Maths Slope Intercept Form
Slope Intercept Form
The slopeintercept form of a linear equation is:
$$y = mx + b$$
where:
 $m$ is the slope of the line
 $b$ is the yintercept of the line
Graphing a Line in Slope Intercept Form
To graph a line in slope intercept form, follow these steps:
 Plot the yintercept on the $y$axis.
 Use the slope to find other points on the line.
 Connect the points to form a line.
Example
The equation $y = 2x + 3$ is in slope intercept form. The slope of the line is 2 and the yintercept is 3.
To graph the line, first plot the yintercept (0, 3) on the $y$axis. Then, use the slope to find other points on the line. For example, when $x = 1$, $y = 2(1) + 3 = 5$. So, the point (1, 5) is on the line.
Connect the points (0, 3) and (1, 5) to form a line.
Applications of Slope Intercept Form
The slope intercept form of a linear equation is used in a variety of applications, including:
 Finding the equation of a line that passes through two points
 Determining the slope of a line
 Finding the yintercept of a line
 Graphing a line
Slope Intercept Form Examples
The slopeintercept form of a linear equation is:
$$y = mx + b$$
Where:
 $m$ is the slope of the line.
 $b$ is the yintercept of the line.
Example 1: Find the equation of the line that passes through the points $(2, 4)$ and $(6, 10)$.
Solution:
First, we need to find the slope of the line. We can use the slope formula:
$$m = \frac{y_2  y_1}{x_2  x_1}$$
Where:
 $(x_1, y_1)$ is the first point.
 $(x_2, y_2)$ is the second point.
Substituting the values into the formula, we get:
$$m = \frac{10  4}{6  2} = \frac{6}{4} = \frac{3}{2}$$
Now that we have the slope, we can use the pointslope form of a linear equation to find the equation of the line. The pointslope form is:
$$y  y_1 = m(x  x_1)$$
Where:
 $(x_1, y_1)$ is a point on the line.
 $m$ is the slope of the line.
Substituting the values into the formula, we get:
$$y  4 = \frac{3}{2}(x  2)$$
Simplifying, we get:
$$y = \frac{3}{2}x  3 + 4$$
$$y = \frac{3}{2}x + 1$$
Therefore, the equation of the line that passes through the points $(2, 4)$ and $(6, 10)$ is $y = \frac{3}{2}x + 1$.
Example 2: Graph the line that has a slope of 2 and a yintercept of 3.
Solution:
To graph the line, we can use the slopeintercept form of a linear equation:
$$y = mx + b$$
Where:
 $m$ is the slope of the line.
 $b$ is the yintercept of the line.
Substituting the values into the formula, we get:
$$y = 2x + 3$$
To graph the line, we can first plot the yintercept, which is $(0, 3)$. Then, we can use the slope to find other points on the line. The slope tells us that for every 1 unit we move to the right, we move down 2 units. So, we can plot the points $(1, 1)$ and $(2, 1)$. Connecting these points, we get the graph of the line.
Slope intercept Formula
The slopeintercept formula is a mathematical equation that describes a straight line. It is written in the form:
$$y = mx + b$$
where:
 y is the dependent variable (the variable that is being measured)
 m is the slope of the line
 x is the independent variable (the variable that is being changed)
 b is the yintercept of the line (the point where the line crosses the yaxis)
Slope
The slope of a line is a measure of how steep it is. It is calculated by dividing the change in y by the change in x.
$$m = (y2  y1) / (x2  x1)$$
where:
 m is the slope of the line
 (y2, x2) is a point on the line
 (y1, x1) is another point on the line
YIntercept
The yintercept of a line is the point where the line crosses the yaxis. It is found by setting x = 0 in the slopeintercept formula.
$$b = y  mx$$
where:
 b is the yintercept of the line
 y is the value of y when x = 0
 m is the slope of the line
Graphing a Line
To graph a line, you can use the slopeintercept formula to find two points on the line. Then, you can connect the two points with a straight line.
For example, to graph the line y = 2x + 1, you would first find two points on the line. You can do this by setting x = 0 and x = 1.
When x = 0, y = 2(0) + 1 = 1. When x = 1, y = 2(1) + 1 = 3.
So, the two points on the line are (0, 1) and (1, 3). You can then connect these two points with a straight line to graph the line.
Applications of the SlopeIntercept Formula
The slopeintercept formula has many applications in mathematics and science. For example, it can be used to:
 Find the equation of a line that passes through two points
 Determine the slope of a line
 Find the yintercept of a line
 Graph a line
 Solve systems of equations
The slopeintercept formula is a powerful tool that can be used to solve a variety of problems.
Derivation of Slope Intercept Form
The slopeintercept form of a linear equation is:
$$y = mx + b$$
where:
 $m$ is the slope of the line
 $b$ is the yintercept of the line
To derive the slopeintercept form, we can start with the pointslope form of a linear equation:
$$y  y_1 = m(x  x_1)$$
where:
 $(x_1, y_1)$ is a point on the line
 $m$ is the slope of the line
We can rearrange this equation to get:
$$y = mx  mx_1 + y_1$$
Factoring out $m$, we get:
$$y = m(x  x_1) + y_1$$
Finally, we can rewrite this equation in the slopeintercept form by letting $b = y_1  mx_1$:
$$y = mx + b$$
Example
To find the slopeintercept form of the line that passes through the points $(2, 4)$ and $(6, 10)$, we can use the following steps:
 Calculate the slope of the line:
$$m = \frac{y_2  y_1}{x_2  x_1} = \frac{10  4}{6  2} = \frac{6}{4} = \frac{3}{2}$$
 Substitute the slope and one of the points into the pointslope form of a linear equation:
$$y  4 = \frac{3}{2}(x  2)$$
 Rearrange this equation to get:
$$y = \frac{3}{2}x  3 + 4$$
 Finally, we can rewrite this equation in the slopeintercept form by letting $b = 4  3 = 1$:
$$y = \frac{3}{2}x + 1$$
Equation of Line with Given Inclination
In geometry, a line is a straight onedimensional figure that extends infinitely in both directions. It is defined by two points, called endpoints, and can be represented by an equation. The equation of a line with a given inclination can be derived using the slopeintercept form.
SlopeIntercept Form
The slopeintercept form of a line is given by:
$$y = mx + b$$
where:
 $y$ is the dependent variable (the output)
 $x$ is the independent variable (the input)
 $m$ is the slope of the line
 $b$ is the yintercept of the line
The slope of a line is a measure of how steep it is. It is calculated by dividing the change in $y$ by the change in $x$. The yintercept is the point where the line crosses the $y$axis.
Inclination
The inclination of a line is the angle it makes with the positive $x$axis. It is measured in degrees or radians. The inclination of a line can be determined using the following formula:
$$inclination = arctan(m)$$
where:
 $inclination$ is the inclination of the line in degrees or radians
 $m$ is the slope of the line
Equation of Line with Given Inclination
To find the equation of a line with a given inclination, we can use the slopeintercept form and the formula for inclination.
 First, we need to find the slope of the line. We can do this by using the given inclination and the formula:
$$m = tan(inclination)$$
 Once we have the slope, we can substitute it into the slopeintercept form to find the equation of the line.
$$y = mx + b$$
 Finally, we need to find the yintercept of the line. We can do this by substituting the values of $m$ and $b$ into the equation of the line and setting $x = 0$.
$$y = mx + b$$
$$y = (tan(inclination))x + b$$
$$y = 0 + b$$
$$b = yintercept$$
Example
Let’s find the equation of a line with an inclination of 30 degrees.

First, we need to find the slope of the line. We can do this by using the formula:
m = tan(inclination) m = tan(30 degrees) m = 0.57735

Once we have the slope, we can substitute it into the slopeintercept form to find the equation of the line.
y = mx + b y = 0.57735x + b

Finally, we need to find the yintercept of the line. We can do this by substituting the values of $m$ and $b$ into the equation of the line and setting $x = 0$.
y = mx + b y = 0.57735x + b y = 0 + b b = yintercept
Therefore, the equation of the line with an inclination of 30 degrees is:
y = 0.57735x + b
Slope Intercept Form with xintercept
The slopeintercept form of a linear equation is:
$$y = mx + b$$
where:
 $m$ is the slope of the line
 $b$ is the yintercept of the line
The xintercept of a line is the point where the line crosses the xaxis. To find the xintercept of a line in slopeintercept form, we set $y = 0$ and solve for $x$.
$$0 = mx + b$$
$$mx = b$$
$$x = \frac{b}{m}$$
Therefore, the xintercept of a line in slopeintercept form is $\frac{b}{m}$.
Example
Find the xintercept of the line $y = 2x  3$.
$$x = \frac{b}{m} = \frac{3}{2} = \frac{3}{2}$$
Therefore, the xintercept of the line $y = 2x  3$ is $\frac{3}{2}$.
Conversion of Standard Form to Slope Intercept Form
The standard form of a linear equation is: $$Ax + By = C$$ where A, B, and C are real numbers and A and B are not both zero.
The slopeintercept form of a linear equation is: $$y = mx + b$$ where m is the slope of the line and b is the yintercept.
To convert a standard form equation to slopeintercept form, follow these steps:
 Solve the equation for y.
 Divide both sides of the equation by A.
 Simplify the equation.
Example:
Convert the equation $$3x + 4y = 12$$ to slopeintercept form.
Solution:

Solve the equation for y. $$3x + 4y = 12$$ $$4y = 3x + 12$$ $$y = \frac{3}{4}x + 3$$

Divide both sides of the equation by A. $$y = \frac{3}{4}x + 3$$ $$y = \frac{3}{4}x + \frac{12}{4}$$ $$y = \frac{3}{4}x + 3$$

Simplify the equation. $$y = \frac{3}{4}x + 3$$
The slope of the line is $\frac{3}{4}$ and the yintercept is 3.
Slope Intercept Form of Perpendicular or Parallel Lines
Understanding Parallel and Perpendicular Lines
In geometry, parallel lines are lines that never intersect, while perpendicular lines are lines that intersect at a right angle (90 degrees). The slopeintercept form of a linear equation is a convenient way to represent the equation of a line and determine its slope and yintercept.
SlopeIntercept Form
The slopeintercept form of a linear equation is given by:
y = mx + b
where:
 $m$ is the slope of the line
 $b$ is the yintercept of the line
The slope of a line is a measure of its steepness, while the yintercept is the point where the line crosses the yaxis.
Parallel Lines
Two lines are parallel if they have the same slope. In other words, if the slopes of two lines are equal, the lines are parallel.
For example, the lines $y = 2x + 1$ and $y = 2x  3$ are parallel because they both have a slope of 2.
Perpendicular Lines
Two lines are perpendicular if their slopes are negative reciprocals of each other. In other words, if the product of the slopes of two lines is 1, the lines are perpendicular.
For example, the lines $y = 2x + 1$ and $y = \frac{1}{2}x + 3$ are perpendicular because the product of their slopes is 1.
Determining Parallel or Perpendicular Lines
To determine if two lines are parallel or perpendicular, you can use the following steps:
 Find the slope of each line.
 If the slopes are equal, the lines are parallel.
 If the product of the slopes is 1, the lines are perpendicular.
The slopeintercept form of a linear equation is a useful tool for representing lines and determining their properties, such as parallelism and perpendicularity. By understanding the concept of slope and yintercept, you can easily identify parallel and perpendicular lines.
Slope Intercept Form Solved Examples
The slopeintercept form of a linear equation is:
$$y = mx + b$$
Where:
 $m$ is the slope of the line
 $b$ is the yintercept of the line
To find the slopeintercept form of a linear equation, you need to:
 Put the equation in slopeintercept form.
 Identify the slope and yintercept.
Example 1:
Find the slopeintercept form of the equation $3x + 2y = 8$.
Solution:
 To put the equation in slopeintercept form, we need to solve for $y$.
$$3x + 2y = 8$$
$$2y = 3x + 8$$
$$y = \frac{3}{2}x + 4$$
 The slope of the line is $\frac{3}{2}$ and the yintercept is $4$.
Example 2:
Find the slopeintercept form of the equation $y  5 = 2(x + 3)$.
Solution:
 To put the equation in slopeintercept form, we need to solve for $y$.
$$y  5 = 2(x + 3)$$ $$y  5 = 2x + 6$$ $$y = 2x + 11$$
 The slope of the line is $2$ and the yintercept is $11$.
Example 3:
Find the slopeintercept form of the equation $2x  3y = 12$.
Solution:
 To put the equation in slopeintercept form, we need to solve for $y$.
$$2x  3y = 12$$
$$3y = 2x + 12$$
$$y = \frac{2}{3}x  4$$
 The slope of the line is $\frac{2}{3}$ and the yintercept is $4$.
Slope Intercept Form FAQs
What is the slopeintercept form of a linear equation?
The slopeintercept form of a linear equation is an algebraic expression that represents a straight line in a twodimensional coordinate plane. It is written in the form:
$$y = mx + b$$
where:
 y is the dependent variable (the variable whose value depends on the value of the independent variable).
 x is the independent variable (the variable whose value can be changed without affecting the value of the dependent variable).
 m is the slope of the line (the ratio of the change in y to the change in x).
 b is the yintercept of the line (the value of y when x is equal to 0).
How do you find the slope and yintercept of a line from its equation in slopeintercept form?
To find the slope and yintercept of a line from its equation in slopeintercept form, simply identify the values of m and b in the equation.
 The slope is the coefficient of x, which is the number that comes before the x variable.
 The yintercept is the constant term, which is the number that comes after the x variable.
For example, in the equation y = 2x + 3, the slope is 2 and the yintercept is 3.
What is the difference between the slope and the yintercept of a line?
The slope of a line is a measure of how steep the line is. The steeper the line, the greater the slope. The yintercept of a line is the point where the line crosses the yaxis.
How do you graph a line from its equation in slopeintercept form?
To graph a line from its equation in slopeintercept form, follow these steps:
 Plot the yintercept on the yaxis.
 Use the slope to find additional points on the line. For example, if the slope is 2, you can move up 2 units and over 1 unit to find another point on the line.
 Connect the points with a straight line.
What are some applications of the slopeintercept form of a linear equation?
The slopeintercept form of a linear equation has many applications in mathematics and science. For example, it can be used to:
 Find the equation of a line that passes through two points.
 Determine the slope of a line.
 Find the yintercept of a line.
 Graph a line.
 Solve systems of linear equations.
Conclusion
The slopeintercept form of a linear equation is a powerful tool that can be used to represent and analyze straight lines. By understanding the slope and yintercept of a line, you can gain valuable insights into its behavior.