Complete the point-slope equation of the line through $(8,-8)$ and $(9,8)$. Use exact numbers. $y-8=$ $\square$
Answer (1)
Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 with points ( 8 , − 8 ) and ( 9 , 8 ) , which gives m = 16 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 9 , 8 ) and the calculated slope m = 16 .
Substitute the values into the point-slope equation: y − 8 = 16 ( x − 9 ) .
The completed point-slope equation is 16 ( x − 9 ) .
Explanation
Understanding the Problem We are given two points, ( 8 , − 8 ) and ( 9 , 8 ) , and we want to find the point-slope equation of the line passing through these points. The point-slope form of a linear equation is given by y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. We are asked to complete the equation in the form y − 8 = □ . This means we will use the point ( 9 , 8 ) as ( x 1 , y 1 ) . The only thing we need to find is the slope m .
Calculating the Slope To find the slope m , we use the formula m = x 2 − x 1 y 2 − y 1 , where ( x 1 , y 1 ) = ( 8 , − 8 ) and ( x 2 , y 2 ) = ( 9 , 8 ) . Plugging in the values, we get:
m = 9 − 8 8 − ( − 8 ) = 1 8 + 8 = 1 16 = 16
Writing the Point-Slope Equation Now that we have the slope m = 16 , we can write the point-slope equation using the point ( 9 , 8 ) :
y − 8 = 16 ( x − 9 )
We are asked to complete the equation y − 8 = □ . Therefore, the expression that goes in the box is 16 ( x − 9 ) .
Final Answer Thus, the completed point-slope equation is y − 8 = 16 ( x − 9 ) .
Examples
Point-slope form is useful in many real-world scenarios. For example, if you know the rate at which a savings account is growing (the slope) and the amount in the account at a particular time (a point), you can use the point-slope form to determine the amount in the account at any other time. Similarly, in physics, if you know the velocity of an object at a certain time and the constant acceleration (the slope), you can find the velocity at any other time using the point-slope form.
