- Start with the point-slope equation \(y + 7 = -\frac{2}{5}(x - 10)\).
- Multiply both sides by 5 to clear the fraction, resulting in \(5(y + 7) = -2(x - 10)\).
- Expand and rearrange the terms into standard form \(Ax + By = C\), yielding \(2x + 5y = -15\).
- Verify the equation with the given points to confirm the final result, which is \(\boxed{2x + 5y = -15}\).
Explanation
- Problem Analysis
To determine the standard form of the linear equation, we begin with the provided point-slope equation:
\[y + 7 = -\frac{2}{5}(x - 10)\]
Our goal is to rearrange this equation into the standard form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A > 0\).
- Eliminating the Fraction
To eliminate the fraction on the right side, we multiply both sides of the equation by 5:
\[5(y + 7) = 5 \times \left(-\frac{2}{5}\right)(x - 10)\]
This simplifies the equation to:
\[5(y + 7) = -2(x - 10)\]
- Rearranging to Standard Form
Next, we distribute the values on both sides of the equation:
\[5y + 35 = -2x + 20\]
Now, we rearrange the equation to gather the \(x\) and \(y\) terms on the left side and the constant terms on the right side:
\[2x + 5y = 20 - 35\]
\[2x + 5y = -15\]
This equation is now in the required standard form \(Ax + By = C\), where \(A = 2\), \(B = 5\), and \(C = -15\). Since \(A = 2\) is a positive integer, the condition for standard form is satisfied.
- Conclusion and Verification
Finally, we verify the equation using the given points \((-5, -1)\) and \((10, -7)\):
For \((-5, -1)\): \(2(-5) + 5(-1) = -10 - 5 = -15\).
For \((10, -7)\): \(2(10) + 5(-7) = 20 - 35 = -15\).
Both points satisfy the equation \(2x + 5y = -15\), which corresponds to the third option. Thus, the standard form of the equation is \(\boxed{2x + 5y = -15}\).
Examples
Linear equations in standard form are useful in fields like computer graphics and geometry for defining lines and boundaries. For example, if you are designing a digital workspace where a boundary must be represented by a linear function, converting between different forms (slope-intercept, point-slope, and standard) helps ensure the equation fits the specific format required by the software or algorithm. Understanding this conversion process is essential for simplifying and organizing data in any coordinate-based system.