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Chapter 8: Problem 46

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of \(-1\) and \(y\) intercept of \(-3\)

Short Answer

Expert verified
The equation in standard form is \(3x + y = -3\).

Step by step solution

01

Understand Intercepts

The x-intercept of -1 means the line crosses the x-axis at the point (-1,0), and the y-intercept of -3 means it crosses the y-axis at the point (0,-3).
02

Use Two-Point Formula

With points (-1, 0) and (0, -3), calculate the slope (m) using the formula: \( m = \frac{y_2-y_1}{x_2-x_1} \). This gives \( m = \frac{-3-0}{0-(-1)} = -3 \).
03

Write Slope-Intercept Form

Using the slope -3 and y-intercept (0, -3), the slope-intercept form is \( y = -3x - 3 \).
04

Convert to Standard Form

Rearrange the equation \( y = -3x - 3 \) to standard form, \( Ax + By = C \) by moving all terms to one side: \(3x + y = -3\).

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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 is the point where the line crosses the x-axis. This means that at the x-intercept, the y-coordinate is always 0. In our example, the x-intercept is given as (-1, 0). This point indicates that when you move along the line, it will touch the x-axis at x = -1.
Understanding the x-intercept is crucial because it helps describe the position of the line on a graph. Moreover:
  • The x-intercept can be found by setting y to 0 in the line equation and solving for x.
  • It provides useful information for sketching the line quickly.
Knowing the x-intercept, along with the slope, allows for writing the line's equation, which can then be transformed into any desired form, such as the slope-intercept form or standard form.
y-intercept
The y-intercept is the point where the line crosses the y-axis, which means at this point, the x-coordinate is always zero. For our problem, the y-intercept is at (0, -3), indicating that the line touches the y-axis at y = -3.
This concept is essential because it gives us an anchor point on the graph where the line will pass through. Some important aspects include:
  • At the y-intercept, you set x to 0 in the equation and solve for y.
  • This helps in writing the slope-intercept equation directly as it's the constant term.
With the y-intercept known, along with another point or the slope, the equation of the line can be developed and manipulated to any desired form.
standard form
The standard form of a linear equation is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative. It provides a structured way to present line equations and is particularly useful in specific calculations involving solving systems of equations or identifying parallel or perpendicular lines.
To convert a line equation into standard form, like in our example, follow these steps:
  • Ensure that all terms involving x and y are on the same side of the equation.
  • Rearrange the equation by moving the x-term to the left and adjusting other terms accordingly, ensuring integer coefficients.
  • In our case, \(y = -3x - 3\) becomes \(3x + y = -3\).
Converting to standard form allows easier interpretations and solutions that involve multiple equations.
slope-intercept form
The slope-intercept form of a linear equation is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept. This form is very intuitive and easy to use for graphing, as it directly shows the line's tilt (slope) and where it crosses the y-axis (y-intercept).
For our exercise, the slope was determined as \(m = -3\) from the changes in y and x, and with the y-intercept being \(-3\), this directly gives us the equation \(y = -3x - 3\). Here are some pertinent features:
  • The "m" value shows how steep the line is and its direction.
  • The "b" value provides the exact point where the line crosses the y-axis.
Using the slope-intercept form simplifies the process of sketching and understanding the behavior and trajectory of the line immediately from its equation.

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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) $$(-6,-2), m=3$$

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=\frac{3}{5} \text { and } b=2 $$

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 \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-\frac{5}{9} \text { and } b=-\frac{1}{2} $$

Explain the importance of the slope-intercept form \((y=m x+b)\) of the equation of a line.
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