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Chapter 8: Problem 47

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of \(-3\) and slope of \(-\frac{5}{8}\)

Short Answer

Expert verified
The equation is \(5x + 8y = -15\) in standard form.

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Our slope, \( m \), is given as \(-\frac{5}{8}\).
02

Use x-intercept to find y-intercept

The x-intercept is \(-3\), meaning the point \((-3, 0)\) lies on the line. Substitute into the slope-intercept form to find the y-intercept (\(c\)). Set \( y = 0 \), \( x = -3 \), and \( m = -\frac{5}{8} \):\[ 0 = -\frac{5}{8}(-3) + c \]which simplifies to:\[ 0 = \frac{15}{8} + c \]\[ c = -\frac{15}{8} \].
03

Write the Slope-Intercept Equation

Substitute \( m = -\frac{5}{8} \) and \( c = -\frac{15}{8} \) into the slope-intercept form:\[ y = -\frac{5}{8}x - \frac{15}{8} \].
04

Convert to Standard Form

The standard form of a linear equation is \( Ax + By = C \). To convert our equation, eliminate the fractions:Multiply through by 8 to clear the denominators:\[ 8y = -5x - 15 \].Rearrange to get integer coefficients:\[ 5x + 8y = -15 \].

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Key Concepts

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

Slope-Intercept Form
When we talk about the slope-intercept form, we refer to one of the most convenient ways to describe a linear equation. This form is expressed as \( y = mx + c \), where:
  • \( m \) represents the slope of the line
  • \( c \) is the y-intercept, the point where the line crosses the y-axis
The slope \( m \) indicates the angle or steepness of the line. A positive slope means the line rises as it moves from left to right, whereas a negative slope means it falls. In our problem, the slope is \(-\frac{5}{8}\), indicating a decreasing line. The intercept \( c \) tells us where the line touches the y-axis, which helps to anchor the line鈥檚 position on the graph.
X-Intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this specific point, the value of \( y \) is zero. Knowing the x-intercept is crucial for plotting and understanding the line鈥檚 behavior.
In the given exercise, the x-intercept is \(-3\), meaning the line passes through the point \((-3, 0)\). Using this point, we can determine other characteristics of the line, such as the y-intercept, by substituting the x-value into the slope-intercept equation and solving for the y-intercept \( c \). This x-intercept also helps us visualize the line on a graph, as it provides a concrete point that the line passes through.
Standard Form
The standard form of a linear equation is another way to express a line, written as \( Ax + By = C \). In this form:
  • \( A \), \( B \), and \( C \) are integers
  • \( A \) is usually positive
This form is useful for quickly identifying certain line properties and is often used in algebra and geometry.
To convert an equation from the slope-intercept form \( y = mx + c \) to standard form, follow these steps:
  • Move all terms involving variables to one side of the equation
  • Eliminate fractional coefficients by multiplying through by their denominators
  • Rearrange the equation to ensure \( A \), \( B \), and \( C \) are integers
This process provides a balanced and structured format for analyzing equations, as shown in our exercise where the final standard form is \( 5x + 8y = -15 \).
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. At this particular point, the value of the variable \( x \) is zero. Understanding the y-intercept is essential for locating the line on a graph quickly.
In our problem, after using the slope and the x-intercept to calculate, we found the y-intercept \( c \) to be \(-\frac{15}{8}\). This means when \( x = 0 \), \( y = -\frac{15}{8} \). Knowing the y-intercept allows us to initiate sketching the line on a graph since it gives another direct point of reference besides the slope.
  • Helps determine the vertical position of the line
  • Used with the slope for plotting the complete linear path
Thus, the y-intercept together with the slope gives a full picture of how the line behaves across a graph.

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Most popular questions from this chapter

Find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective 1a) $$(5,-6), m=\frac{3}{5}$$

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=\frac{5}{9} \text { and } b=4 $$

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((-3,-7)\) and is parallel to the \(x\) axis

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ -5 x-13 y=26 $$

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ -4 x+9 y=18 $$
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