Problem 18 - Algebra for College Students

Question

Determine the linear function whose graph is a line with a slope of \(-\frac{3}{5}\) and contains the point \((4,-5)\). \(f(x)=-\frac{3}{5} x-\frac{13}{5}\)

Step-by-Step Solution

Answer

The linear function is \( f(x) = -\frac{3}{5}x - \frac{13}{5} \).

Step 1: Understand the Structure of a Linear Function

The general form of a linear function is given by \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Plug in the Given Slope

We know the slope \( m = -\frac{3}{5} \). Substitute this value into the linear function formula to get \( f(x) = -\frac{3}{5}x + b \).

Step 3: Use the Given Point to Find the Y-intercept

The point (4, -5) satisfies the linear function. Substitute \( x = 4 \) and \( f(x) = -5 \) into the equation: \(-5 = -\frac{3}{5}(4) + b\).

Step 4: Solve for the Y-intercept

Simplify the equation: \(-5 = -\frac{12}{5} + b\). Add \(\frac{12}{5}\) to both sides to isolate \( b \): \(b = -5 + \frac{12}{5}\).

Step 5: Convert Mixed Numbers to Improper Fractions

Convert \(-5\) to an improper fraction which is \(-\frac{25}{5}\), so we have: \(b = -\frac{25}{5} + \frac{12}{5}\).

Step 6: Add the Fractions

Combine the fractions: \(b = \frac{-25 + 12}{5} = \frac{-13}{5}\).

Step 7: Write the Linear Function

Now that we have both the slope and the y-intercept, substitute them back into the general formula: \( f(x) = -\frac{3}{5}x - \frac{13}{5}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear function is one of the most common and useful ways to express a linear equation. It is given by the formula \( f(x) = mx + b \), where:

• \( m \) is the slope of the line, describing its steepness or tilt.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

Understanding this format is essential because it allows you to quickly graph a line, by using \( m \) to determine the direction and steepness, and \( b \) to locate its position on the graph. Rather than beginning with cumbersome calculations, the slope-intercept form directly connects algebraic expressions to graphical representations. Thus, it simplifies the process of interpreting linear relationships.

Finding the Y-Intercept

To find the y-intercept in a linear function, we need to determine the value of \( b \) in the equation \( f(x) = mx + b \). Knowing just the slope is not enough to fully describe a line, unless you also have information about one specific point on that line.For example, if given a slope and a point, as in the exercise with slope \(-\frac{3}{5}\) and point \((4, -5)\), you can substitute the point into the linear equation to solve for \( b \). This is how it works:

• Insert \( x = 4 \) and \( f(x) = -5 \) into \( f(x) = -\frac{3}{5}x + b \).
• This forms the equation: \(-5 = -\frac{3}{5}(4) + b \).
• Simplify and solve for \( b \) by isolating it on one side of the equation.

This process reveals the y-intercept, giving you a complete picture of the line's equation.

Solving Linear Equations

Solving linear equations is an essential skill involving finding the value of the variable that makes an equation true. Here's how it works in the context of the exercise:You use the point \((4, -5)\) given; substitute \( x = 4 \) and \( f(x) = -5 \) to form the equation \(-5 = -\frac{3}{5}(4) + b \). Here’s a step-by-step guide:

• First, compute \(-\frac{3}{5} \times 4\) to get \(-\frac{12}{5}\).
• Reformulate the equation as \(-5 = -\frac{12}{5} + b \).
• Add \( \frac{12}{5} \) to both sides to solve for \( b \).
• Simplify \(-5 + \frac{12}{5} \) by expressing \(-5\) as an improper fraction: \(-\frac{25}{5}\), resulting in \( b = \frac{-13}{5} \).

This solving process is how you determine missing parts of the equation, ensuring you have all components to define the linear function.