Problem 76 - 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. $$y=-\frac{5}{2} x+1$$

Step-by-Step Solution

Answer

The points (-2,6), (-1,3.5), (0,1), (1,-1.5), (2,-4) are all solutions of the equation, and, when plotted on a graph and connected, they form the line of the equation \(y=-\frac{5}{2} x+1\).

Step 1: Identify the slope and y-intercept

From the given equation \(y=-\frac{5}{2} x+1\), it is evident that the slope 'm' is \(-\frac{5}{2}\) and the y-intercept 'b' is 1.

Step 2: Prepare a Table of values

Select five different values of x and calculate the corresponding y values using the equation \(-\frac{5}{2} x+1\). For instance, choosing -2, -1, 0, 1 and 2 as values for 'x' provides the values of 'y' as 6, 3.5, 1, -1.5 and -4 respectively.

Step 3: Plotting the graph

Plot the y-intercept (0,1) on the graph first. Then use the slope \(-\frac{5}{2}\) to move 5 units down (because of the negative sign) and 2 units to the right. Plot your next point here. Repeat with all points from the table of values. After all points have been plotted, draw a straight line through all these points to represent the linear equation.

Key Concepts

Slope

The slope of a line is a crucial concept in understanding linear equations. It tells you how steep the line is and the direction it is going.
The formula for the slope, usually represented by the letter "m," describes the vertical change (rise) for a unit horizontal change (run) between any two points on a line.
This is often known as "rise over run."
In the equation given, \( y = -\frac{5}{2}x + 1 \), the slope is \(-\frac{5}{2}\).

• The negative sign indicates that the line is decreasing, slanting downwards from left to right.
• The fraction \(\frac{5}{2}\) means that for every 2 units moved horizontally to the right, the line moves 5 units downwards.

Understanding the slope helps predict how the line will behave without graphing it, simply by looking at the numbers.

Y-intercept

The y-intercept is another fundamental aspect of linear equations. It indicates where the line crosses the y-axis. This point is determined when the value of \( x \) is zero.
In our example, the y-intercept is represented by the value "1," given directly in the equation as the constant term, \(b\).

• The point (0,1) indicates the line crosses the y-axis at 1.
• It is a starting point for graphing a line and shows the value of \(y\) when \(x\) is 0.

Having a clear understanding of the y-intercept allows you to begin plotting the line and serves as a guideline to determine if a point lies on the given line.

Table of Values

Creating a Table of Values is a simple yet effective way to comprehend and graph linear equations.
It involves selecting different values for \(x\) and computing their corresponding \(y\) values using the linear equation.
This shows multiple points that lie on the line.
For our equation \( y = -\frac{5}{2}x + 1 \), choosing five values for \(x\), such as -2, -1, 0, 1, and 2, yields five corresponding \(y\) values, 6, 3.5, 1, -1.5, and -4.

• These pairs of \(x\) and \(y\) values can be plotted as points on a graph.
• This method helps verify the linearity of the equation by seeing that all points align on a straight line.

Using a table of values offers a tangible method for understanding how each \(x\) value affects \(y\), reinforcing the equation's behavior.

Graphing Linear Equations

Graphing Linear Equations is often a visual way of comprehending the relationship displayed by the equation.
Involves using the slope and y-intercept to plot the line on a graph.
Start by plotting the y-intercept, since it is a known point, in our case (0, 1). Then, utilize the slope to identify additional points. For the equation \( y = -\frac{5}{2}x + 1 \),

• Start from the y-intercept (0,1) and move 5 units down and 2 units right to plot the next point.
• Repeat this to plot additional points according to the pre-identified values from the Table of Values.
• Once all points are plotted, draw a straight line through these points.

Graphing these points will ensure they all lie on the same line, confirming the linear nature of the equation. This visual representation aids in understanding how the equation behaves across different values.