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Chapter 3: Problem 38

Find the slope and the -intercept of the line with the given equation. See Example 1 $$ 4 x+5 y=25 $$

Short Answer

Expert verified
The slope is \(-\frac{4}{5}\) and the y-intercept is 5.

Step by step solution

01

Identify the Standard Form

The given equation is in standard form: \(4x + 5y = 25\). The standard form of a linear equation is \(Ax + By = C\). Here, \(A = 4\), \(B = 5\), and \(C = 25\).
02

Rewriting in Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To convert the given equation to this form, solve for \(y\).
03

Solve for y

Isolate \(y\) on one side:\[5y = -4x + 25\]Divide each term by 5:\[y = -\frac{4}{5}x + 5\]
04

Identify the Slope and y-Intercept

From the equation \(y = -\frac{4}{5}x + 5\), identify the slope \(m\) and y-intercept \(b\). Here, \(m = -\frac{4}{5}\) and \(b = 5\).

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Key Concepts

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

Slope-Intercept Form of a Linear Equation
The slope-intercept form is a popular way of expressing a linear equation. It follows the structure \(y = mx + b\), where:
  • \(m\) represents the slope of the line
  • \(b\) is the y-intercept
This form is beneficial because it immediately shows the slope and the y-intercept, making it easier to graph the equation or understand its behavior. To convert an equation from another form鈥攍ike standard form鈥攖o slope-intercept, you'll need to solve for \(y\). Once \(y\) is by itself on one side of the equation, you've successfully rewritten it in slope-intercept form. The conversion showcases the slope, indicating how steep the line is, and the y-intercept, highlighting where the line crosses the y-axis.
Standard Form of Linear Equation
The standard form of a linear equation is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. This form offers a neat, tidy representation that emphasizes the balance between \(x\) and \(y\). Here are some points to remember:
  • \(A\), \(B\), and \(C\) are often whole numbers
  • \(A\) and \(B\) should not both be zero
  • Generally, \(A\) is non-negative
The standard form is helpful for analyzing the equation in a straightforward manner or when dealing with systems of equations. It's also a favored way when computers parse equations due to its balanced nature.
Y-Intercept
The y-intercept is a fundamental element of a linear equation and refers to the point where the line crosses the y-axis. The notation \(b\) in the slope-intercept form \(y = mx + b\) signifies the y-intercept. This is simply the value of \(y\) when \(x = 0\). Understanding the y-intercept helps visualize where the line penetrates the y-axis on a graph. Knowing this point is crucial when plotting or interpreting data:
  • It gives an immediate start point on a graph for drawing the line
  • It's useful for interpreting real world scenarios where initial values play a role
When a problem asks to "find the y-intercept," it seeks the \(y\)-value at the point where the line meets the y-axis, providing a pivotal piece of identifying the line's behavior.

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Most popular questions from this chapter

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \(\left(-2, \frac{1}{2}\right)\) and \(\left(-1, \frac{3}{2}\right)\)

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((8,-3)\) and \((8,-8)\) \((11,3)\) and \((22,3)\)

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \((5,-4)\) and \((-1,-7)\)

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((5,3)\) and \((1,4)\) \((-3,-4)\) and \((1,-5)\)

Halloween Candy. A candy maker wants to make a 60 -pound mixture of two candies to sell for $$ 2\( per pound. If black licorice bits sell for \) 1.90 per pound and orange gumdrops sell for $$ 2.20$ per pound, how many pounds of each should be used?
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