Equation of a line can be written in various different forms if we know a point through which the line is crossing through. Slope intercept form is one of the linear equations which is look like as y = mx + c, where "m" is the slope of the line and "b" is the y-intercept or the y - coordinate of the point at which the line crosses the y-axis. Slope intercept form is convenient for reading off the slope and the y intercept of the line. For example, the slope (m) and y-intercept (c) of the line with equation y = 15x + 10 is m = 15 and c = 10.

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If a linear equation is written in the form y = mx + b. The graph of the equation is a straight line with slope m and y-intercept (0, b).

### Slope Intercept form of a Line

When an equation of a line is written in slope intercept form, the coefficient of the x-term is the line's slope and the constant term gives the y - coordinate of the y intercept.

It can be written as Ax + By = - C

Dividing both side be - C, we get

$\frac{Ax}{- C}$ x + $\frac{By}{- C}$ = 1

=> $\frac{x}{\frac{- C}{A}}$ x + $\frac{y}{\frac{- C}{B}}$ = 1

which is in the required intercept form $\frac{x}{a} + \frac{y}{a}$ = 1

where a = $\frac{- C}{A}$ and b = $\frac{- C}{B}$

x-intercept = - $\frac{\text{constant term}}{\text{coefficient of x}}$

and y-intercept = - $\frac{\text{constant term}}{\text{coefficient of y}}$

The slope intercept form of a line is y = mx + c

Put m = $\frac{-A}{B}$ (Let slope of line)

Then y = $\frac{-A}{B}$ x + c

By = - Ax + c

Ax + By = c

Ax + By - c = 0

Ax + By + C = 0 (where C = - c)

Which is general equation of the line.

Write the equation of a line in standard form, if the slope is -1 and a point on the line is (1, -2).

The equation of a line passing through (1, -2) is

y - (-2) = m(x - 1)

y + 2 = m (x - 1)

y + 2 = -1 (x - 1) (Given m = -1)

y + 2 = - x + 1

y = - x - 1 (Slope intercept form)

To change it to the standard form, add the x term each side

y + x = -1

or y + x + 1 = 0

The slope-intercept form of the equation of a line is y = mx + c, where 'm' is the slope and 'c' is the y-intercept. The graph of this equation is always a line. To explore the relationship between a linear equation and its graph, let us consider a example.

Let y = 3x + 5

Plot a table for the values of x and the corresponding values of y

To find the slope of the line, pick two points on the line, (-1, 2) and (0, 5), and draw a slope triangle and count grid squares.

Slope of line = $\frac{Rise}{Run}$ = $\frac{3}{1}$ = 3

If a line with slope m passes through the point ($x_1, y_2$), the equation of the line is given below:

y - y$_1$ = m(x - x$_1$)

Which is point-slope form of the equation of a line.

To convert point slope form to slope intercept form, we need to know the value of m. Let us study with the help of example.

Write the equation of a line in slope intercept form, if the slope is 6 and a point on the line is (4, 8).

Use the point-slope form and substitute the given values

y - y$_1$ = m(x - x$_1$)

y - 8 = 6(x - 4)

To write this equation in slope intercept form, solve for y

y - 8 = 6x - 24

y = 6x - 24 + 8

y = 6x - 16

This is the slope intercept form.

Below are some examples based on slope-intercept form:

**Example 1:** Find the y-intercept and slope of the graph of equation: 4y + 6x = 8

**Solution: **

Step 1: Convert given equation in slope-intercept form

Slope intercept form formula : y = mx + c

4y + 6x = 8

4y = 8 - 6x

y = 2 - $\frac{3}{2}$ x

or y = - $\frac{3}{2}$ x + 2

Since m = - $\frac{3}{2}$ and b = 2, the slope of the line is - $\frac{3}{2}$ and y-intercept is (0, 2).**Example 2:** Write an equation in slope intercept form for a line with slope 10 and f(-3) = 5.

**Solution:** Begin with slope intercept form

y = m x + c

Substitute the value for the slope

y = 10 x + b

To solve b, put x = -3 and y = 5

5 = 10 (-3) + b

5 = -30 + b

b = 35

Rewrite the equation

y = 10 x + 35

Which is the required equation.

**Problem 1:** Write the line equation 3x - 6y -11 = 0 in slope intercept form.

**Problem 2:** Find the y-intercept from the equation y = 8x - 12.

**Problem 3:** Find the slope from the equation $\frac{x}{3} + \frac{y}{4} = 1$.

When an equation of a line is written in slope intercept form, the coefficient of the x-term is the line's slope and the constant term gives the y - coordinate of the y intercept.

**To find the equation of a straight line passing through a fixed point and having a given slope.**

Consider a straight line passes through the fixed point A($x_1$, $y_1$) with slope m.

We know that the slope intercept form equation with slope m is

y = mx + b

where b is unknown constant.

Since the line passes through the point A ($x_1$, $y_1$) we get

y - $y_1$ = m(x - $x_1$), which is the required equation of a straight line passing through the fixed point ($x_1$, $y_1$) and having slope m.

This is known as point-slope form.

**Alternatively** :

Let l be the given line passing the fixed point A($x_1$, $y_1$) and having slope m. Let P ($x_1$, $y_1$) be any point on l.

Then slope of the line l = $\frac{y - y_1}{x - x_1}$.

But the slope of line l = m ( given)

$\Rightarrow$ $\frac{y - y_1}{x - x_1}$ = m

$\Rightarrow$ (y - $y_1$) = m (x - $x_1$)

To find the equation of a (non-vertical) straight line passing through two fixed points. Let us consider a straight line passing through the fixed points A($x_1$, $y_1$) and B($x_2$, $y_2$). Since the line is non-vertical $x_2$ = $x_1$.

Slope of the line AB = $\frac{y_2 - y_1}{x_2 - x_1}$

Therefore the equation of the line through A($x_1$, $y_1$) and having slope $\frac{y_2 - y_1}{x_2 - x_1}$ is

y - $y_1$ = $\frac{y_2 - y_1}{x_2 - x_1}$ (x - $x_1$) which is the required equation of the line passing through two fixed points A ($x_1$, $y_1$) and B($x_2$, $y_2$).

This is also known as two-point form or slope intercept form from two points.

It can be written as Ax + By = - C

Dividing both side be - C, we get

$\frac{Ax}{- C}$ x + $\frac{By}{- C}$ = 1

=> $\frac{x}{\frac{- C}{A}}$ x + $\frac{y}{\frac{- C}{B}}$ = 1

which is in the required intercept form $\frac{x}{a} + \frac{y}{a}$ = 1

where a = $\frac{- C}{A}$ and b = $\frac{- C}{B}$

x-intercept = - $\frac{\text{constant term}}{\text{coefficient of x}}$

and y-intercept = - $\frac{\text{constant term}}{\text{coefficient of y}}$

The slope intercept form of a line is y = mx + c

Put m = $\frac{-A}{B}$ (Let slope of line)

Then y = $\frac{-A}{B}$ x + c

By = - Ax + c

Ax + By = c

Ax + By - c = 0

Ax + By + C = 0 (where C = - c)

Which is general equation of the line.

Write the equation of a line in standard form, if the slope is -1 and a point on the line is (1, -2).

The equation of a line passing through (1, -2) is

y - (-2) = m(x - 1)

y + 2 = m (x - 1)

y + 2 = -1 (x - 1) (Given m = -1)

y + 2 = - x + 1

y = - x - 1 (Slope intercept form)

To change it to the standard form, add the x term each side

y + x = -1

or y + x + 1 = 0

The slope-intercept form of the equation of a line is y = mx + c, where 'm' is the slope and 'c' is the y-intercept. The graph of this equation is always a line. To explore the relationship between a linear equation and its graph, let us consider a example.

Let y = 3x + 5

Plot a table for the values of x and the corresponding values of y

x | y = 3x + 5 |

0 | 5 |

1 | 8 |

-1 | 2 |

To find the slope of the line, pick two points on the line, (-1, 2) and (0, 5), and draw a slope triangle and count grid squares.

Slope of line = $\frac{Rise}{Run}$ = $\frac{3}{1}$ = 3

If a line with slope m passes through the point ($x_1, y_2$), the equation of the line is given below:

y - y$_1$ = m(x - x$_1$)

Which is point-slope form of the equation of a line.

To convert point slope form to slope intercept form, we need to know the value of m. Let us study with the help of example.

Write the equation of a line in slope intercept form, if the slope is 6 and a point on the line is (4, 8).

Use the point-slope form and substitute the given values

y - y$_1$ = m(x - x$_1$)

y - 8 = 6(x - 4)

To write this equation in slope intercept form, solve for y

y - 8 = 6x - 24

y = 6x - 24 + 8

y = 6x - 16

This is the slope intercept form.

Below are some examples based on slope-intercept form:

Step 1: Convert given equation in slope-intercept form

Slope intercept form formula : y = mx + c

4y + 6x = 8

4y = 8 - 6x

y = 2 - $\frac{3}{2}$ x

or y = - $\frac{3}{2}$ x + 2

Since m = - $\frac{3}{2}$ and b = 2, the slope of the line is - $\frac{3}{2}$ and y-intercept is (0, 2).

y = m x + c

Substitute the value for the slope

y = 10 x + b

To solve b, put x = -3 and y = 5

5 = 10 (-3) + b

5 = -30 + b

b = 35

Rewrite the equation

y = 10 x + 35

Which is the required equation.

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