Standard form to slope intercept form answer key

Earlier in this chapter we have expressed linear equations using the standard form Ax + By = C and also y= mx +b. Now we're going to focus on the slope-intercept form y = mx + b.

In the slope-intercept form you use the slope of the line and the y-intercept to express the linear function.

$$y=mx+b$$

Where m is the slope and b is the y-intercept.

Example

Graph the equation

$$y-2x=1$$

rewrite in slope-intercept form

$$y=2x+1$$

Identify the slope and the y-intercept

m = 2 and b = 1

Plot the point corresponding to the y-intercept, (0,1)

The m-value, the slope, tells us that for each step to the right on the x-axis we move 2 steps upwards on the y-axis (since m = 2)

And once you have your second point you can just draw a line through the two points and extend it in both directions.

You can check to see that the line you've drawn is the correct one by substituting the coordinates of the second point into the original equation. If the equation holds true than the second point is correct.

Our second point = (1, 3)

$$y-2x=1$$

$$3-2\cdot 1=3-2=1$$

Our second point is a solution to the equation i.e. the line we drew is correct.

A line that passes through the origin has a y-intersect of zero, b = 0, and represents a direct variation.

$$y=mx$$

In a direct variation the nonzero number m is called the constant of variation.

You can name a function, f by using the function notion

$$f\left ( x \right )=mx+b$$

f(x) is another name for y and is read as "the value of f at x" or "f of x". You can use other letters than f to name functions.

A group of functions that have similar characteristics are called a family of functions. All functions that can be written on the form f(x) = mx + b belong to the family of linear functions.

The most basic function in a family of functions is called the parent function. The parent function of all linear functions is

$$f\left ( x \right )=x$$

Video lesson

Graph y = 3x - 2

In geometry, the equation of a line can be written in different forms and each of these representations is useful in different ways. The equation of a straight line is written in either of the following methods:

1. 1. Point slope form
   2. Two point form
   3. Slope intercept form
   4. Intercept form

In this article, you will learn about one of the most common forms of the equation of lines called slope-intercept form along with derivation, graph and examples.

Learn what is the intercept of a line here.

Let’s have a look at the slope-intercept form definition.

What is the Slope Intercept Form of a Line?

The graph of the linear equation y = mx + c is a line with m as slope, m and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers.

The slope, m, represents the steepness of a line. The slope of the line is also termed as gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis.

In this section, you will learn the derivation of the equation of a line in the slope-intercept form.

Consider a line L with slope m cuts the y-axis at a distance of c units from the origin.

Here, the distance c is called the y-intercept of the given line L.

So, the coordinate of a point where the line L meets the y-axis will be (0, c).

That means, line L passes through a fixed point (0, c) with slope m.

We know that, the equation of a line in point slope form, where (x1, y1) is the point and slope m is:

(y – y1) = m(x – x1)

Here, (x1, y1) = (0, c)

Substituting these values, we get;

y – c = m(x – 0)

y – c = mx

y = mx + c

Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c

Note: The value of c can be positive or negative based on the intercept is made on the positive or negative side of the y-axis, respectively.

Slope Intercept Form Formula

As derived above, the equation of the line in slope-intercept form is given by:

y = mx + c

Here,

(x, y) = Every point on the line

m = Slope of the line

c = y-intercept of the line

Usually, x and y have to be kept as the variables while using the above formula.

Slope Intercept Form x Intercept

We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:

y = m(x – d)

Here, 

m = Slope of the line

d = x-intercept of the line

Sometimes, the slope of a line may be expressed in terms of tangent angle such as:

m = tan θ

Also, try: Slope Intercept Form Calculator

Derivation of Slope-Intercept Form from Standard Form Equation

We can derive the slope-intercept form of the line equation from the equation of a straight line in the standard form as given below:

As we know, the standard form of the equation of a straight line is:

Ax + By + C = 0

Rearranging the terms as:

By = -Ax – C

⇒y = (-A/B)x + (-C/B)

This is of the form y = mx + c

Here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.

Slope Intercept Form Graph

When we plot the graph for slope-intercept form equation we get a straight line. Slope-intercept is the best form. Since it is in the form “y=”, hence it is easy to graph it or solve word problems based on it. We just have to put the x-values and the equation is solved for y.

The best part of the slope-intercept form is that we can get the value of slope and the intercept directly from the equation.

Solved Examples

Example 1:

Find the equation of the straight line that has slope m = 3 and passes through the point (–2, –5).

Solution:

By the slope-intercept form we know;

y = mx+c

Given,

m = 3

As per the given point, we have;

y = -5 and x = -2

Hence, putting the values in the above equation, we get;

-5 = 3(-2) + c

-5 = -6+c

c = -5 + 6 = 1

Hence, the required equation will be;

y = 3x+1

Example 2:

Find the equation of the straight line that has slope m = -1 and passes through the point (2, -3).

Solution:

By the slope-intercept form we know;

y = mx+c

Given,

m = -1

As per the given point, we have;

y = -3 and x = 2

Hence, putting the values in the above equation, we get;

-3 = -1(2) + c

-3 = -2 + c

c = -3+2 = -1

Hence, the required equation will be;

y = -x-1

Example 3:

Find the equation of the lines for which tan θ = 1/2, where θ is the inclination of the line such that: 

(i) y-intercept is -5 

(ii) x-intercept is 7/3

Solution:

Given, tan θ = 1/2

So, slope = m = tan θ = 1/2

(i) y-intercept = c = -5

Equation of the line using slope intercept form is:

y = mx + c

y = (1/2)x + (-5)

Or

2y = x – 10

x – 2y – 10 = 0

(ii) x-intercept = d = 7/3

Equation of slope intercept form with x-intercept is:

y = m(x – d)

y = (1/2)[x – (7/3)]

Or

2y = (3x – 7)/3

6y = 3x – 7

3x – 6y – 7 = 0

Practice Problems

1. Find the slope of the line y = 5x + 2.
2. Find the slope of the line which crosses the line at point (-2,6) and have an intercept of 3.
3. What is the equation of the line whose angle of inclination is 45 degrees and x-intercept is -⅗?
4. Write the equation of the line passing through the point (0, 0) with slope -4.