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Chapter 2: Problem 34

Exer. 33-36: Find the slope-intercept form of the line that satisfles the given conditions. \(x\) -intercept \(-5, \quad y\) -intercept \(-1\)

Short Answer

Expert verified
The slope-intercept form is \( y = -\frac{1}{5}x - 1 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line is provided by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
02

Identify the y-intercept

From the problem, the y-intercept \( b \) is given as \(-1\). This means the line crosses the y-axis at \( (0, -1) \).
03

Understand the x-intercept

The x-intercept is given as \(-5\), meaning the line crosses the x-axis at \(-5, 0\). This point can help to calculate the slope \( m \).
04

Calculate the Slope

The slope \( m \) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, the two points are \((x_1, y_1) = (-5, 0)\) and \((x_2, y_2) = (0, -1)\). Substitute these into the slope formula: \[ m = \frac{-1 - 0}{0 + 5} = \frac{-1}{5} \].
05

Write the Slope-Intercept Form Equation

Substitute the calculated slope \( m = -\frac{1}{5} \) and y-intercept \( b = -1 \) into the slope-intercept form \( y = mx + b \). The equation becomes: \[ y = -\frac{1}{5}x - 1 \].

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Key Concepts

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

x-intercept
The x-intercept of a line represents the point where it crosses the x-axis. This occurs when the y-value is zero. In simpler terms, it's where the line "touches" the x-axis. For the given problem, the x-intercept is -5. This informs us that when the line crosses the x-axis, its coordinates are (-5, 0).
Knowing the x-intercept is crucial because it helps in understanding the orientation and positioning of the line on the graph. It is also a vital component in calculating the slope, which we will discuss in a later section.
y-intercept
The y-intercept is the point where the line crosses the y-axis. Here, the x-value is zero. In this problem, the y-intercept is given as -1. Therefore, the point is (0, -1).
The y-intercept is essential for finding the equation of the line in slope-intercept form. When we say slope-intercept form, we mean the equation \( y = mx + b \), where \( b \) is the y-intercept. Hence, this point directly gives us half of what we need to write the equation of the line.
slope calculation
Calculating the slope is the next piece of the puzzle in finding the line's equation. The slope of a line measures its steepness and is calculated by dividing the change "rise" in y-coordinates by the change "run" in x-coordinates.
In formula terms, the slope \( m \) is determined by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  • For the x-intercept (-5, 0) and y-intercept (0, -1), \( m = \frac{-1 - 0}{0 - (-5)} \)
  • Simplify to get \( \frac{-1}{5} \)
      • Thus, the slope \(-\frac{1}{5}\) implies the line descends gently as we move from left to right on the graph.
linear equations
A linear equation describes a straight line in mathematical terms. The slope-intercept form, \( y = mx + b \), is a common way to express such an equation, where \( m \) represents the slope and \( b \) is the y-intercept.
For this problem, we calculated the slope to be \(-\frac{1}{5}\) and the y-intercept is \(-1\). Hence, when we substitute these values into the equation, it becomes:
  • \( y = -\frac{1}{5}x - 1 \)
      • This is the slope-intercept form of the equation for the line fulfilling the given conditions. Such equations are incredibly useful for quickly sketching the line's graph and understanding the relationship between x and y-values.

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