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There is a line that includes the point $(10,-9)$ and has a slope of $\frac{1}{5}$. What is its equation in point-slope form?



Use the specified point in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.

$y-\square=\square(x-\square)$

Verified Answer :

  • Identify the point \((10, -9)\) and the slope \(\frac{1}{5}\).
  • Use the point-slope form \(y - y_1 = m(x - x_1)\).
  • Substitute the values to get \(y - (-9) = \frac{1}{5}(x - 10)\).
  • Simplify to arrive at the final form: \(y - (-9) = \frac{1}{5}(x - 10)\).
  • The final equation is \(\boxed{y - (-9) = \frac{1}{5}(x - 10)}\).

Explanation

  1. Identifying Given Information and the Formula

To determine the equation of a line in point-slope form, we need to know a point \((x_1, y_1)\) on the line and the slope \(m\) of the line. The point-slope formula is given by:
\[y - y_1 = m(x - x_1)\]
For this problem, we are provided with:

  1. The point \((x_1, y_1) = (10, -9)\).
  2. The slope \(m = \frac{1}{5}\).
  1. Substituting and Simplifying the Equation

Now, we substitute the known values into the point-slope formula.
Using \(x_1 = 10\), \(y_1 = -9\), and \(m = \frac{1}{5}\):
\[y - (-9) = \frac{1}{5}(x - 10)\]
Since subtracting a negative number is equivalent to adding its positive counterpart, we simplify the left side of the equation:
\[y + 9 = \frac{1}{5}(x - 10)\]
This is the equation of the line in the requested point-slope form.

Examples

The point-slope form is a fundamental tool for modeling linear relationships in the real world. Imagine you are tracking the height of a plant that grows at a constant rate. If you know the plant's height at a specific time (your point) and its growth rate (the slope), you can easily write an equation that predicts its height at any other time. This method is used in economics, physics, and daily budgeting to understand how a dependent variable changes relative to an independent variable.

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