Complete the point-slope equation of the line through $(1,0)$ and $(6,-3)$. Use exact numbers. $y-(-3)=$ $\square$
Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 with points ( 1 , 0 ) and ( 6 , − 3 ) , which gives m = − 5 3 .
Substitute the slope and the point ( 6 , − 3 ) into the point-slope form y − y 1 = m ( x − x 1 ) .
The equation becomes y − ( − 3 ) = − 5 3 ( x − 6 ) .
The completed point-slope equation is y − ( − 3 ) = − 5 3 ( x − 6 ) .
Explanation
Understanding the Problem We are given two points, ( 1 , 0 ) and ( 6 , − 3 ) , and we want to find the point-slope equation of the line that passes through them. 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 y − ( − 3 ) = □ . This means we will use the point ( 6 , − 3 ) as ( x 1 , y 1 ) .
Finding the Slope First, we need to find the slope of the line. The slope, m , can be calculated using the formula m = x 2 − x 1 y 2 − y 1 , where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the given points. In our case, ( x 1 , y 1 ) = ( 1 , 0 ) and ( x 2 , y 2 ) = ( 6 , − 3 ) .
Calculating the Slope Plugging in the coordinates of the points into the slope formula, we get: m = 6 − 1 − 3 − 0 = 5 − 3 = − 5 3 So, the slope of the line is − 5 3 .
Writing the Point-Slope Equation Now we can write the point-slope equation of the line using the point ( 6 , − 3 ) and the slope m = − 5 3 . The equation is: y − ( − 3 ) = − 5 3 ( x − 6 ) Thus, the missing expression is − 5 3 ( x − 6 ) .
Final Answer Therefore, the completed point-slope equation is: y − ( − 3 ) = − 5 3 ( x − 6 ) .
Examples
Point-slope form is incredibly useful in various real-world scenarios. For instance, imagine you're tracking the descent of an airplane. If you know the plane's altitude at two different times, you can determine the rate of descent (slope) and use the point-slope form to predict its altitude at any given time. Similarly, in business, if you know the cost of producing a certain number of items, you can use the point-slope form to model the cost function and predict the cost of producing different quantities. This method provides a simple yet powerful way to model linear relationships and make predictions based on available data.
