Find the slope-intercept form of the line that satisfies the given conditions:
Through $A(-9,7)$ and $B(8,13)$
Answer (1)
Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 17 6 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with point A ( − 9 , 7 ) to get y − 7 = 17 6 ( x + 9 ) .
Simplify the equation to slope-intercept form: y = 17 6 x + 17 54 + 7 .
Convert 7 to 17 119 and combine constants to get the final equation: y = 17 6 x + 17 173 .
Explanation
Problem Analysis We are given two points, A ( − 9 , 7 ) and B ( 8 , 13 ) , and we need to find the equation of the line passing through these points in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope First, we need to calculate the slope m of the line using the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) = ( − 9 , 7 ) and ( x 2 , y 2 ) = ( 8 , 13 ) .
Slope Calculation Result Substituting the coordinates of points A and B into the slope formula, we get: m = 8 − ( − 9 ) 13 − 7 = 8 + 9 6 = 17 6 So, the slope of the line is 17 6 .
Using Point-Slope Form Now, we use the point-slope form of a line, which is: y − y 1 = m ( x − x 1 ) We can use either point A or B . Let's use point A ( − 9 , 7 ) .
Substituting Values Substituting the values of x 1 , y 1 , and m into the point-slope form, we get: y − 7 = 17 6 ( x − ( − 9 )) y − 7 = 17 6 ( x + 9 )
Simplifying to Slope-Intercept Form Now, we simplify the equation and solve for y to get the slope-intercept form: y − 7 = 17 6 x + 17 6 × 9 y − 7 = 17 6 x + 17 54 y = 17 6 x + 17 54 + 7 To add the fractions, we need a common denominator, which is 17. So, we rewrite 7 as 17 7 × 17 = 17 119 :
y = 17 6 x + 17 54 + 17 119 y = 17 6 x + 17 54 + 119 y = 17 6 x + 17 173 Thus, the equation of the line in slope-intercept form is y = 17 6 x + 17 173 .
Final Answer The slope-intercept form of the line that passes through the points A ( − 9 , 7 ) and B ( 8 , 13 ) is: y = 17 6 x + 17 173
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a scenario where a taxi charges a fixed fee plus a per-mile rate. If the taxi charges $7 for a trip of -9 miles and $13 for a trip of 8 miles, we can use the points (-9, 7) and (8, 13) to determine the taxi's rate per mile and fixed fee. The slope represents the rate per mile, and the y-intercept represents the fixed fee. This problem demonstrates how linear equations can model real-world pricing and cost structures.
