Problem 47 - College Algebra

Question

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. \(0.537 x-2.19 y=100\)

Step-by-Step Solution

Answer

The slope-intercept form is \(y = 0.245x - 45.662\), with \(ymin = -50\) and \(ymax = 0\).

Step 1: Understand the Equation

The given equation is in standard form: \(0.537x - 2.19y = 100\). We need to convert this to slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Isolate the y-Term

Move the \(x\) term to the other side of the equation: \(-2.19y = -0.537x + 100\). This sets us up to solve for \(y\).

Step 3: Solve for y

Divide every term by \(-2.19\) to solve for \(y\): \[y = \frac{-0.537}{-2.19}x + \frac{100}{-2.19}\].

Step 4: Simplify the Equation

Calculate the division to simplify the equation, rounding to the thousandths place: \[y = 0.245x - 45.662\]. This is your slope-intercept form.

Step 5: Set Graphing Calculator Window

Since the y-intercept is \(-45.662\), choose a \(ymin\) value less than \(-46\) and a \(ymax\) value higher than 0 to ensure the y-intercept is visible. For example, set \(ymin = -50\) and \(ymax = 0\).

Key Concepts

Understanding the y-intercept

In linear equations, the y-intercept is where the line crosses the y-axis. This point occurs when the value of x is zero. It's a vital aspect as it provides a starting point for graphing the line. By identifying the y-intercept, you can determine that particular point's location without plotting multiple points. In our example, after converting the given equation to slope-intercept form, the y-intercept is

• -45.662.

The y-intercept is aligned with the constant term in the equation, following the structure of the slope-intercept form, which is:

• y = mx + b

Where:

• m is the slope.
• b is the y-intercept.

Thus, if you plug 0 for x in the equation, y will equal

• the y-intercept.

Using a graphing calculator

A graphing calculator is an essential tool for visualizing linear equations. It allows you to input the slope-intercept form into its graphing function to see how the line appears on a coordinate grid. After converting the standard form equation to the slope-intercept form for calculation, enter it as Y1 in the graphing calculator. This helps you easily adjust your viewing window to include critical points such as the y-intercept. To see our line that crosses

• the y-axis at -45.662,

you’ll need to set ymin to a number smaller than this value, such as

• -50,

and ymax to a number greater than zero, such as

• 0.

This setup ensures the visibility of the y-intercept and other significant sections of the graph.

Explaining standard form

Standard form is one of the ways to express linear equations. It follows this structure:

• Ax + By = C,

where A, B, and C are constants. This form is particularly useful for identifying and simplifying the process of finding intercepts. In our problem, the equation

• 0.537x - 2.19y = 100

is initially in standard form.
To convert it to the slope-intercept form (y = mx + b), you need to:

• Isolate y by moving the x term to the opposite side.
• Divide every coefficient by the y-term's factor.

Once you’ve done those steps, this equation will help more easily identify and graph the line on a coordinate plane.

The essentials of linear equations

Linear equations are used to create straight lines when graphed. They can be written in several forms, such as slope-intercept form and standard form. When you have a linear equation, you're working with an expression that describes a line’s characteristics using its slope and y-intercept.

• Slope-intercept form is expressed as y = mx + b.
• Standard form is expressed as Ax + By = C.

Each form suits different analytical purposes.
The slope-intercept form is best for easily graphing and understanding the steepness and direction of a line. On the other hand, standard form can simplify communication between equations to find particular points like intercepts more easily. After converting one form to another, you can utilize tools such as a graphing calculator to visualize and carry out related mathematical tasks.