Graph each linear function. 6x-5f(x) - 20 = 0

Verified step by step guidance

1. Rewrite the given equation to isolate the function \( f(x) \). The original equation is \( 6x - 5f(x) - 20 = 0 \). Add \( 5f(x) \) and subtract \( 6x \) from both sides to get \( -5f(x) = -6x + 20 \).
2. Divide both sides of the equation by \( -5 \) to solve for \( f(x) \). This gives \( f(x) = \frac{-6x + 20}{-5} \).
3. Simplify the right-hand side by dividing each term separately: \( f(x) = \frac{-6x}{-5} + \frac{20}{-5} \), which simplifies to \( f(x) = \frac{6}{5}x - 4 \).
4. Identify the slope and y-intercept from the equation \( f(x) = \frac{6}{5}x - 4 \). The slope \( m = \frac{6}{5} \) and the y-intercept \( b = -4 \).
5. To graph the function, start by plotting the y-intercept at \( (0, -4) \). Then use the slope \( \frac{6}{5} \) to find another point by rising 6 units and running 5 units to the right. Draw a straight line through these points to complete the graph.

Key Concepts

Linear Functions

A linear function is a function whose graph is a straight line, typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. Understanding linear functions helps in identifying how changes in x affect the output f(x).

Rearranging Equations

Rearranging equations involves manipulating the given equation to isolate the dependent variable, usually f(x) or y, on one side. This step is essential to rewrite the function in slope-intercept form, making it easier to graph.

Graphing Linear Equations

Graphing linear equations requires plotting points or using the slope and y-intercept to draw the line on the coordinate plane. Recognizing the slope and intercept from the equation allows for accurate and efficient graphing.