Problem 42 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) With slope \(-4 ; y\) -intercept \(\left(0, \frac{2}{9}\right) ;\) use slope- intercept form. slope-intercept form.

Step-by-Step Solution

Answer

The equation is \( y = -4x + \frac{2}{9} \).

Step 1: Review Slope-Intercept Form

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

Step 2: Examine Given Values

Identify the given values in the problem. The slope \( m \) is \(-4\) and the y-intercept \( b \) is \( \frac{2}{9} \). These values will be used in the equation.

Step 3: Substitute Values into Slope-Intercept Form

Insert the given slope and y-intercept values into the slope-intercept form equation: \( y = -4x + \frac{2}{9} \).

Step 4: Write the Final Equation

After substitution, the equation of the line in slope-intercept form is \( y = -4x + \frac{2}{9} \). This is the answer in the desired form.

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra. They represent relationships between two variables with a constant rate of change. Simply put, a linear equation can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
The most commonly used form of a linear equation in many problems is the slope-intercept form, \( y = mx + b \). This form makes it easy to identify the slope and y-intercept of the line, two crucial characteristics that define the line's behavior and position in a coordinate plane.

• **Standard Form:** Typically appears as \( ax + by = c \).
• **Slope-Intercept Form:** Written as \( y = mx + b \), highlighting the slope and y-intercept.
• **Point-Slope Form:** Useful for quick equation derivation, expressed as \( y - y_1 = m(x - x_1) \).

Linear equations are straight lines when graphed. The numbers attached to \( x \), \( y \), and other constants decide the line's steepness and where it crosses the y-axis.

Slope

Slope is a measure of how steep a line is. It indicates the rate at which \( y \) changes with respect to \( x \). Calculating slope involves comparing the vertical change (rise) to the horizontal change (run).
The slope \( m \) is calculated as \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \).
This formula is derived by identifying two distinct points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), and calculating the change in \( y \) over the change in \( x \).

• A **positive slope** means the line goes upwards from left to right.
• A **negative slope** means the line goes downwards from left to right.
• A **zero slope** indicates a horizontal line, while an **undefined slope** indicates a vertical line.

In the exercise, the slope \( m = -4 \), tells us that for every one unit increase in \( x \), \( y \) decreases by 4 units. That's what gives the line its downward slant.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It's an essential component in defining the linear equation using slope-intercept form. In the equation \( y = mx + b \), \( b \) represents the y-intercept. It gives a starting value for \( y \) when \( x \) is zero.

• The y-intercept is expressed as a point on the graph, \((0, b)\).
• In practical terms, it represents the value of \( y \) when there has been no movement along the x-axis.
• Knowing the y-intercept helps in quickly graphing the equation without needing a table of values.

For the given problem, the y-intercept is \( \frac{2}{9} \). This means the line crosses the y-axis at \( y = \frac{2}{9} \). Understanding y-intercepts allows us to easily see where the line starts in relation to the origin.