Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{3}{2} x+1$$

Step-by-Step Solution

Answer

The graph of the equation \( y=-\frac{3}{2}x + 1 \) is a straight line with a negative slope that crosses the y-axis at the point (0,1). The five solutions derived from the table of values are (-2,4), (-1,2.5), (0,1), (1,-0.5), (2,-2).

Step 1: Understand the Slope-Intercept Form

The given equation is \( y= -\frac{3}{2}x+1 \) which is given in the slope-intercept form. The slope is -3/2 and the y-intercept where the graph crosses the y-axis is 1.

Step 2: Create a Table of Values

Let's choose some x-values and then use the equation to solve for y-values to create a table which will give us the points to plot on the graph. The numbers -2, -1, 0 ,1, and 2 are typically good starting points.

Step 3: Compute y-values and Plot the Points

Plugging these x-values into the equation: for x = -2, y = -3/2(-2)+1 = 4; for x = -1, y = -3/2(-1)+1 = 2.5; for x = 0, y = -3/2(0)+1 = 1; for x = 1, y = -3/2(1) + 1 = -0.5; and for x = 2, y = -3/2(2) + 1 = -2. Now, we can plot these points on the graph and draw the line.

Step 4: Drawing the Line

After plotting the points, draw a line through them to represent the equation \( y=-\frac{3}{2}x + 1 \).

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is one of the most straightforward ways to express a linear equation, which is \( y = mx + b \). In this form, \( m \) represents the slope, and \( b \) represents the y-intercept. The slope \( m \) indicates how steep the line is, and whether it leans upward or downward as we move from left to right.

• A positive slope means the line inclines upward.
• A negative slope means the line declines downward.
• A slope of zero means the line is horizontal.

The y-intercept, identified by \( b \), is the point where the line crosses the y-axis. This is a crucial piece of information because it gives us an exact starting point on the graph. In our example equation, \( y = -\frac{3}{2}x + 1 \), the slope is \(-\frac{3}{2}\), telling us the line will decline, and it crosses the y-axis at \( y = 1 \).

Building a Table of Values

A table of values helps us find specific points that lie on our line. By choosing various x-values, we can compute the corresponding y-values using the given equation. This makes plotting on a graph a piece of cake. For our equation \( y = -\frac{3}{2}x + 1 \), choosing x-values such as -2, -1, 0, 1, and 2 can simplify calculations because they are close to zero and mostly lead to simple arithmetic.
For each x-value:

• When \( x = -2 \), \( y = 4 \)
• When \( x = -1 \), \( y = 2.5 \)
• When \( x = 0 \), \( y = 1 \)
• When \( x = 1 \), \( y = -0.5 \)
• When \( x = 2 \), \( y = -2 \)

List these points as \((-2, 4), (-1, 2.5), (0, 1), (1, -0.5), (2, -2)\). This gives us clear coordinates to plot on the graph, ensuring accuracy as we draw our line.

Tips for Graphing Linear Equations

Graphing linear equations becomes intuitive with practice and the right approach. Let's focus on an easy way to transform our table into a visual representation on the plane. Start by labeling your axes and draw evenly spaced ticks.
Steps to plot the points:

• Locate each calculated point from your table of values.
• Mark each point clearly on the graph.
• Connect these points with a straight line.

Remember, the line should extend in both directions beyond the plotted points because linear equations continue infinitely. Check your graph to guarantee that it passes through each plotted point, reflecting the consistency of the equation \( y = -\frac{3}{2}x + 1 \).

The Role of Y-Intercept

The y-intercept plays a unique role in quickly identifying where the line begins on the y-axis. In our equation \( y = -\frac{3}{2}x + 1 \), the y-intercept is 1. This means, regardless of the slope, the graph will always cross the y-axis at the point (0, 1). Since it is expressed with \( x = 0 \), it’s one of the first coordinates you should plot when graphing.
The y-intercept can greatly simplify plotting:

• Start by marking \( y = 1 \) on the y-axis.
• This serves as a reference for other points you calculate.
• By first plotting the y-intercept, you anchor the line on the graph effectively.

Recognizing the y-intercept also verifies the accuracy of your other points since the line must pass through this exact point for the equation to hold true.