Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Put the equation in slope-intercept form by solving for \(y .\) b. Identify the slope and the \(y\) -intercept. c. Use the slope and y-intercept to graph the equation. $$3 x+y=4$$

Step-by-Step Solution

Answer

The line represented by the equation \(3x + y = 4\) has a slope of -3 and y-intercept of 4. The graph of the line starting from (0,4) slopes downward.

Step 1: Transforming to Slope-Intercept Form

To put the equation \(3x + y = 4\) into slope-intercept form, isolate \(y\) on one side. You can do this by subtracting \(3x\) from both sides, which will give \(y = -3x + 4\).

Step 2: Identifying the Slope and Y-intercept

In the equation \(y = -3x + 4\), the slope is the coefficient of \(x\), which is -3, and the y-intercept is the constant term, which is 4.

Step 3: Plotting the Graph Using the Slope and Y-intercept

Plot the y-intercept on the y-axis, which is at point (0,4). From there, since the slope is -3, for each step right along the x-axis, move three steps down along the y-axis. Draw a line through these points to represent the equation.

Key Concepts

Understanding Slope in Slope-Intercept Form

The slope is a crucial concept whenever you're working with linear equations. Imagine it as the 'steepness' of the line on a graph. For the equation form \( y = mx + b \), the slope is represented by \( m \). It tells you how much \( y \) changes for a change in \( x \).

• If the slope \( m \) is positive, the line ascends (goes upwards) as you move from left to right.
• If the slope \( m \) is negative, the line descends (goes downwards) as you move from left to right.

In the given equation \( y = -3x + 4 \), the slope is \( -3 \). This tells us that for each unit you move to the right on the x-axis, you move three units downwards on the y-axis. This is because the slope is negative, thus the line is descending.

Discovering the Y-Intercept

The y-intercept is the second essential element of the slope-intercept form. This is the point where the line crosses the y-axis. For the equation \( y = mx + b \), the y-intercept is signified by \( b \). It gives you a starting point to begin graphing your line.

• The y-intercept is always on the y-axis, which means the x-value at this point is zero.
• In \( y = -3x + 4 \), the y-intercept is \( 4 \), meaning the line crosses the y-axis at the point (0,4).

By knowing the y-intercept, you can easily begin plotting the line on a graph and ascertain its initial point before considering the slope.

Graphing Linear Equations Using Slope and Y-Intercept

Once you have identified the slope and the y-intercept, you can graph the linear equation. Here’s how you do it step-by-step:
1. Start by plotting the y-intercept on the graph. For \( y = -3x + 4 \), you will place a point at (0, 4) on the y-axis.
2. Use the slope to determine another point. With a slope of \( -3 \), you can count three units down on the y-axis for each unit you move right on the x-axis. This will give the next point in line.
3. Draw a straight line through these points, extending in both directions, which represents the equation on the graph.
These steps will help you visualize and better understand how changes in slope affect the direction and steepness of the line, and how the y-intercept places the line on the graph.