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Chapter 11: Problem 52

Find an equation of the line satisfying the given conditions. Containing the points \((2,3)\) and \((4,-5)\)

Short Answer

Expert verified
y = -4x + 11

Step by step solution

01

Identify the coordinates

The problem provides two points: \(2, 3\) and \(4, -5\). Label these points as \(x_1, y_1\) = \(2, 3\) and \(x_2, y_2\) = \(4, -5\).
02

Calculate the slope

Use the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] to find the slope of the line. Substitute the given points into the equation: \[ m = \frac{-5 - 3}{4 - 2} = \frac{-8}{2} = -4 \]
03

Use the point-slope form

The point-slope form of the equation of a line is \[ y - y_1 = m (x - x_1) \]. Using the point \(2, 3\) and the calculated slope \(m = -4\), substitute into the equation: \[ y - 3 = -4 (x - 2) \]
04

Simplify to slope-intercept form

Distribute and simplify the equation to get it into slope-intercept form \(y = mx + b\): \[ y - 3 = -4x + 8 \] \[ y = -4x + 8 + 3 \] \[ y = -4x + 11 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
To start understanding the equation of a line, it's essential to grasp the concept of slope. Slope indicates how steep a line is and is calculated using the coordinates of two points on the line.
The formula for slope calculation is written as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \(m\) represents the slope, and \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points.
Let's go through the steps given in the problem:
  • Identify the coordinates. We have \( (2,3) \) and \( (4,-5) \).
  • Substitute these points into the formula.
    \[ m = \frac{-5 - 3}{4 - 2} = \frac{-8}{2} = -4 \]
    As shown, the slope of the line is \( -4\).
    This negative slope signifies that the line is descending.
Understanding the slope is vital because it directly influences the line's steepness and direction on a graph. The next step uses this slope to find the line's equation.
Point-Slope Form
The point-slope form of a line's equation is handy. This form uses a known point and the slope to describe the whole line. The formula for point-slope form is:
\[ y - y_1 = m (x - x_1) \]
Here, \( y_1 \) and \( x_1 \) are the coordinates of the given point, and \( m \) is the slope.
Using the given point \( (2,3) \) and our calculated slope \( -4 \), we substitute these into the formula:
  • \[ y - 3 = -4 (x - 2) \]
    By doing so, we have mapped out the line using this specific point and the slope.
This form is particularly useful when you need to quickly write the equation of a line but don鈥檛 necessarily want to rearrange it into a slope-intercept form. However, to better visualize the line, converting it into slope-intercept form can be more illustrative.
Slope-Intercept Form
The slope-intercept form is a standard way to represent a line in mathematics. It is written as:
\[ y = mx + b \]
Where \( m \) is the slope and \( b \) is the y-intercept (the point where the line crosses the y-axis).
To convert our point-slope form equation to slope-intercept form, simplify it as follows:
  • Starting with:
    \[ y - 3 = -4 (x - 2) \]
  • Distribute the slope:
    \[ y - 3 = -4x + 8 \]
  • Isolate \( y \):
    \[ y = -4x + 8 + 3 \]
    \[ y = -4x + 11 \]
Now, this equation \( y = -4x + 11 \) clearly shows the slope \( m = -4 \) and the y-intercept \( b = 11 \). This form makes it easier to graph a line because you can quickly identify the slope and y-intercept. Thus, understanding slope-intercept form simplifies visualizing and working with linear equations.

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