The point-slope form of the equation of the line that passes through $(-4,-3)$ and $(12,1)$ is $y-1=\frac{1}{4}(x-12)$. What is the standard form of the equation for this line?
A. $x-4 y$
B. $x-4 y=2$
C. $4 x-y=8$
D. $4 x-y=2$
Answer (1)
Distribute the 4 1 : y − 1 = 4 1 x − 3 .
Move the x term to the left: − 4 1 x + y = − 2 .
Multiply by -4 to eliminate the fraction: x − 4 y = 8 .
The standard form of the equation is x − 4 y = 8 .
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 y − 1 = 4 1 ( x − 12 ) . The standard form is A x + B y = C , where A, B, and C are integers, and A is non-negative.
Distribute First, distribute the 4 1 on the right side of the equation:
y − 1 = 4 1 x − 4 1 ( 12 )
y − 1 = 4 1 x − 3
Rearrange Terms Next, move the x term to the left side of the equation:
− 4 1 x + y = − 3 + 1
− 4 1 x + y = − 2
Eliminate Fraction Now, multiply both sides of the equation by -4 to eliminate the fraction and make the coefficient of x positive:
( − 4 ) ( − 4 1 x + y ) = ( − 4 ) ( − 2 )
x − 4 y = 8
Final Answer The standard form of the equation is x − 4 y = 8 .
Examples
Understanding standard form equations helps in various real-world applications. For example, when planning a budget, you might use a linear equation in standard form to represent the relationship between the amount spent on two different categories of expenses, such as housing and food. By converting the equation to standard form, you can easily analyze the trade-offs between these expenses and make informed decisions about how to allocate your resources effectively. This skill is also useful in physics, where standard form equations can describe relationships between variables like force and acceleration.
