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Chapter 7: Problem 102

Find the slope and \(y\) -intercept of the graph of \(3 x+4 y=-36\)

Short Answer

Expert verified
Slope: \(-\frac{3}{4}\), y-intercept: \(-9\).

Step by step solution

01

Rearrange the Equation

The first step is to rearrange the equation into the slope-intercept form, which is \(y = mx + b\). Start with the given equation: \[3x + 4y = -36\]We want to solve for \(y\). Begin by subtracting \(3x\) from both sides:\[4y = -3x - 36\]
02

Solve for y

Now that the equation is \(4y = -3x - 36\), divide each term by 4 to isolate \(y\):\[y = -\frac{3}{4}x - 9\]This is now in slope-intercept form \(y = mx + b\).
03

Identify the Slope

The slope-intercept form of a line is \(y = mx + b\) where \(m\) is the slope. From the equation \(y = -\frac{3}{4}x - 9\), we identify the slope \(m\) as \(-\frac{3}{4}\).
04

Identify the y-intercept

In the equation \(y = -\frac{3}{4}x - 9\), the \(b\) term represents the y-intercept. Therefore, the y-intercept is \(-9\).

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Key Concepts

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

Understanding the Slope
The slope is a fundamental concept in understanding linear equations. It essentially measures the steepness or incline of a line on a graph. This concept is crucial because it tells us how much the line rises or falls as we move along it.

In the slope-intercept form, indicated by the equation \(y = mx + b\), the slope is represented by the variable \(m\).
  • A positive slope indicates that the line rises as it moves from left to right.
  • A negative slope means that the line falls as it moves from left to right.
  • If the slope is zero, the line is perfectly horizontal, meaning there is no rise or fall.
In our exercise, the slope is \(-\frac{3}{4}\), signaling that for each unit we move to the right on the x-axis, the line moves down 0.75 units on the y-axis. This negative fraction clearly depicts a falling line.
Exploring the Y-Intercept
The y-intercept is another essential aspect of linear equations. It is the point where the line crosses the y-axis. This occurs when the x-value is zero. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by the constant \(b\).

Consider its importance:
  • The y-intercept gives us a starting point for plotting the graph of the line.
  • It helps in understanding the relationship between variables when x is zero.
  • Changes in the y-intercept shift the line up or down on the graph without affecting its slope.
In the given equation \(y = -\frac{3}{4}x - 9\), the y-intercept is identified as \(-9\). This tells us the point at which the line will cross the y-axis.
Defining the Linear Equation
A linear equation is a type of equation that forms a straight line on a graph. It is characterized by having no exponents higher than one and a consistent rate of change. The slope-intercept form, \(y = mx + b\), is the most common representation of a linear equation.

Key features of linear equations:
  • They provide a consistent rate of increase or decrease, dictated by the slope \(m\).
  • The equation translates algebraic expressions to graphical lines efficiently.
  • Adjusting the slope or y-intercept changes the line's steepness or vertical position, respectively.
In our example, the equation \(3x + 4y = -36\) was converted to \(y = -\frac{3}{4}x - 9\). This showcases how a standard linear equation can be reshaped into slope-intercept form to better understand and visualize the relationship between the variables.

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