The point-slope form of the equation of the line that passes through $(-5,-1)$ and $(10,-7)$ is $y+7=-\frac{2}{5}(x-10)$. What is the standard form of the equation for this line?
A. $2 x-5 y=-15$
B. $2 x-5 y=-17$
C. $2 x+5 y=-15$
D. $2 x+5 y=-17$
Answer (1)
Distribute the term on the right side of the equation: y + 7 = − 5 2 x + 4 .
Isolate y : y = − 5 2 x − 3 .
Multiply by 5 to eliminate the fraction: 5 y = − 2 x − 15 .
Convert to standard form: 2 x + 5 y = − 15 , so the answer is 2 x + 5 y = − 15 .
Explanation
Understanding the Problem We are given the point-slope form of a line: y + 7 = − 5 2 ( x − 10 ) . Our goal is to convert this to standard form, which looks like A x + B y = C , where A , B , and C are integers, and A is non-negative.
Distributing the fraction First, distribute the − 5 2 on the right side of the equation:
y + 7 = − 5 2 x + 5 2 ( 10 )
y + 7 = − 5 2 x + 4
Isolating y Next, subtract 7 from both sides to isolate the y term:
y = − 5 2 x + 4 − 7
y = − 5 2 x − 3
Eliminating the fraction Now, multiply both sides of the equation by 5 to eliminate the fraction:
5 y = 5 ( − 5 2 x − 3 )
5 y = − 2 x − 15
Converting to Standard Form Finally, rearrange the equation to the standard form A x + B y = C by adding 2 x to both sides:
2 x + 5 y = − 15
Final Answer The standard form of the equation is 2 x + 5 y = − 15 .
Examples
Understanding how to convert between different forms of linear equations is useful in many real-world scenarios. For example, if you are tracking the cost of a service that has a fixed initial fee and a per-unit charge, you can model this with a linear equation. Converting between point-slope form and standard form allows you to easily analyze different aspects of the cost, such as the total cost for a given number of units or the per-unit charge. This skill is also valuable in fields like physics, engineering, and economics, where linear models are frequently used to represent relationships between variables.
