# Slope Intercept Form

• Definition of slope-intercept form
• The Slope-Intercept form
• Derivation of the Slope Intercept Formula
• Straight Line Equation Using Slope-Intercept Form
• Standard Form to Slope Intercept Form
• Important Notes on the Slope Intercept Form
• Solved Examples
• Practice Problems

## Definition of Slope-Intercept Form

The slope-intercept form is one of the forms used to write the equation of a straight line. It is a preferred way of writing the equation of a straight line when the slope of the line and the y-intercept of the line are given. The slope indicates how steep the line is and whether it rises or falls as it moves from left to right. The y-intercept is the point where the line crosses the y-axis. This form is useful in graphing linear equations and understanding their behavior.

## The Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b. In this form:

m represents the slope of the line, indicating its steepness and direction.

b represents the y-intercept, the point where the line crosses the y-axis.

Example: Find the equation of the line passing through the point (3, 5) with a slope of 2.

Solution:

We can use the slope-intercept form of a linear equation, which is y = mx + b.

Given:

• Slope $$m = 2$$
• Point $$(x, y) = (3, 5)$$

We can substitute the values of $$m$$ into $$y = mx + b$$ and write the equation of the line as

$$y = 2x + b$$

To find the value of $$b$$, we use the given point $$(3, 5)$$ and substitute its coordinates into the equation:

$$5 = 2(3) + b$$

$$5 = 6 + b$$

$$b = 5 - 6$$

$$b = -1$$

Now, we have found the value of $$b$$, which is $$-1$$.

Therefore, the equation of the line passing through the point (3, 5) with a slope of 2 is:

$$\boxed{y = 2x - 1}$$

## Derivation of the Slope Intercept Formula

Let's derive the slope intercept formula for a straight line.

Consider a line with a slope $$m$$ that intersects the y-axis at the point $$(0, b)$$, where $$b$$ represents the y-intercept. Also, let's take an arbitrary point $$(x, y)$$ on this line.

Use the slope formula, which states that the slope $$m$$ of a line joining two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is given by:

$$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$$

Let's assign $$(x_1, y_1) = (0, b)$$ and $$(x_2, y_2) = (x, y)$$. Substituting these values into the slope formula, we get:

$$m = \frac{{y - b}}{{x - 0}}$$

$$\Rightarrow m = \frac{{y - b}}{{x}}$$

Multiplying both sides by $$x$$, we have:

$$mx = y - b$$

Adding $$b$$ to both sides, we obtain:

$$y = mx + b$$

## Straight Line Equation Using Slope-Intercept Form

The equation y = mx + b represents the equation of a straight line involving its slope $$m$$ and its y-intercept $$b$$. Hence, this form of the equation is termed the slope-intercept form, which we derived from the given conditions.

To determine the equation of a line using the slope-intercept form, two crucial pieces of information are needed:

• The inclination of the line (or its slope, represented by $$m$$, or the angle it makes with the x-axis, denoted by $$\theta$$)
• The line's placement (specified by its y-intercept, $$b$$, or the point on the y-axis through which it passes).

These parameters uniquely define any line.

The steps to find the equation of a line using the slope-intercept form are as follows:

Step 1: Identify the y-intercept, denoted as $$b$$, and the slope of the line, denoted as $$m$$. If the slope is not directly given, you can determine it using the angle it makes with the x-axis, provided it is available.

Step 2: Apply the slope-intercept formula: $$y = mx + b$$.

Example: A line is inclined at an angle of $$60^\circ$$ to the horizontal axis and passes through the point $$(0, -1)$$. Find the equation of this line.

Solution:

Using $$m = \tan(60^\circ) = \sqrt{3}$$, we can use the slope-intercept formula $$y = mx + b$$.

Thus, the equation of the line is $$y = \sqrt{3}x - 1$$.

## Standard Form to Slope Intercept Form

Standard form is yet another form of writing the equation of any straight line. Converting a linear equation from standard form $$Ax + By = C$$ to slope-intercept form $$y = mx + b$$ involves rearranging the terms to isolate $$y$$ on one side of the equation. Here are the steps to convert from standard form to slope-intercept form:

1. Identify the values of $$A$$, $$B$$, and $$C$$: These coefficients represent the parameters of the linear equation in standard form.

2. Solve for $$y$$: To isolate $$y$$, subtract $$Ax$$ from both sides of the equation:

$$Ax + By = C$$

$$By = -Ax + C$$

3. Divide by $$B$$: Divide every term by $$B$$ to isolate $$y$$ on one side:

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

4. Identify $$m$$ and $$b$$: The coefficient of $$x$$, $$-\frac{A}{B}$$, represents the slope $$m$$, while $$\frac{C}{B}$$ represents the y-intercept $$b$$.

5. Write the equation in slope-intercept form: Substitute the values of $$m$$ and $$b$$ into the equation:

$$y = mx + b$$

$$y = \left(-\frac{A}{B}\right)x + \frac{C}{B}$$

## Important Notes on the Slope Intercept Form

1. Slope and Y-Intercept: The slope-intercept form of a linear equation $$y = mx + b$$ uniquely defines a line by its slope $$m$$ and y-intercept $$b$$.

2. Graphical Representation: This form simplifies graphing. To graph a linear equation given in slope intercept form, start graphing by plotting the y-intercept first. Next, starting from the y-intercept, use the slope value and follow the rise over run to plot the second point. Connect the two points using a straight line.

3. Versatility: The slope-intercept form allows easy identification of key properties such as slope and intercepts, making it invaluable for understanding and analyzing linear relationships.

## Solved Examples

Example 1. Find the equation of a line passing through the point $$(2, 4)$$ with a slope of $$3$$.

Solution:

Given the point $$(2, 4)$$ and the slope $$m = 3$$, we can use the slope-intercept form $$y = mx + b$$.

Substituting the given values, we get:

$$4 = 3(2) + b$$

$$4 = 6 + b$$

$$b = 4 - 6$$

$$b = -2$$

Therefore, the equation of the line is $$y = 3x - 2$$.

Example 2. Determine the equation of a line with a slope of $$-\frac{1}{2}$$ passing through the point $$(5, 3)$$.

Solution:

Given the slope $$m = -\frac{1}{2}$$ and the point $$(5, 3)$$, we use the slope-intercept form $$y = mx + b$$.

Substituting the given values:

$$3 = -\frac{1}{2}(5) + b$$

$$3 = -\frac{5}{2} + b$$

$$b = 3 + \frac{5}{2}$$

$$b = \frac{11}{2}$$

Therefore, the equation of the line is $$y = -\frac{1}{2}x + \frac{11}{2}$$.

Example 3. What is the y-intercept of the line with the equation 2y−4x=10x−20?

Solution:

To find the y-intercept of the line with the equation $$2y - 4x = 10x - 20$$, we need to rewrite the equation in the slope-intercept form $$y = mx + b$$.

Given equation: $$2y - 4x = 10x - 20$$

First, let's isolate $$y$$-term to one side of the equation:

$$2y = 4x + 10x - 20$$

$$2y = 14x - 20$$

Next, divide both sides by 2 to solve for $$y$$:

$$y = 7x - 10$$

Now, we have the equation in slope-intercept form, where $$m = 7$$ (the slope) and $$b = -10$$ (the y-intercept).

Therefore, the y-intercept of the line is $$-10$$.

Example 4. What is the equation of the line between (0,10) and (4,20)?

Solution:

To find the equation of the line passing through the points $$(0, 10)$$ and $$(4, 20)$$, we first need to determine the slope ($$m$$) using the formula:

$$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$$

Given the points $$(0, 10)$$ and $$(4, 20)$$, we can substitute their coordinates into the formula:

$$m = \frac{{20 - 10}}{{4 - 0}}$$

$$m = \frac{{10}}{{4}}$$

$$m = 2.5$$

Now that we have the slope ($$m$$), we can use one of the given points and the slope to find the equation of the line using the point-slope form:

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

Let's use the point $$(0, 10)$$:

$$y - 10 = 2.5(x - 0)$$

$$y - 10 = 2.5x$$

Now, let's isolate $$y$$ :

$$y = 2.5x + 10$$

Therefore, the equation of the line passing through the points $$(0, 10)$$ and $$(4, 20)$$ is $$y = 2.5x + 10$$.

Example 5. Rewrite the equation in slope-intercept form: 2/3y=1/2x+3

Solution:

To rewrite the equation $$\frac{2}{3}y = \frac{1}{2}x + 3$$ in slope-intercept form $$y = mx + b$$, we need to isolate $$y$$ on one side of the equation.

Given equation: $$\frac{2}{3}y = \frac{1}{2}x + 3$$

First, let's isolate $$y$$ by multiplying both sides by $$\frac{3}{2}$$ to get rid of the fraction:

$$\frac{2}{3}y \times \frac{3}{2} = \frac{1}{2}x \times \frac{3}{2} + 3 \times \frac{3}{2}$$

$$y = \frac{3}{4}x + \frac{9}{2}$$

Now, we have the equation in slope-intercept form, where $$m = \frac{3}{4}$$ (the slope) and $$b = \frac{9}{2}$$ (the y-intercept).

Therefore, the equation in slope-intercept form is:

$$y = \frac{3}{4}x + \frac{9}{2}$$

## Practice Problems

Q1. What is the equation of a line with a slope of $$2$$ passing through the point $$(3, 7)$$?

1. $$y = 2x + 1$$
2. $$y = 2x - 1$$
3. $$y = 2x + 3$$
4.  $$y = 2x - 3$$

Q2. Select the equation of a line with a slope of $$-\frac{3}{4}$$ passing through the point $$(6, 2)$$.

1.  $$y = -\frac{3}{4}x + 8$$
2.  $$y = -\frac{3}{4}x + \frac{13}{2}$$
3.  $$y = -\frac{4}{3}x + \frac{13}{2}$$
4.  $$y = -\frac{4}{3}x + 8$$

Q3. Write the equation $$3x - 6y = 10$$ in slope-intercept form.

1.  $$y = \frac{1}{2}x - \frac{5}{3}$$
2.  $$y = \frac{1}{2}x + \frac{5}{3}$$
3.  $$y = -\frac{1}{2}x - \frac{5}{3}$$
4.  $$y = -\frac{1}{2}x + \frac{5}{3}$$

Q4. Identify the equation given in slope-intercept form from the following choices:

1. $$x = \frac{1}{4}y - 2$$
2. $$y = 4x + 2$$
3. $$4x - y + 2 = 0$$
4. $$4x - y = -2$$

Q5. Rewrite the equation $$2 + 4y = 3x + 10$$ in slope-intercept form.

1.  $$y = \frac{3}{4}x + 2$$
2. $$y = \frac{3}{4}x - \frac{1}{2}$$
3. $$y = \frac{3}{4}x + \frac{5}{2}$$
4. $$y = \frac{3}{4}x - \frac{5}{2}$$

Q1. What is the slope-intercept form of a linear equation?

Answer: The slope-intercept form of a linear equation is $$y = mx + b$$, where $$m$$ represents the slope of the line and $$b$$ represents the y-intercept.

Q2. How do you find the slope-intercept form of a line given its slope and a point?

Answer: You can use the formula $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. Substitute the values of the slope and the given point to find the equation of the line.

Q3. What information does the slope-intercept form provide about a line?

Answer: The slope-intercept form allows you to easily identify the slope of the line, which represents its steepness and direction, as well as the y-intercept, which indicates where the line intersects the y-axis.

Q4. Can a line have a negative slope in the slope-intercept form?

Answer: Yes, a line can have a negative slope. In the slope-intercept form, if $$m$$ is negative, the line slopes downward from left to right.

Q5. How does the slope-intercept form simplify graphing linear equations?

Answer: The slope-intercept form directly provides the slope and y-intercept of the line, making it straightforward to plot the line on a graph without needing to calculate additional points.