Problem 40 - A Graphical Approach to College Algebra

Question

Write equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. \(-0.23 x-0.46 y=0.82\) \([-5,5]\) by \([-5,5]\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -0.5x - 1.783\).

Step 1: Identify the standard form

The given equation is \(-0.23x - 0.46y = 0.82\) which is in standard form \(Ax + By = C\). In this case, \(A = -0.23\), \(B = -0.46\), and \(C = 0.82\).

Step 2: Rearrange to solve for y

Rearrange the equation to solve for \(y\): Start with the equation:\[-0.23x - 0.46y = 0.82\]Add \(0.23x\) to both sides:\[-0.46y = 0.23x + 0.82\]

Step 3: Isolate y

Divide every term by \(-0.46\) to solve for \(y\):\[y = \frac{0.23}{-0.46}x + \frac{0.82}{-0.46}\]Simplify the fractions to express \(x\) and the constant term:

Step 4: Simplify the slope and intercept

Calculate \(\frac{0.23}{-0.46}\) which simplifies to \(-0.5\), and \(\frac{0.82}{-0.46}\) which simplifies to approximately \(-1.783\).Thus, the equation becomes: \[y = -0.5x - 1.783\].

Step 5: Verify the equation form

Ensure the format is \(y = mx + b\) where \(m = -0.5\) and \(b = -1.783\). This fits the y-intercept form perfectly, confirming the conversion from standard form.

Key Concepts

Standard Form

The standard form of a linear equation is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are real numbers and \(A\) and \(B\) are not both zero. This form is very useful because it presents the equation in a way that highlights the relationship between \(x\) and \(y\) in a straightforward manner. The coefficients \(A\) and \(B\) can be used to easily identify the x- and y-intercepts, which are the points where the line crosses the axes. To determine these intercepts:

• X-intercept: Set \(y = 0\) and solve for \(x\).
• Y-intercept: Set \(x = 0\) and solve for \(y\).

This method gives a quick insight into the graph of the equation by showing where it crosses the axes. Keep in mind that this form may be less intuitive for graphing compared to the slope-intercept form.

Slope-Intercept Form

When a linear equation is expressed in the slope-intercept form, it looks like \(y = mx + b\). Here, \(m\) represents the slope of the line, indicating its steepness and direction, while \(b\) shows where the line crosses the y-axis, known as the y-intercept. This form is particularly helpful for quickly sketching or interpreting the graph of a line.

• Slope \(m\): It is calculated as the change in \(y\) over the change in \(x\) \( (\Delta y / \Delta x)\), and it tells us how the line moves upward or downward.
• Y-intercept \(b\): This is the point on the graph where the line touches the y-axis, making it easy to start plotting the line.

For example, converting the given equation to \(y = -0.5x - 1.783\) identifies the slope as \(-0.5\) and the y-intercept as approximately \(-1.783\). This makes it straightforward to draw the line starting at \(-1.783\) on the y-axis and moving with a rise over run pattern of \(-0.5\).

Equation Conversion

Converting between different forms of linear equations is a valuable skill in algebra. It allows for flexibility in how linear relationships are represented and analyzed. The most common conversion is from the standard form to the slope-intercept form, as this can greatly simplify the process of graphing a line.To convert an equation from standard form \(Ax + By = C\) to slope-intercept form \(y = mx + b\):

• First, solve for \(y\) by isolating it on one side of the equation. Add or subtract terms as necessary to get all \(y\) terms on one side.


• Then, divide each term by \(B\) to ensure \(y\) stands alone, making the equation clear in \(y = mx + b\) format.


• Finally, simplify any fractions or constants.

In our given example, \(-0.23x - 0.46y = 0.82\) is transformed to \(y = -0.5x - 1.783\). This process not only reaffirmed the function's form but also made graphing and analyzing the equation more accessible.