Complete the point-slope equation of the line through $(-5,4)$ and $(1,6)$. Use exact numbers.
$y-6=$ $\square$
Answer (1)
Calculate the slope using the formula: m = x 2 − x 1 y 2 − y 1 = 1 − ( − 5 ) 6 − 4 = 3 1 .
Substitute the slope and the point ( 1 , 6 ) into the point-slope form: y − y 1 = m ( x − x 1 ) .
The point-slope equation is: y − 6 = 3 1 ( x − 1 ) .
The completed equation is y − 6 = 3 1 ( x − 1 ) .
Explanation
Understanding the Problem We are given two points on a line, ( − 5 , 4 ) and ( 1 , 6 ) , and we want to find the point-slope equation of the line using the point ( 1 , 6 ) . The point-slope form of a line is given by y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope of the line.
Calculating the Slope First, we need to calculate 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 two points on the line. In our case, ( x 1 , y 1 ) = ( − 5 , 4 ) and ( x 2 , y 2 ) = ( 1 , 6 ) . Substituting these values into the formula, we get: m = 1 − ( − 5 ) 6 − 4 = 1 + 5 2 = 6 2 = 3 1
Writing the Point-Slope Equation Now that we have the slope m = 3 1 , we can write the point-slope equation of the line using the point ( 1 , 6 ) . The point-slope form is: y − y 1 = m ( x − x 1 ) Substituting x 1 = 1 , y 1 = 6 , and m = 3 1 , we get: y − 6 = 3 1 ( x − 1 )
Finding the Missing Expression The question asks us to complete the equation y − 6 = □ . From our calculation, we found that the equation is y − 6 = 3 1 ( x − 1 ) . Therefore, the missing expression is 3 1 ( x − 1 ) .
Final Answer Thus, the completed point-slope equation is: y − 6 = 3 1 ( x − 1 )
Examples
Point-slope form is useful in many real-world scenarios. For example, if you are tracking the distance you've traveled over time and know your speed (slope) and your location at a specific time (point), you can use the point-slope form to determine your location at any other time. Similarly, in construction, if you know the slope of a ramp and a point it must pass through, you can define the entire ramp's path using this equation.
