How do you find slope-intercept form?

Answer 1

What is slope-intercept form?

Slope intercept form of a line is y = mx +b. In the equation, m is the slope, and b is the y-intercept. (The line crosses the y-axis at the point (0,b)). For example, in the equation y = -(2/3)x - 5, the slope of the line is -2/3, and the line crosses the y-axis at the point (0,-5).

The "solve for b" method

(This can be used if you are given two points on a line.)

The formula for slope intercept is y = mx + b.

m is the slope

First you need to find the slope of the line (m). The formula for slope is

(y2-y1)/(x2-x1).

For example, let's take the points (3, -2) and (5,1). The slope would be

(1 - -2)/(5-3), which is 3/2, which is 11/2or 1.5. For this problem, we will leave it as an improper fraction, so m, or the slope, is 3/2.

Now we have y = (3/2)x + b.

b is the y-intercept, or the point where the line crosses the x-axis. (The line crosses the x-axis at the point (0,b). b is the number we need to find next.

In order to find b, plug in one of the original points for x and y. In this example, we will use the point (5,1). (You could use (3,-2) if you wanted to.) When we plug in the point (5,1), we will get:

1 = 5(3/2) + b

Solve for b: First combine like terms:

1 = 15/2 + b

Subtract 15/2 from each side:

-13/2 = b

We have now found out that b = -13/2. so the complete formula for a line that passes through (3,-2) and (5,1) is:

y = (3/2)x - 13/2

(You could also write it as y = 1.5x - 6.5)

(Below is another method)

The point-slope to slope intercept method:

(This can be used if you are given two points on a line.)

First you write it in point slope form and then convert it to slope intercept y=mx+b form.

Point-Slope form is y-y1 = m(x-x1)

m is the slope

In this example, we will use the points (3, -2) and (5,1), to show that both methods arrive at the same answer.

To start, calculate the slope, (the same way as above):

The formula for slope is

(y2-y1)/(x2-x1).

The slope would be (1 - -2)/(5-3), which is 3/2, which is 11/2or 1.5. We will leave it as an improper fraction, so m, or the slope, is 3/2.

Next, plug in one of the points into x1 and x2­ in the point slope equation. In this example, the point (5,1) is used. Now, we have:

y - 1 = (3/2)(x- 5)

Now, the equation needs to be put into the form y = mx + b

First, distribute:

y-1 = (3/2)x - 15/2

Next, add 1 to each side:

y = (3/2)x -13/2

The final answer is y = (3/2)x - 13/2, which can also be written as y = 1.5x - 6.5

How to find slope-intercept form if you are given the equation of a line in standard form:

The formula for standard form is ax + by + c. You are converting this to slope intercept form, which is y = mx + b. This process is a matter of using algebra. Here is an example: Convert the line 3x + 5y = 25 to slope intercept form:

3x + 5y = 25

First, the x must be moved to the other side, so we will subtract 3x from both sides:

5y = 25 - 3x

In order to match the formula y = mx + b, the terms 25 and -3x need to be switched around:

5y = -3x + 25

Remember to keep the negatives signs in front of their terms when you move them around!

Now, both sides of the equation need to be divided by 5. (All 3 terms must be divided by 5):

(5/5)y = -(3/5)x + (25/5)

After simplifying the fractions, the final equation is

y = -(3/5)x + 5

So, the line 3x + 5y = 25 is y = -(3/5)x + 5 in slope-intercept form. (This can also be written as y = -.6x + 5)

How to find slope-intercept form if you are given the equation of a line in point-slope form:

Point-slope form of a line is y-y1 = m(x - x1). You are converting this to slope intercept form, which is y = mx + b. This process is a matter of using algebra. Here is an example: Convert the line

y - 5 = (2/3)(x+4) to slope intercept form:

y - 5 = (2/3)(x+4)

First, distribute the terms the right side of the equation:

y - 5 = (2/3)x + (2/3)(4)

After simplifying, the equation is

y - 5 = (2/3)x + (8/3)

(8/3) can be rewritten as 22/3:

y - 5 = (2/3)x + 22/3

Now, add five to each side of the equation:

y = (2/3)x + 72/3

So, the equation y - 5 = (2/3)(x+4) is y = (2/3)x + 72/3 in slope-intercept form. (This can also be written as y = (2/3)x + 23/3 or y = (2/3)x + 7.666)

How to find slope-intercept form if you are given the graph of a line:

The first thing to do is pick two points on the line that are at whole numbers, (such as (2,3) and (4,7)). If you can't find two points at whole numbers, you can estaimate where you think a point on that line is, and use those. (For example, you may see a point on the line around (1,2.5).)

Once you have found those two points, you can use one of the methods above, either the "solve for b" method or the "point-slope to slope intercept method". You can also use the visual method below:

In this example, we will find the formula of a line that passes through the points (2,3) and (4,7). (It may help to draw out the line so that you can better visualize this example.) First, we will find the slope using the "rise over run: method. If you put your finger on the point (2,3), you will need to go four spaces to get level with the point (4,7). So 4 is your "rise". You will need to go right 2 spaces to get to the point (4,7), so 2 is your run. The line climbs so the slope is positive. (If the line were to drop, such as a line that passed through the points (1,1) and (0,2), the slope would be negative.) Now you have found that your "rise" is 4, your "run" is 2, and the slope is positive. The slope is "rise over run", which, in this case, is 4/2, which can be simplified to 2. In the equation y = mx + b, m is the slope, so we now have

y = 2x + b

and we need to find b. b is the point where the line crosses the y-axis. You may be able to see that the line in this example crosses the y-axis at the point (0,1), so in this case b is 1. (If the line had crossed the y-axis at (0,-5), b would have been -5). Now we have the complete equation:

y = 2x + 1

So, the equation of the line that passes through the points (2,3) and (4,7) is y = 2x + 1.