Answer :
Certainly! Let's write the equation of the line in point-slope form step-by-step.
### Step 1: Understand the Problem
We need to find the equation of a line that falls 7 units for every 11 unit increases in \(x\) and passes through the point (-2, 5).
### Step 2: Determine the Slope
Since the line falls 7 units for every 11 units it increases in \(x\), the rise (\( \Delta y \)) is -7 (because it falls), and the run (\( \Delta x \)) is 11. The slope \(m\) of the line is given by:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{-7}{11} = -0.6363636363636364 \][/tex]
### Step 3: Use the Point-Slope Form
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where \((x_1, y_1)\) is a point on the line and \( m \) is the slope.
Given the point \((-2, 5)\) and the slope \( -0.6363636363636364 \), we can substitute these values into the point-slope form:
[tex]\[ y - 5 = -0.6363636363636364(x - (-2)) \][/tex]
### Step 4: Simplify the Equation
To make the equation clearer:
[tex]\[ y - 5 = -0.6363636363636364(x + 2) \][/tex]
### Summary
So, the equation of the line in point-slope form is:
[tex]\[ y - 5 = -0.6363636363636364(x + 2) \][/tex]
This is the detailed step-by-step solution to writing the equation of the line in point-slope form based on the given conditions.
### Step 1: Understand the Problem
We need to find the equation of a line that falls 7 units for every 11 unit increases in \(x\) and passes through the point (-2, 5).
### Step 2: Determine the Slope
Since the line falls 7 units for every 11 units it increases in \(x\), the rise (\( \Delta y \)) is -7 (because it falls), and the run (\( \Delta x \)) is 11. The slope \(m\) of the line is given by:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{-7}{11} = -0.6363636363636364 \][/tex]
### Step 3: Use the Point-Slope Form
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where \((x_1, y_1)\) is a point on the line and \( m \) is the slope.
Given the point \((-2, 5)\) and the slope \( -0.6363636363636364 \), we can substitute these values into the point-slope form:
[tex]\[ y - 5 = -0.6363636363636364(x - (-2)) \][/tex]
### Step 4: Simplify the Equation
To make the equation clearer:
[tex]\[ y - 5 = -0.6363636363636364(x + 2) \][/tex]
### Summary
So, the equation of the line in point-slope form is:
[tex]\[ y - 5 = -0.6363636363636364(x + 2) \][/tex]
This is the detailed step-by-step solution to writing the equation of the line in point-slope form based on the given conditions.
