Problem 39 - 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. \(-2 x+7 y=4\) \([-5,5]\) by \([-5,5]\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = \frac{2}{7}x + \frac{4}{7}\).

Step 1: Understand the Given Equation

The provided linear equation is \(-2x + 7y = 4\). Our task is to rearrange this equation into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Isolate the y-term

Begin by isolating the term containing \(y\). You can do this by adding \(2x\) to both sides of the equation. This results in the equation \(7y = 2x + 4\).

Step 3: Solve for y

Next, divide every term in the equation by 7 to solve for \(y\). This gives \(y = \frac{2}{7}x + \frac{4}{7}\).

Step 4: Write in Slope-Intercept Form

The equation \(y = \frac{2}{7}x + \frac{4}{7}\) is now in the slope-intercept form \(y = mx + b\), where \(m = \frac{2}{7}\) and \(b = \frac{4}{7}\).

Step 5: Verify the Slope and Intercept

Check to make sure the coefficients are correct and the equation reflects the slope \(m = \frac{2}{7}\) and y-intercept \(b = \frac{4}{7}\).

Key Concepts

Linear Equation

A linear equation is a type of equation where the highest power of the variable is 1. It creates a straight line when graphed on a coordinate plane. Linear equations often appear in the form of \(ax + by = c\), where \(x\) and \(y\) are variables, and \(a\), \(b\), and \(c\) are constants. To convert from a standard form equation like \(-2x + 7y = 4\) to a slope-intercept form, you aim to isolate \(y\) on one side of the equation. This transformation helps in quickly identifying the slope and y-intercept of the line, which are essential for graphing.

• Standard form: \(ax + by = c\)
• Slope-intercept form: \(y = mx + b\)

Understanding linear equations is crucial because they model relationships with a constant rate of change, making them relevant in real-world applications such as calculating speed, predicting trends, and more.

Slope

The slope of a linear equation represents the steepness or the tilt of the line. In the slope-intercept form \(y = mx + b\), the slope is denoted by \(m\). The slope is calculated as the change in \(y\) over the change in \(x\), often described as "rise over run." A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. For example, in the equation \(y = \frac{2}{7}x + \frac{4}{7}\), the slope \(m\) is \(\frac{2}{7}\):

• It signifies that for every 7 units the line moves horizontally, it rises 2 units.
• A slope of 0 indicates a horizontal line, showing no vertical change as \(x\) changes.

Having a clear understanding of slope helps in predicting the trajectory of the line and can be powerful in analyzing linear relationships.

Y-intercept

The y-intercept is the point where the line crosses the y-axis on a graph. In the slope-intercept format \(y = mx + b\), the y-intercept is represented by \(b\). It is the value of \(y\) when \(x\) is zero. This point gives us a starting value from which the line extends, using the slope to guide its direction.In our example equation \(y = \frac{2}{7}x + \frac{4}{7}\), the y-intercept \(b\) is \(\frac{4}{7}\):

• This means the line cuts the y-axis at \(y = \frac{4}{7}\).
• Understanding the y-intercept aids not only in graphing the equation but also in interpreting scenarios in practical contexts, such as initial conditions in a problem.

Recognizing the y-intercept is crucial for drawing and understanding the graph of a linear function fully.