Problem 30 - Elementary Algebra

Question

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-2 \text { and } b=\frac{7}{3} $$

Step-by-Step Solution

Answer

y = -2x + \frac{7}{3}

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form shows how the variables \( y \) and \( x \) are related in a straight line.

Step 2: Identify Given Values

From the problem, the given values are the slope \( m = -2 \) and the y-intercept \( b = \frac{7}{3} \). We will use these values to formulate the equation of the line.

Step 3: Substitute Values into Slope-Intercept Form

Substitute the given slope \( m = -2 \) and y-intercept \( b = \frac{7}{3} \) into the slope-intercept form equation. This gives:\[ y = -2x + \frac{7}{3} \]

Step 4: Finalize the Equation

The equation \( y = -2x + \frac{7}{3} \) already represents the slope-intercept form, where the slope \( m \) and y-intercept \( b \) are properly integrated. No further simplification is needed.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed on a coordinate plane. These equations are typically in the form \( y = mx + b \), where each variable and coefficient has a specific function.

• \( y \): Represents the dependent variable whose value changes based on the value of \( x \).
• \( x \): The independent variable, which is typically chosen or set.
• \( m \): The slope of the line, indicating how steep the line is.
• \( b \): The y-intercept, showing where the line crosses the y-axis.

Understanding linear equations involves knowing how each component affects the overall line. It's important to note that linear equations can graph any line as long as it is straight, from a vertical line to a shallow angle.

Slope

The slope of a line quantifies the steepness and direction of a line. In the equation \( y = mx + b \), the slope is represented by \( m \).

• If \( m \) is positive, the line rises as it moves from left to right.
• If \( m \) is negative, the line falls from left to right.

The slope is calculated as the "rise over the run," which means the vertical change divided by the horizontal change between two points on the line:\[m = \frac{\Delta y}{\Delta x}\]This characteristic is crucial because it defines the angle and direction of the line compared to the x-axis. The slope is a critical factor when determining the relationship between the variables in a linear equation.

Y-intercept

The y-intercept is where the line crosses the y-axis on a graph. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \).

• Explanation: It indicates the value of \( y \) when \( x \) is zero.
• On a graph: This is where you start plotting the line if you were drawing it by hand.

To find the y-intercept, you can look at the constant term in the slope-intercept form. It's very useful because it gives you a starting point to draw your line before applying the slope to find other points. The y-intercept helps in understanding the initial value of the dependent variable without the influence of the independent variable.