Complete the point-slope equation of the line through $(-8,-1)$ and $(-6,5)$. Use exact numbers.
[tex]y-5=[/tex] $\square$
Answer (1)
Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 .
Substitute the given points ( − 8 , − 1 ) and ( − 6 , 5 ) into the slope formula to find m = 3 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( − 6 , 5 ) and the slope m = 3 .
The point-slope equation is y − 5 = 3 ( x + 6 ) , so the missing expression is 3 ( x + 6 ) .
Explanation
Understanding the Problem We are given two points on a line, ( − 8 , − 1 ) and ( − 6 , 5 ) , and we want to find the point-slope equation of the line in the form y − 5 = m ( x − x 1 ) .
Finding the Slope First, we need to find the slope m of the line. The slope is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In our case, ( x 1 , y 1 ) = ( − 8 , − 1 ) and ( x 2 , y 2 ) = ( − 6 , 5 ) .
Calculating the Slope Substituting the coordinates into the slope formula, we get: m = − 6 − ( − 8 ) 5 − ( − 1 ) = − 6 + 8 5 + 1 = 2 6 = 3 So, the slope of the line is m = 3 .
Writing the Point-Slope Equation Now we can write the point-slope equation of the line using the point ( − 6 , 5 ) and the slope m = 3 . The point-slope form is: y − y 1 = m ( x − x 1 ) Substituting the values, we get: y − 5 = 3 ( x − ( − 6 )) y − 5 = 3 ( x + 6 ) Thus, the equation is y − 5 = 3 ( x + 6 ) .
Final Answer The point-slope equation of the line through ( − 8 , − 1 ) and ( − 6 , 5 ) is y − 5 = 3 ( x + 6 ) . Therefore, the missing expression is 3 ( x + 6 ) .
Examples
The point-slope form is useful in many real-world applications. For example, if you know the rate at which a car is traveling (slope) and its position at a certain time (point), you can determine its position at any other time using the point-slope equation. Similarly, in business, if you know the rate of change of profit (slope) and the profit at a certain time (point), you can predict the profit at any other time.
