Problem 36 - Elementary Algebra

Question

For the following problems, graph the equations. $$ 626.01 x-506.73 y=2443.50 $$

Step-by-Step Solution

Answer

Question: Graph the equation \(626.01x - 506.73y = 2443.50\) and identify the slope and y-intercept. Answer: The graph of the equation \(y = \frac{626.01}{506.73}x - \frac{2443.50}{506.73}\) is a straight line with slope \(m = \frac{626.01}{506.73}\) and y-intercept \(b = -\frac{2443.50}{506.73}\).

Step 1: Transform the equation into the slope-intercept form (y = mx+b)

To transform the equation into the slope-intercept form, we need to isolate y on one side of the equation. Currently, the equation is given by: $$ 626.01x - 506.73y = 2443.50 $$ First, we will subtract \(626.01x\) from both sides of the equation to isolate the \(y\) term: $$ -506.73y = -626.01x + 2443.50 $$ Now, we should divide both sides of the equation by \(-506.73\) to get \(y\) by itself: $$ y = \frac{626.01}{506.73}x - \frac{2443.50}{506.73} $$

Step 2: Identify the slope and y-intercept of the equation

Now that the equation is in the form \(y = mx+b\), we can identify the slope and \(y\)-intercept. Here, the slope is \(m = \frac{626.01}{506.73}\) and the \(y\)-intercept is \(b = -\frac{2443.50}{506.73}\).

Step 3: Graph the equation using the slope and y-intercept

Now, we can graph the equation using the slope and \(y\)-intercept we found in Step 2. 1. Plot the \(y\)-intercept on the graph, which is the point \((0, -\frac{2443.50}{506.73})\). 2. From the \(y\)-intercept, move the slope (\(\frac{626.01}{506.73}\)) by going up/down the rise and right/left the run to find another point on the line. In this case, it's easier to use decimals, so using the slope of approximately \(1.24\), we can go up 1 unit and right 1 unit. 3. Connect the points to create a straight line representing the equation \(y =\frac{626.01}{506.73}x - \frac{2443.50}{506.73}\). Now, the graph of the equation is efficiently drawn using the identified slope and \(y\)-intercept.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a line. It helps us graph linear equations easily. The basic form is:

• \( y = mx + b \)

Where \( y \) is the dependent variable and \( x \) is the independent variable.
The slope of the line is represented by \( m \), showing how much \( y \) changes for a change in \( x \). The constant \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
To convert an equation like \( 626.01x - 506.73y = 2443.50 \) into this form, isolate \( y \) on one side. This involves moving terms around and solving for \( y \), so it looks like \( y = mx + b \). This form is beneficial because once transformed, identifying slope and y-intercept becomes a breeze!

Identifying Slope and Y-Intercept

Once the equation is in the slope-intercept form \( y = mx + b \), it's easy to identify both the slope and y-intercept. These two components tell us a lot about how the graph behaves.

• Slope (\( m \)): In the problem, \( m = \frac{626.01}{506.73} \). This ratio indicates the steepness of the line. After calculating this, you find that the slope is approximately \( 1.24 \), meaning for every unit moved horizontally, the line rises 1.24 units vertically.


• Y-intercept (\( b \)): The intercept is \( b = -\frac{2443.50}{506.73} \). This evaluates to a specific point on the y-axis, showing where the line crosses it. Calculating this gives us a numeric value that marks the starting point on the vertical axis, which is where we start plotting.

Understanding these components is crucial, as they directly correlate to drawing the line on a graph by showing the starting point and direction.

Plotting Points on a Graph

With the slope and y-intercept identified, plotting the graph becomes easier. Start by plotting the y-intercept on the graph, this is your first point where the line touches the y-axis. In our example:

• Plot the point \( (0, -\frac{2443.50}{506.73}) \).

Once you have this initial point, you can use the slope to find additional points. The slope tells us how to "move" from one point to the next:

• From the y-intercept, apply the slope \( 1.24 \) to find another point. This means move up 1.24 units vertically for a 1 unit horizontal move.
• By plotting this second point, continue using the slope to find more points along the line if necessary.
• Connect these points with a straight line.

Remember, all points found this way should lie on the line. The more points you check, the clearer your line becomes, confirming its accuracy. This method ensures your line reflects the original equation correctly.