Write the equation from points (3,1) & (4,2) in slope intercept form.

Question

In order, all the step to finding the slope intercept form.

Answer 1

Recall the slope-intercept form of a linear equation is as follows:

     y = mx + b ,   where m represents the slope of the line and b represent the y-intercept (the point where the line intersects the y-axis).

Given two points on a line, (x₁, y₁) and (x₂, y₂), we can find the slope (m) of the line using the following formula:

             m = (y₂ - y₁)/(x₂ - x₁)

Therefore, for a line with the points (x₁, y₁)=(3, 1) and (x₂, y₂)=(4, 2), the slope is as follows:

             m = (2 - 1)/(4 - 3) = 1/1 = 1

Using the slope of the line (m=1) and one of the points on the line (e.g., (3, 1)), plug in these values into the slope-intercept form of a linear equation to find the y-intercept (b):

     y = mx + b

     1 = (1)(3) + b

     1 = 3 + b          (subtract 3 from both sides of the equation)

     -2 = b

Thus, given that the slope of the line is 1 (m=1) and the y-intercept is -2 (b=-2), then the equation of the line in slope-intercept form is as follows:

         y = mx + b

         y = (1)x + (-2)

         y = x - 2

Answer 2

To answer Nataliya's question regarding the difference between "Slope Intercept Form" and "Point-Slope Form": They're the same thing; only written differently.

Let's work the problem using Point-Slope Form.

Start with slope formula m = (y2 - y1) / (x2 - x1 ).  

m = ( 2 - 1 ) / ( 4 - 3)

m = 1 / 1 = 1.

 

Now, using the Point-Slope form, we plug into as such using point (3,1):

y - y1 = m (x - x1)

y - (1) = 1 ( x - 3 )

y - 1 = x - 3           then add 1 to each side to get y by itself.

y = x - 2             FINAL ANSWER IN SLOPE-INTERCEPT FORM.

 

HOW THE TWO FORMS ARE RELATED:

Assume, the point you use is in the form (0,b) where the b is the y-intercept.

If you plug it into the Point-Slope Form, you would obtain:

y - b = m(x - 0)

y - b = mx              Then add b to each side.

y = mx + b

If you picked any other point, the form would modify itself as such:

y - y1 = m (x - x1)

y - y1 = mx - mx1

y = mx - mx1 + y1

So, basically (y1 - mx1) is the same as the y-intercept.

Answer 3

Find the equation of the line between points (3,1) and (4,2)

Notes:
Points are written as (x-coordinate, y-coordinate) where x is the horizontal axis and y is the vertical axis

Slope-intercept form: y = mx + b

• y (point on y-axis)
• m (slope)
• x (point on x-axis)
• b (height crossing the y-axis)
  

Examples of lines: y = 4x - 8, y = (3/4)x + 1, y = -5x -2

Solve:

Start by finding the slope of the line between those 2 points.  You can also do this visually/on graph paper for simple lines.

The slope, or 'm' is the steepness of the line:

m = (y₂ - y₁) / (x₂ - x₁) = (change in y) / (change in x)

So choose from points (3,1) & (4,2):

                m = (2 - 1) / (4 - 3)   

                m = 1 / 1  

So the slope is 1.

Now take this slope (m) and your points, and plug it into the equation to solve for where the line crosses the y-intercept (b).

y = mx + b

So if you have m = 1, and choose from (3,1) x = 3, and y = 1:

(1) = (1)(3) + b

1 = 3 + b

1 - 3 = b

b = -2

Therefore your equation is:

y = (1)x +(-2)

y = x - 2

 

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