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 (2)
Multiply both sides of the equation by 5: 5 ( y + 7 ) = − 2 ( x − 10 ) .
Distribute: 5 y + 35 = − 2 x + 20 .
Rearrange to standard form: 2 x + 5 y = 20 − 35 .
Simplify: 2 x + 5 y = − 15 . The standard form of the equation is 2 x + 5 y = − 15 .
Explanation
Understanding the Problem We are given the point-slope form of a line and asked to convert it to standard form. The point-slope form is given by y + 7 = − 5 2 ( x − 10 ) . The standard form of a linear equation is A x + B y = C , where A, B, and C are integers, and A is non-negative.
Eliminate the Fraction First, we multiply both sides of the equation by 5 to eliminate the fraction: 5 ( y + 7 ) = 5 ( − 5 2 ( x − 10 ) ) 5 ( y + 7 ) = − 2 ( x − 10 )
Distribute Next, we distribute on both sides of the equation: 5 y + 35 = − 2 x + 20
Rearrange to Standard Form Now, we rearrange the equation to get the standard form A x + B y = C . We add 2 x to both sides and subtract 35 from both sides: 2 x + 5 y = 20 − 35
Simplify and Conclude Finally, we simplify the equation: 2 x + 5 y = − 15 Thus, 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 applications. 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 and standard form can help you easily determine the total cost for a given number of units or find the number of units you can afford with a certain budget. This is also applicable in physics, where linear equations can describe motion or relationships between physical quantities.
The standard form of the equation derived from the point-slope form y + 7 = − 5 2 ( x − 10 ) is 2 x + 5 y = − 15 . Therefore, the chosen option is C: 2 x + 5 y = − 15 .
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