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Chapter 0: Problem 38

Find the slope and \(y\) -intercept of each line. $$ 4 x+5 y=-20 $$

Short Answer

Expert verified
Slope: \(-\frac{4}{5}\), y-intercept: \(-4\).

Step by step solution

01

Understanding the Equation

We start with the equation of the line given as \(4x + 5y = -20\). This is in standard form, \(Ax + By = C\), where \(A = 4\), \(B = 5\), and \(C = -20\).
02

Transform to Slope-Intercept Form

To find the slope and y-intercept, we need the equation in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We solve for \(y\) by first isolating the \(y\) term: \(5y = -4x - 20\).
03

Solve for y

Divide each term by 5: \(y = -\frac{4}{5}x - 4\). Now the equation is in the form \(y = mx + b\).
04

Identify the Slope

In the equation \(y = -\frac{4}{5}x - 4\), the coefficient of \(x\) is the slope, \(m\). Thus, \(m = -\frac{4}{5}\).
05

Identify the y-intercept

The \(b\) term in the equation \(y = -\frac{4}{5}x - 4\) represents the y-intercept. Here, \(b = -4\).

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Key Concepts

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

Equation of a Line
An equation of a line is an algebraic expression representing all the points that lie on the line in a two-dimensional plane. In the standard form of the line, the equation is usually written as \[ Ax + By = C \]where \(A\), \(B\), and \(C\) are constants. Each term in this equation contributes to specifying a particular line.
  • The \(A\) and \(B\) coefficients determine the slope's direction and steepness.
  • An equation can be manipulated into different forms to make various characteristics of the line more apparent.
  • Working with the standard form can be useful but sometimes switching to another form, like the slope-intercept form, helps in directly identifying the slope and y-intercept.
Understanding this concept allows us to flexibly switch between forms depending on what we need to know about the line.
Standard Form to Slope-Intercept Form
The transformation from the standard form to slope-intercept form is crucial when we need to identify the slope and y-intercept easily. The slope-intercept form is expressed as \[ y = mx + b \]where \(m\) represents the slope and \(b\) the y-intercept of the line. To convert from \(Ax + By = C\) into this form, follow these steps:
  • Isolate the \(y\)-term on one side of the equation. Start by moving the \(x\)-term to the opposite side by subtracting or adding as needed.
  • Divide every term by the coefficient of \(y\) to solve for \(y\). This will get \(y\) by itself.
  • Once in the form \(y = mx + b\), identifying the slope and y-intercept is straightforward.
This conversion helps provide a deeper understanding of the line's behavior and allows you to see how it will graph in the coordinate plane.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis in a coordinate plane. This occurs where the value of \(x\) is zero. In the equation\[ y = mx + b \]the y-intercept is represented by the \(b\) term.
  • It provides a starting value for the graph of the line.
  • The y-intercept indicates the specific point the line will hit the y-axis, making it a vital part of graph plotting.
  • Finding the y-intercept helps in constructing a graph more accurately, as it provides a fixed point through which the line will definitely pass.
Understanding y-intercepts allows you to not just plot precise graphs, but to analyze how changes in the equation affect the graph's position and rotation on a Cartesian plane.

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Most popular questions from this chapter

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}+y^{2}=4\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(2 x^{2}-4 x+3 y^{2}+12 y=-2\)

After being in business for \(t\) years, a manufacturer of cars is making \(120+2 t+3 t^{2}\) units per year. The sales price in dollars per unit has risen according to the formula \(6000+700 t\). Write a formula for the manufacturer's yearly revenue \(R(t)\) after \(t\) years.

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(y=x^{3}-x\)

Calculate each of the following without using a calculator. (a) \(\sin 570^{\circ}\) (b) \(\cos \frac{9 \pi}{2}\) (c) \(\cos \left(\frac{-13 \pi}{6}\right)\)
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