Finding the equation of a line

Question

Find an equation of the line that passes through the given points.

(-13, 13) and (20, -53)

Answer 1

The equation of a line is given by the formula y= mx+b.

This is known as the slope-intercept form of a linear equation. It is called a linear equation because when you graph this equation all of the points make a straight line.

In this equation m = slope. The slope = rise over run or "change in y over change in x.
To find the slope you would use the formula m=y2-y1/x2-x1

We will label our points: X1=-13, X2=20, Y1= 13, Y2=-53

We will insert the points into our formula , m= (-53)- 13/20-(-13) = -66/33= -2
Therefore our slope is m= -2

Next we need to find b which is the y-intercept. This is the value of y when x = 0

To find the y intercept, we can use 2 methods:

• Method 1:
Since we know the points and the slope, we can plug one pair of them (it doesn't matter which pair) into our equation y-mx +b, We will use the points (-13,13)

13 =-2(-13) +b)
13= 26+b
We will subtract 26 both sides to isolate b.
13-26= 26-26+b
-13 =b (Therefore b or the "y-intercept=-13)

• Method 2
Since we know the points and the slope, we can use the point-slope formula to find the y -intercept.
The point-slope formula is y-y1= m(x-x1) We will use the points (13,-13, the points we have identified as (x1,y1), Again we could have used either pair.

Y-13= -2(x-(-13) )
y-13= -2x-26
We will add 13 to both sides to isolate Y
Y-13+13= -2x-26+13
Y= -2x-13 (Therefore b or the "y-intercept=-13)

Therefore the equation of the line is: Y= -2X-13

To verify our solution we will substitute both set of points into the equation.

             (-13,13)
             • 13 = -2(-13)-13
             • 13= 26-13
             • 13=13

            (20,-53)

• -53 = -2(20)-13
• -53 = -40 - 13
• -53= -53

We have verified our equation of the line, y= -2x-13

M (Slope) =   -2

B (Y-Intercept) = -13

 

Answer 2

The equation of a line is given by y=mx+b where m is the slope and b is the y-intercept.

First, to solve for the slope, use "rise over run," or "y over x:"  m=(y2-y1)/(x2-x1)  where  (-13, 13) is the first point and (20, -53) is the second point.  Thus, after substituting the values into the equation, you would have m=(-53-13)/[20-(-13)].

Next, to solve for the y-intercept, insert the value found for m into y=mx+b.  Then, substitute the x and y values of just one of the given points into the same equation.  For example, -53=m(20)+b or 13=m(-13)+b are both viable options.  After solving for b, you can then substitute that value into y=mx+b.

To check your answer, plug one of the given points back into y=mx+b.  Both sides of the equal sign should be equal.

Answer 3

We are looking to fill in the basic equation of a line: y=mx+b

m=slope and b=y-intercept

 

First, we will find the slope by using the equation:  (y2-y1)/(x2-x1)

In this case y2=-53  y1=13  x2=20  x1=-13

Fill in those points into the slope equation.  (-53-13)/(20-(-13)) = -66/33 = -2

Therefore, m=-2

 

Now, we will plug in one of the points listed above either (-13, 13) or (20, -53) and plug it into the equation: y=-2x+b

So, 13=-2(-13)+b

13=26+b

-13=b

 

Finally, the equation for the line that passes through the given points (-13, 13) and (20, -53) is:

y=-2x-13