Problem 30 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Slope \(\frac{1}{3},\) passing through \(\left(\frac{1}{2},-2\right)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{3}x - \frac{13}{6} \).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Insert the Given Slope

We are given that the slope \( m \) is \( \frac{1}{3} \). Substitute \( m = \frac{1}{3} \) into the slope-intercept form equation, giving us \( y = \frac{1}{3}x + b \).

Step 3: Substitute the Point into the Equation

Use the point \( \left( \frac{1}{2}, -2 \right) \) to find \( b \). Substitute \( x = \frac{1}{2} \) and \( y = -2 \) into the equation \( y = \frac{1}{3}x + b \). This gives us \( -2 = \frac{1}{3} \cdot \frac{1}{2} + b \).

Step 4: Solve for the Y-Intercept

Calculate \( \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6} \), so \( -2 = \frac{1}{6} + b \). Rewrite this as \( b = -2 - \frac{1}{6} \). Convert \( -2 \) to a fraction with a denominator of 6: \( -2 = \frac{-12}{6} \). Now, \( b = \frac{-12}{6} - \frac{1}{6} = \frac{-13}{6} \).

Step 5: Write the Final Equation

Substitute \( b = \frac{-13}{6} \) back into the equation \( y = \frac{1}{3}x + b \). The equation of the line in slope-intercept form is \( y = \frac{1}{3}x - \frac{13}{6} \).

Key Concepts

Linear Equations

Linear equations are foundational in algebra and represent a straight line when graphed on a coordinate plane. A typical linear equation can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. The most common and user-friendly form to work with, however, is the slope-intercept form, which is \( y = mx + b \). In this equation, \( m \) represents the slope of the line, and \( b \) is the y-intercept.

The slope \( m \) is a measure of the line's steepness, indicating how much \( y \) changes for a unit change in \( x \). If the slope is positive, the line rises as it moves to the right; if negative, it falls.

Understanding linear equations is crucial because they model real-world scenarios where one quantity depends linearly on another.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero. In the slope-intercept form equation \( y = mx + b \), the y-intercept is represented by \( b \).

To find the y-intercept, you simply need to evaluate the equation at \( x = 0 \). The y-intercept provides a starting point on the graph and helps in sketching the line quickly. It's particularly useful as it tells you where the line starts in relation to the origin.

In our problem, after solving the equation with the given point, we found that the y-intercept is \( \frac{-13}{6} \). This tells us that as the line intersects the y-axis, it does so at the point \( (0, \frac{-13}{6}) \).

Algebraic Expression

An algebraic expression comprises numbers, variables, and arithmetic operations. It does not have an equality sign, unlike algebraic equations.

In the context of linear equations, expressions like \( \frac{1}{3}x + b \) represent the components of a linear equation before solving for specific values of \( b \). Each part of the expression plays a vital role:

• The number \( \frac{1}{3} \) is a coefficient that determines the slope.
• The variable \( x \) represents the dependent variable.
• The constant \( b \) (found to be \( \frac{-13}{6} \) in this case) is the y-intercept.

These elements together form equations that depict relationships between variables in a mathematical and visually interpretable format. Simplifying and solving these expressions is at the core of understanding linear relationships.