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Chapter 3: Problem 5

Determine the slope and \(y\) -intercept. $$ 4 x-5 y=15 $$

Short Answer

Expert verified
The slope is \( \frac{4}{5} \) and the \( y \)-intercept is \(-3\).

Step by step solution

01

Rearrange the Equation

First, let's rearrange the equation into the slope-intercept form of a line, which is \( y = mx + b \). Start by moving terms around to solve for \( y \). Subtract \( 4x \) from both sides of the equation: \[ -5y = -4x + 15 \]
02

Solve for y

Now, solve for \( y \) by dividing every term by \(-5\):\[ y = \frac{-4}{-5}x + \frac{15}{-5} \] This simplifies to:\[ y = \frac{4}{5}x - 3 \]
03

Identify the Slope

From the equation in slope-intercept form \( y = \frac{4}{5}x - 3 \), identify the slope \( m \). It is the coefficient of \( x \):\( m = \frac{4}{5} \).
04

Determine the y-intercept

Identify the \( y \)-intercept \( b \) from the equation \( y = \frac{4}{5}x - 3 \). It is the constant term:\( b = -3 \).

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Key Concepts

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

Understanding Linear Equations
Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. These equations graph as straight lines on the Cartesian plane, hence the term "linear." The most commonly used form for linear equations is the slope-intercept form:
  • Standard Form: Ax + By = C
  • Slope-Intercept Form: y = mx + b
In the equation 4x - 5y = 15, it is initially in standard form. To convert it to slope-intercept form, you solve for y, resulting in the form y = mx + b. This allows us to easily identify the slope and the y-intercept from the equation.
What is Slope?
Slope is a measure of how steep a line is. In mathematical terms, it represents the change in the y-values relative to the change in the x-values between two distinct points on a line. The slope is often denoted by the letter 'm'. In the slope-intercept form equation y = mx + b:
  • m = slope
  • It indicates the rise (change in y) over the run (change in x)
For instance, in the equation y = \( \frac{4}{5} \)x - 3, the slope is \( \frac{4}{5} \). This means that for every 5 units the line moves horizontally (x-direction), it moves 4 units vertically (y-direction). A line with a positive slope slants upwards from left to right, while a negative slope slants downwards.
Understanding the Y-Intercept
The y-intercept is a critical component in graphing linear equations. It gives you the point where the line crosses the y-axis. This is represented by the 'b' in the slope-intercept form y = mx + b.
  • When x = 0, the value of y is the y-intercept
  • It is expressed as a single point (0, b)
In our rearranged equation y = \( \frac{4}{5} \)x - 3, the y-intercept is -3. This tells us that when the line crosses the y-axis, it does so at the point (0, -3). This intercept is instrumental in graphing because it gives you a starting point to draw the line on the graph, along with the slope to guide its direction. Understanding these components allows for accurate graphing and analysis of linear relationships.

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Most popular questions from this chapter

Find the equation of the line given two points on the line. (10,1) and (10,-3)

The perimeter of an isosceles triangle with a base measuring 10 centimeters is given by the function \(P(x)=2 x+10\), where \(x\) represents the length of each of the equal sides. Find the length of each side if the perimeter is 40 centimeters.

Find the equation of the line in slope-intercept form. Parallel to \(5 x-y=15\) and passing through \((-10,-1) .\)

Graph using the slope and \(y\) -intercept. $$ y=-52 x+3 $$

Determine if the lines are parallel, perpendicular, or neither. $$ \\{2 x+3 y=182 x-3 y=36 $$
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