Interpreting Lines:
This is an introduction to drawing lines when given the slope and the y-intercept in an equation form. Remember that the y-intercept is where the graph crosses the y-axis; this is where we usually start. First, find the y-intercept, then determine the slope. For now, just focus on whether the slope is positive or negative.
Here are the variables we will start using in our function:
- m = slope
- b = y-intercept
The equation is y = mx + b. The x and y variables remain as letters, but m and b are replaced by numbers (ex: y = 2x + 4, slope = 2 and y-intercept = 4). The following video will show a few examples of understanding how to use the slope and intercept from an equation.
Video Source (03:53 mins) | Transcript
y = mx + b
This equation is called the slope-intercept form because the two numbers in the equation are the slope and the intercept. Remember, the slope (m) is the number being multiplied to x and the intercept (b) is the number being added or subtracted.
Additional Resources
- Khan Academy: Intro to Slope-Intercept Form (08:59 mins; Transcript)
- Khan Academy: Worked Examples: Slope-Intercept Intro (04:39 mins; Transcript)
Practice Problems
- Find the slope of the line:
\(\text{y}=6\text{x}+2\) - Find the y-intercept of the line:
\({\text{y}}=-7{\text{x}}+4\) - Find the slope of the line:
\({\text{y}}=-3{\text{x}}+5\) - Find the y-intercept of the
line:
\({\text{y}}=-{\text{x}}-3\)
The equation, y = mx + b, is the slope-intercept form of a straight line. Here, x and y are the coordinates of the points, m is the gradient, and b is the intercept of the y-axis. The equations of lines can be of different forms based on the information we have. Suppose the coordinates of two points are given, which forms a straight line, then the line will form a linear equation
(e.g. y = x + 3, where x and y are the coordinates of the point). The general form of the equation of the straight line is given by Ax + By + C = 0, for a line. y = mx + b is the slope-intercept form of the equation of a straight line. In the equation y = mx + b, m is the slope of the line and b is the intercept. x and y represent the distance of the line from the x-axis and y-axis, respectively. The value of b is equal to y when x = 0, and m shows how steep the
line is. The slope of the line is also called the gradient. The formula to find the slope, m, of the line is given by: m = (difference in y coordinates)/(difference in x coordinates) \(\begin{array}{l}m = \frac{y_2-y_1}{x_2-x_1}\end{array} \)What is y = mx + b?
The equation of a line passing through a point (x1, y1) is given by:
y – y1 = m(x – x1)
The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
\(\begin{array}{l}\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\end{array} \)
How To Find y = mx + b?
To find the equation of the straight line, we use the slope-intercept form, y = mx + b, where m is the slope of the line, b is the y-intercept of the line.
We can find the equation of a line in the form of y = mx + b, if the coordinates of points forming the line are known to us.
The slope of the line, m can also be written as:
m = (y-b)/x
So, the formula to find the slope of the straight line is:
m = change in y/change in x
Now, suppose we have two points on a straight line whose coordinates are (x1, y1) and (x2, y2). Thus, we can write:
y1 = mx1 + b and y2 = mx2 + b
Since, m is the ratio of change in y to change in x, thus;
\(\begin{array}{l}\frac{y_{2}-y_{1}}{x_{2}-x_{1}} =\frac{(m x_{2}+b)-(m x_{1}+b)}{x_{2}-x_{1}}\end{array} \)
\(\begin{array}{l}=\frac{m x_{2}-m x_{1}}{x_{2}-x_{1}}\end{array} \)
Taking m common, we get,
\(\begin{array}{l}=\frac{m( x_{2}- x_{1})}{x_{2}-x_{1}}\end{array} \)
= m
Hence,
\(\begin{array}{l}m = \frac{y_2-y_1}{x_2-x_1}\end{array} \)
I.e., m= Difference in y coordinates / Difference in x coordinates.
Y = mx + b at Origin
The equation of a straight line with slope m passing through the origin (0,0) is given by:
y = mx
Hence, the y-intercept at the origin is zero.
Solved Examples
Example 1:
Find the slope and y-intercept of the equation, y = 3x – 2.
Solution:
If we compare the given equation with y = mx + b, where m is the slope and b is the y-intercept, then we get,
Slope, m = 3
y-intercept, b = -2
Example 2:
What is the slope and y-intercept of the equation, y= 5x?
Solution:
If we compare the given equation with y = mx + b, where m is the slope and b is the y-intercept, then we get,
Slope, m = 5
y-intercept, b = 0
Here, the y-intercept is zero, which proves that the slope of the line passes through the origin.
Example 3:
If the slope of a straight line is 5 and the y-intercept is 3, then find the equation of the line.
Solution:
We know the equation of the line in slope-intercept form is given by:
y = mx + b
Given, m = 5 and b = 3.
Thus, the required equation is:
y = 5x + 3