form, ढाल-Intercept, Two Points, Point + ढाल, or Parallel & Perpendicular.
Enter your values in the input fields. Decimal values are accepted for non-integer ढालs and intercepts.
Click गणना करें. Results appear instantly below the inputs with stat cards, both equation forms, and step-by-step working.
Scroll down to the Line Graph to see the line plotted with labeled intercepts and a rise/run ढाल triangle.
Use the Sample button to auto-fill an example if you are unsure what values to enter.

Conversion Formulas with Worked Examples

Standard Form to ढाल-Intercept — Example

Given 2x + 3y = 12:

Subtract 2x from both sides: 3y = −2x + 12
Divide every term by B = 3: y = −(2/3)x + 4
Result: ढाल m = −2/3, Y-अंतःखंड b = 4, X-अंतःखंड = 6

Two Points to Equation — Example

Given (1, 3) and (4, 9):

ढाल: m = (9 − 3) / (4 − 1) = 6/3 = 2
Use point (1, 3): 3 = 2(1) + b ⇒ b = 1
ढाल-intercept form: y = 2x + 1
Standard form: −2x + y = 1 ⇒ multiply by −1: 2x − y = −1

Point-ढाल to ढाल-Intercept — Example

Given point (3, 5) and ढाल m = 2:

Point-ढाल form: y − 5 = 2(x − 3)
Expand: y − 5 = 2x − 6
Add 5: y = 2x − 1. So b = −1

Graph Interpretation Guide

Positive ढाल (m > 0): Line rises from left to right. Steeper as m increases.
Negative ढाल (m < 0): Line falls from left to right. Steeper as |m| increases.
Zero ढाल (m = 0): Horizontal line y = b. No rise, only runs.
Undefined ढाल: Vertical line x = a. Cannot be written in y = mx + b form.
Y-अंतःखंड (0, b): Where the line meets the y-axis (set x = 0).
X-अंतःखंड (−b/m, 0): Where the line meets the x-axis (set y = 0 and solve for x).
ढाल triangle: The orange rise/run triangle on the graph shows ढाल visually — a rise of m units for every 1 unit of horizontal run.

Real-World Examples of ढाल-Intercept Form

Cost function: A plumber charges a $50 call-out fee plus $80 per hour. Total cost: y = 80x + 50. ढाल = 80 (cost per hour), Y-अंतःखंड = 50 (fixed fee).
Distance-time: A car travelling at 60 km/h starting 20 km from home: d = 60t + 20. ढाल = 60 (speed in km/h), Y-अंतःखंड = 20 (starting distance).
Temperature conversion: Celsius from Fahrenheit: C = (5/9)F − 160/9. ढाल = 5/9, b ≈ −17.78 (the value of 0°F in Celsius).
Depreciation: A machine worth $10,000 depreciating $1,000 per year: V = −1000t + 10000. ढाल = −1000, Y-अंतःखंड = 10,000 (initial value).

अक्सर पूछे जाने वाले प्रश्न

What is ढाल-intercept form?

ढाल-intercept form is y = mx + b, where m is the ढाल (rise over run) and b is the Y-अंतःखंड (the y-value when x = 0). It is the most common linear equation format because you can read ढाल and intercept directly without any additional algebra, making it ideal for graphing and analysing linear relationships in science, engineering, and everyday problem-solving.

What does m represent in y = mx + b?

m is the ढाल of the line — the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. A positive m means the line goes up from left to right; a negative m means it descends. When m = 0 the line is horizontal. A larger absolute value of m produces a steeper line.

How do I find the ढाल from two points?

Use the ढाल formula m = (y₂ − y₁) / (x₂ − x₁). For (1, 3) and (4, 9): m = (9 − 3) / (4 − 1) = 6 / 3 = 2. Then substitute back into y = mx + b with either point to find b. Using (1, 3): 3 = 2(1) + b, so b = 1, giving y = 2x + 1.

What is the difference between ढाल-intercept and standard form?

ढाल-intercept (y = mx + b) directly shows ढाल m and Y-अंतःखंड b, making it ideal for graphing. Standard form (Ax + By = C) uses integer coefficients with A positive and is preferred for finding integer solutions, writing clean equations, and certain algebraic manipulations. Both forms represent the same straight line — they are fully algebraically equivalent.

What is an undefined ढाल (vertical line)?

A vertical line (e.g., x = 3) has an undefined ढाल because the horizontal change (run) between any two points is zero, making rise/run a division by zero. Vertical lines cannot be expressed in ढाल-intercept form y = mx + b at all. In standard form they appear simply as x = constant (e.g., 1x + 0y = 3).

What is a zero ढाल (horizontal line)?

A zero ढाल (m = 0) means the line is perfectly horizontal. The equation simplifies to y = b, where b is the constant y-value of every point on the line. There is no rise as x increases. For example, y = 5 is a horizontal line 5 units above the x-axis. Horizontal lines have no X-अंतःखंड unless b = 0.

How are parallel and perpendicular ढालs related?

Parallel lines have identical ढालs (m₁ = m₂) but different Y-अंतःखंडs. Perpendicular lines have ढालs that are negative reciprocals of each other: m₁ × m₂ = −1. For example, if a line has ढाल 3, a perpendicular line has ढाल −1/3. If a line is horizontal (m = 0), its perpendicular is a vertical line with undefined ढाल.

How do I convert standard form Ax + By = C to ढाल-intercept form?

Subtract Ax from both sides: By = −Ax + C. Then divide each term by B: y = (−A/B)x + (C/B). So ढाल m = −A/B and Y-अंतःखंड b = C/B. Example: 2x + 3y = 12 ⇒ 3y = −2x + 12 ⇒ y = (−2/3)x + 4. If B = 0, the line is vertical and has no ढाल-intercept form.