## What Is the Equation of a Line?

**The equation of a line is the algebraic equation that defines the relationship between ****coordinates**** (x, y) of any point on the line. It gives us important information such as slope, intercepts, etc.**

Through the equation of a line, we can easily figure out if the point lies on the line or not. All the points located on the line must satisfy the equation. The equation of a line is a linear equation in both x and y.

There are different forms of writing the equation of a line, depending on the information provided.

The general form to** **define the equation of a line is $ax + by + c = 0$, where

- x and y are variables that represent the coordinates of any point on the line,
- a,b,c are constants (integers) such that “a” and “b” cannot be simultaneously 0, and “a” is non-negative.

The standard form is given as $ax + by = c$, where

- x and y are variables that represent the coordinates of any point on the line,
- a,b,c are constants (integers) such that “a” and “b” cannot be simultaneously 0, and “a” is non-negative.

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## Equation of a Line Formulas

Depending on the known parameters for the line, five distinct formulas can be used to determine the **equation of a line**.

**Point Slope Form of Equation of a Line**

The slope point form of equation of a line that has slope m and passes through a point $(x_{1}, y_{1})$ is expressed as:

$(y – y_{1}) = m(x – x_{1})$

**Two-Point Form of Equation of a Line**

The two-point form of the equation of a line that passes through the points $(x_{1},\; y_{1})$ and $(x_{2},\; y_{2})$ is expressed as:

$(y – y_{1}) = \frac{(y_{2} – y_{1})}{(x_{2} – x_{1})}(x – x_{1})$

**Slope-Intercept Form**

The slope-intercept form of the equation of a straight line is expressed as:

$y = mx + b$

where

m is the slope of the line,

b is the y-intercept (the point where the line intersects the y-axis.)

**Intercept Form**

The intercept form of equation of a straight line is expressed as:

$\frac{x}{a} + \frac{y}{b} = 1$

where

a is the x-intercept of the line, and

b is the y-intercept of the line.

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## How to Find the Equation of a Line

Depending on the information provided, you can choose the form to determine the equation of a line.

**Step 1: **Notice the data provided in the given problem.

**Step 2: **Choose the appropriate form of the equation.

- If the coordinates of one point on the line $(x_{1}, y_{1})$ and the slope m are given, use the point-slope form.
- If the coordinates of two points lying on the line are given, use the two-point form equation of a line formula
- If the slope m and the y-intercept b are given, use the slope-intercept form of the equation of a straight line.
- If both the x-intercept and y-intercept are given, use the intercept form of the equation of a line.

## Facts about Equation of a Line

- The equation of a straight line with slope m, passing through the origin is given by
- y = mx.
- You can find parallel and perpendicular lines using equations of two lines.
- Two lines are parallel if and only if they have the same slope.
- Two lines are perpendicular if the product of their slopes is -1.
- The equation of a vertical line is of the form x = a, where ‘a’ represents a constant. Vertical lines have an undefined slope.
- The equation of a horizontal line is of the form y = b, where ‘b’ represents a constant. Horizontal lines have a slope of zero.

## Conclusion

In this article, we learned about the equation of a line, its different forms based on the kind of information available to us, and related examples. Let’s solve a few examples and practice MCQs for better comprehension.

## Solved Examples on Equation of Line

**1. Determine the equation of the line passing through the points (1, 3) and (2, 5)****.**

**Solution:**

$(x_{1},\;y_{1}) = (1,\;3)$

$(x_{2},\;y_{2}) = (2,\;5)$

We use the two-point form of the equation of a line given by

$(y – y_{1}) = \frac{(y_{2} – y_{1})}{(x2 – x1)} (x – x_{1})$

$(y – 3) = \frac{(5 – 3)}{(2 – 1)}(x – 1)$

$(y – 3) = (\frac{2}{1})(x – 1)$

$y – 3 = 2 (x – 1)$

$y – 3 = 2x – 2$

$y = 2x – 5$

Thus, the slope-intercept form equation will be $y = 2x – 5$.

**2. Find the equation of a line with slope 3 and y-intercept 2.**

**Solution:**

Slope $= m = 3$

y-intercept $= b = 2$

Slope-intercept form is given by $y = mx + b$**.**

$y = (3)x + 2$

$y = 3x + 2$

**3. Are the lines **$x + y + 1 = 0$** and **$x – y – 1 = 0$** perpendicular? **

**Solution:**

**Line 1:** $x + y + 1 = 0$

We can write it as $y = – x – 1$

Slope $= -1$

**Line 2:** $x – y – 1 = 0$

We can write it as $y = x – 1$

Slope $= 1$

Product of two slopes $= (1) \times (- 1) = -1$

Thus, the two lines are perpendicular.

## Practice Problems on Equation of Line

## How to Find the Equation of a Line? Formula, Examples, FAQs

### What is the point-slope form of the equation of a line?

The point-slope form of the equation of a line is $(y - y_{1}) = m(x - x_{1})$.

### The equation of a line with slope -1 and passing through origin is

The equation of a line with slope -1 and passing through origin is $y = -x$.

We can rewrite is as $x + y = 0$.

### Find the equation of the line parallel to $y = 3x - 1$.

Two lines are parallel if they have the same slope.

$y = 3x -1$ and $y = 3x + 1$ have the same slope $m = 3$.

### Equation of a line having x-intercept 1 and y-intercept -2 is

Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$

Here, $a = 1,\; b = -2$

$\frac{x}{1} + \frac{y}{(- 2)} = 1$

$-2x + y = -2$

$2x - y = 2$

## Frequently Asked Questions about Equation of a Line

**What are the equations of the x-axis and the y-axis?**

The equation of the x-axis is y = 0.

The equation of the y-axis is x = 0.

**How do you find the slope of a line using two points on it?**

The slope of a line that passes through the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is $\frac{(y_{2} – y_{1})}{(x_{2} – x_{1})}$.

**Why do people usually prefer to use the slope-intercept form for the equation of a line?**

The slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept. It is special because you can easily identify the slope (m) and the y-intercept (b) of the line simply by looking at the equation.

**How do you define the equation of a line?**

The equation of a straight line is a linear algebraic equation that defines the relationship between coordinates of every point on the line and provides information such as about the slope, intercepts of the line.

**How do you find the equation of a line if you know the coordinates of two points on the line?**

If you know the coordinates of two points on the line, use the Two-Points form of the equation of a line.