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Chapter 2: Problem 46

Find the slope and \(y\)-intercept of the line and draw its graph. \(-3 x-5 y+30=0\)

Short Answer

Expert verified
The slope is \(-\frac{3}{5}\) and the \(y\)-intercept is 6.

Step by step solution

01

Rearrange the Equation

First, rearrange the given equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Start by moving the terms involving \( x \) and the constant to the other side of the equation: \(-5y = 3x - 30 \).
02

Solve for y

Divide all terms by \(-5\) to solve for \( y \). This gives you the equation in slope-intercept form: \( y = -\frac{3}{5}x + 6 \).
03

Identify the Slope and y-Intercept

Now that you have the equation in the form \( y = mx + b \), identify the slope \( m \) and the \( y \)-intercept \( b \). Here, the slope \( m = -\frac{3}{5} \) and the \( y \)-intercept \( b = 6 \).
04

Graph the Equation

To draw the graph, start by plotting the \( y \)-intercept \( (0, 6) \) on the y-axis. From this point, use the slope \( -\frac{3}{5} \): move down 3 units and to the right 5 units to find another point \( (5, 3) \). Draw a straight line through these points.

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Key Concepts

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

Slope
The slope of a line is a crucial concept in algebra. It describes how steep a line is and its direction on a graph. The slope is often represented by the letter \( m \) in the slope-intercept form of a linear equation, which is \( y = mx + b \). For our specific equation, \( -3x - 5y + 30 = 0 \), we've rearranged this to \( y = -\frac{3}{5}x + 6 \). Here, the slope \( m \) is \(-\frac{3}{5}\).
  • A negative slope, like \(-\frac{3}{5}\), means the line goes downwards from left to right.
  • The absolute value of the slope tells us the steepness: the larger the absolute value, the steeper the line.
The slope \(-\frac{3}{5}\) indicates the line moves down 3 units vertically for every 5 units it moves to the right horizontally. This ratio is always constant for a straight line, showcasing a uniform rate of change.
Graph of a Linear Equation
The graph of a linear equation forms a straight line. It's one of the simplest forms of graphs because of its predictable linearity once you know the slope and the y-intercept.
  • The slope \( m \) determines the line's angle or direction.
  • The y-intercept \( b \) tells where the line crosses the y-axis.
With the slope-intercept equation \( y = -\frac{3}{5}x + 6 \), you can easily draft the graph. You start by plotting the \( y \)-intercept at \((0, 6)\). From there, apply the slope: go down 3 units, and move 5 units right to mark another point at \((5, 3)\). Once you have these two points, draw a line through them, and you have the graph of your linear equation. This straightforward process makes graphing linear equations efficient and clear.
Y-Intercept
The y-intercept is a pivotal part of the slope-intercept form \( y = mx + b \). It is represented by \( b \) and indicates the point where the graph intersects the y-axis. For the equation \( y = -\frac{3}{5}x + 6 \), the y-intercept \( b \) is 6.
  • This means the line crosses the y-axis at the point \((0, 6)\).
  • It's essential because it gives the starting point on the graph before the slope is applied.
When drafting the graph, the y-intercept is the first coordinate you plot on the y-axis. Understanding the y-intercept is critical as it allows you to anchor the line accurately on the graph. The rest of the line is defined by the slope emanating from this y-intercept point.

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Most popular questions from this chapter

13鈥�22 ? Express the statement as an equation. Use the given information to find the constant of proportionality. \(M\) is jointly proportional to \(a, b,\) and \(c,\) and inversely proportional to \(d .\) If \(a\) and \(d\) have the same value, and if \(b\) and \(c\) are both \(2,\) then \(M=128 .\)

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