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Chapter 2: Problem 34

Given each set of information, find a linear equation satisfying the conditions, if possible \(f(-1)=4,\) and \(f(5)=1\)

Short Answer

Expert verified
The linear equation is \(y = -\frac{1}{2}x + 3.5\).

Step by step solution

01

Understand the problem

We need to find a linear equation that passes through the points given by the conditions \(f(-1) = 4\) and \(f(5) = 1\). This means the line should intersect the point \((-1,4)\) and \((5,1)\).
02

Calculate the slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For the points \((-1, 4)\) and \((5, 1)\), the slope is \(m = \frac{1 - 4}{5 - (-1)} = \frac{-3}{6} = -\frac{1}{2}\).
03

Use the point-slope formula

With the slope \(m = -\frac{1}{2}\) and a point \((-1, 4)\), we use the point-slope formula: \(y - y_1 = m(x - x_1)\). Substituting our values, we get \(y - 4 = -\frac{1}{2}(x + 1)\).
04

Simplify the equation

Simplify the equation \(y - 4 = -\frac{1}{2}(x + 1)\) to get it into the slope-intercept form \(y = mx + c\). Distribute the slope: \(y - 4 = -\frac{1}{2}x - \frac{1}{2}\). Add 4 to both sides: \(y = -\frac{1}{2}x + 3.5\).
05

Verify the equation with given points

Check that the equation works for both points. Substitute \(x = -1\) into \(y = -\frac{1}{2}x + 3.5\) and see if \(y = 4\). Substitute \(x = 5\) and check if \(y = 1\). Both calculations confirm the equation is correct.

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Key Concepts

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

Finding the Slope
The slope of a line is a measure of its steepness and direction. Finding the slope is the first step when we have two points on a line. In our case, the points are
  • \((-1, 4)\)
  • \((5, 1)\)
To calculate the slope \( m \), we use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,
  • \( (x_1, y_1) = (-1, 4) \)
  • \( (x_2, y_2) = (5, 1) \)
Substituting these values in, \[ m = \frac{1 - 4}{5 - (-1)} = \frac{-3}{6} = -\frac{1}{2} \]So, the slope \( m \) is \(-\frac{1}{2}\), indicating that for every step in the positive \( x \) direction, the line moves half a step down in the \( y \) direction.
Point-Slope Form
The point-slope form is a way of writing the equation of a line when you know:
  • The slope \( m \)
  • One point on the line \( (x_1, y_1) \)
The formula for point-slope form is \[ y - y_1 = m(x - x_1) \]Let's use the slope \( m = -\frac{1}{2} \) and the point \( (-1, 4) \). Substituting these values into the formula, we get \[ y - 4 = -\frac{1}{2}(x + 1) \]This step is crucial as it represents the equation in terms of the slope and a known point, establishing a direct relationship between \( x \) and \( y \) on the line.
Slope-Intercept Form
The slope-intercept form is one of the most common and easily usable forms of a linear equation. It is represented by \[ y = mx + c \]where:
  • \( m \) is the slope
  • \( c \) is the y-intercept, or the point where the line crosses the y-axis
To convert from point-slope to slope-intercept form, we simplify \[ y - 4 = -\frac{1}{2}(x + 1) \]First, distribute the slope:\[ y - 4 = -\frac{1}{2}x - \frac{1}{2} \]Next, add 4 to both sides to solve for \( y \): \[ y = -\frac{1}{2}x - \frac{1}{2} + 4 \]Simplifying further, we have: \[ y = -\frac{1}{2}x + 3.5 \]Thus, the line's equation is in slope-intercept form, making it easier to quickly graph and understand the line's characteristics.

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