Line $JK$ passes through points $J(-3,11)$ and $K(1,-3)$. What is the equation of line $JK$ in standard form?
A. $7 x+2 y=-1$
B. $7 x+2 y=1$
C. $14 x+4 y=-1$
D. $14 x+4 y=1
Answer (1)
Calculate the slope of the line using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 2 7 .
Use the point-slope form of the line y − y 1 = m ( x − x 1 ) with point J ( − 3 , 11 ) to get y − 11 = − 2 7 ( x + 3 ) .
Convert the equation to standard form A x + B y = C by eliminating fractions and rearranging terms.
The equation of the line in standard form is 7 x + 2 y = 1 .
Explanation
Understanding the Problem We are given two points, J ( − 3 , 11 ) and K ( 1 , − 3 ) , and we need to find the equation of the line J K in standard form, which is A x + B y = C , where A , B , and C are integers, and A is non-negative.
Calculating the Slope First, we need to find the slope of the line J K . The slope m is given by the formula: m = x 2 − x 1 y 2 − y 1 Substituting the coordinates of points J and K , we get: m = 1 − ( − 3 ) − 3 − 11 = 4 − 14 = − 2 7 = − 3.5 So, the slope of the line is − 2 7 .
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 J or K . Let's use point J ( − 3 , 11 ) :
y − 11 = − 2 7 ( x − ( − 3 )) y − 11 = − 2 7 ( x + 3 ) y − 11 = − 2 7 x − 2 21
Converting to Standard Form Next, we convert the equation to standard form A x + B y = C . To do this, we first eliminate the fraction by multiplying the entire equation by 2: 2 ( y − 11 ) = 2 ( − 2 7 x − 2 21 ) 2 y − 22 = − 7 x − 21 Now, we rearrange the terms to get the standard form: 7 x + 2 y = 22 − 21 7 x + 2 y = 1
Final Answer The equation of the line J K in standard form is 7 x + 2 y = 1 . Comparing this with the given options, we find that it matches the second option.
Conclusion Therefore, the equation of line J K in standard form is 7 x + 2 y = 1 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a scenario where you are tracking the depreciation of a car's value over time. If the car's value decreases linearly, you can use a linear equation to model this depreciation. By knowing the initial value and the rate of depreciation (slope), you can predict the car's value at any point in time. This helps in financial planning, insurance assessments, and understanding the long-term cost of ownership.
