- The given point-slope equation is \(y - 1 = \frac{1}{4}(x - 12)\).
- Multiply by 4 to remove the fraction: \(4y - 4 = x - 12\).
- Rearrange variables and constants: \(x - 4y = 8\).
- The correct standard form is \(\boxed{x - 4y = 8}\).
Explanation
- Analyzing the Given Point-Slope Form
To determine the standard form of the linear equation, we begin with the given point-slope form:
\[y - 1 = \frac{1}{4}(x - 12)\]
This equation represents a line that passes through the point \((12, 1)\) with a slope of \(m = \frac{1}{4}\). The standard form of a linear equation is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers.
- Converting to Standard Form
To convert the equation into standard form, we first eliminate the fraction. We multiply both sides of the equation by 4:
\[4 \times (y - 1) = 4 \times \left( \frac{1}{4}(x - 12) \right)\]
This simplifies to:
\[4(y - 1) = x - 12\]
Next, we expand the left side of the equation:
\[4y - 4 = x - 12\]
To rearrange the terms into the form \(Ax + By = C\), we move the variables to one side and the constant term to the other. Subtracting \(4y\) from both sides and adding \(12\) to both sides gives:
\[-4 + 12 = x - 4y\]
Simplifying the expression results in:
\[8 = x - 4y\]
Or, equivalently, \(x - 4y = 8\).
- Verification and Conclusion
We verify the result by checking the original points \((-4, -3)\) and \((12, 1)\) in the equation \(x - 4y = 8\):
For the point \((-4, -3)\):
\[-4 - 4(-3) = -4 + 12 = 8\]
For the point \((12, 1)\):
\[12 - 4(1) = 12 - 4 = 8\]
Both points satisfy the equation \(x - 4y = 8\). Thus, the correct standard form is \(x - 4y = 8\).
Examples
Linear equations are essential in many fields, including finance and physics. For example, if you are tracking the distance an object moves over time and the movement is constant, the relationship between time and distance can be modeled by a linear equation. Converting between different forms, like point-slope and standard form, allows you to more easily analyze intercepts or identify specific properties of the motion, helping you calculate future positions or required speeds for specific tasks.