Problem 32 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. $$ \text { Passing through }\left(\frac{3}{4},-\frac{1}{4}\right) \text { and }\left(\frac{5}{4}, \frac{7}{4}\right) $$

Step-by-Step Solution

Answer

The slope-intercept form of the line is \( y = 4x - \frac{13}{4} \).

Step 1: Identify the given points

The given points are \( \left( \frac{3}{4}, -\frac{1}{4} \right) \) and \( \left( \frac{5}{4}, \frac{7}{4} \right) \). These points will be used to determine the slope \( m \) and later to find the equation of the line.

Step 2: Calculate the slope

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. For the given points, \( x_1 = \frac{3}{4}, \ y_1 = -\frac{1}{4}, \ x_2 = \frac{5}{4}, \ y_2 = \frac{7}{4}\). Substitute these values into the formula: \[ m = \frac{\frac{7}{4} - -\frac{1}{4}}{\frac{5}{4} - \frac{3}{4}} = \frac{\frac{8}{4}}{\frac{2}{4}} = \frac{8}{2} = 4 \]. So, the slope is \( m = 4 \).

Step 3: Use the point-slope formula

Using the slope \( m = 4 \) and one of the points, we can find the line's equation using the point-slope formula \( y - y_1 = m(x - x_1) \). Let's use the point \( \left( \frac{3}{4}, -\frac{1}{4} \right) \). Substitute \( m = 4 \), \( x_1 = \frac{3}{4} \), \( y_1 = -\frac{1}{4} \) into the formula: \[ y + \frac{1}{4} = 4 \left(x - \frac{3}{4}\right) \].

Step 4: Simplify to find slope-intercept form

Distribute and simplify the equation to place it in slope-intercept form \( y = mx + b \). Start with \[ y + \frac{1}{4} = 4(x - \frac{3}{4}) \]. First, expand the expression: \[ y + \frac{1}{4} = 4x - 3 \]. Next, subtract \( \frac{1}{4} \) from both sides: \[ y = 4x - 3 - \frac{1}{4} \]. Simplifying \( -3 - \frac{1}{4} \) gives \( -\frac{13}{4} \). Thus, the slope-intercept form is \( y = 4x - \frac{13}{4} \).

Key Concepts

Equation of a Line

When we talk about the equation of a line, one popular form is the **slope-intercept form**. This form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) indicates the y-intercept. The slope tells you how steep the line is, and the y-intercept shows where the line crosses the y-axis.

Knowing the equation of a line is crucial because it allows you to plot the line on a graph easily. It provides a straightforward way to model relationships in data and analyze trends. For example, if you know the values of \( m \) and \( b \), you can quickly find the value of \( y \) for any given \( x \).

• The 'slope-intercept form' helps to identify the line instantly on a graph by tracing its rise and run and its meeting point with the y-axis.
• This form is very useful in both theoretical aspects and real-world applications since it simplifies analyzing and interpreting the graphical representation of linear relationships.

Slope Calculation

The slope is a vital concept when discussing the equation of a line. It tells us how fast and in what direction our line moves. The formula for finding the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This slope formula helps calculate the rate at which \( y \) increases or decreases as \( x \) changes.

For example, given two points \( \left( \frac{3}{4}, -\frac{1}{4} \right) \) and \( \left( \frac{5}{4}, \frac{7}{4} \right) \), we substitute into the formula:

• The change in \( y \) is \( \frac{7}{4} - \left(-\frac{1}{4}\right) = \frac{8}{4} \).
• The change in \( x \) is \( \frac{5}{4} - \frac{3}{4} = \frac{2}{4} \).

Simplifying gives \( \frac{8}{2} = 4 \), making the slope \( m = 4 \), which implies that for every unit increase in \( x \), \( y \) increases by 4 units. It's straightforward and provides a clear understanding of the steepness of a line.

Point-Slope Formula

The **point-slope formula** is another convenient way of writing the equation of a line. The formula is:\[y - y_1 = m(x - x_1)\]Here, \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope. By using this formula, we can quickly form the equation of a line when we know a point it passes through and its slope.

If we take the point \( \left( \frac{3}{4}, -\frac{1}{4} \right) \) and the slope \( m = 4 \), we substitute these into the formula:

• Replace \( y_1 \) with \(-\frac{1}{4}\) and \( x_1 \) with \( \frac{3}{4} \).
• The formula becomes \( y + \frac{1}{4} = 4(x - \frac{3}{4}) \).

This formula provides a handy bridge to convert from point-slope to slope-intercept form. This process simplifies into the slope-intercept form \( y = 4x - \frac{13}{4} \). The point-slope form is extremely helpful in quickly determining a line's equation when specific information is known.