Problem 8 - Applied Calculus

Question

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$-4 y+2 x+8=0$$$

Step-by-Step Solution

Answer

The slope is \( \frac{1}{2} \) and the \( y \)-intercept is 2.

Step 1: Rearrange the equation

The first step is to rearrange the given equation into the form \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the \( y \)-intercept. Start by isolating the \( y \) term. The given equation is \(-4y + 2x + 8 = 0\). Move \( 2x \) and \( 8 \) to the right side of the equation:\[-4y = -2x - 8\]

Step 2: Solve for y

Now, divide every term by \(-4\) to solve for \( y \).\[y = \frac{-2x}{-4} + \frac{-8}{-4}\]Simplify the fractions:\[y = \frac{1}{2}x + 2\]

Step 3: Identify the slope and y-intercept

From the rearranged equation \(y = \frac{1}{2}x + 2\), identify the slope and \( y \)-intercept. The slope \( m \) is the coefficient of \( x \), which is \( \frac{1}{2}\), and the \( y \)-intercept \( b \) is the constant term, which is \( 2 \).

Key Concepts

Slope

The slope of a line is an essential concept in linear equations. It defines how steep the line is, indicating the rate at which the line ascends or descends.
Mathematically, the slope is represented by the letter \( m \). It is the coefficient of \( x \) when the equation of the line is in the form \( y = mx + b \). This form is known as the slope-intercept form.
There are several key points to remember about slope:

• If the slope (\( m \)) is positive, the line increases as it moves from left to right.
• If the slope (\( m \)) is negative, the line decreases as it moves from left to right.
• A larger absolute value of \( m \) means a steeper line.
• A slope of zero indicates a flat line, parallel to the \( x \)-axis.

In the equation from our exercise, \( y = \frac{1}{2}x + 2 \), the slope \( m = \frac{1}{2} \), meaning the line rises half a unit for every unit it moves horizontally to the right.

Y-intercept

Understanding the \( y \)-intercept is crucial as it indicates where the line crosses the \( y \)-axis in a linear equation.
The \( y \)-intercept is represented by the letter \( b \). When the equation is in the slope-intercept form \( y = mx + b \), \( b \) is simply the constant term.
Here are a few things to note about \( y \)-intercepts:

• The \( y \)-intercept is the value of \( y \) when \( x \) is zero.
• The line touches the \( y \)-axis at this point.
• If the equation has no constant term, the \( y \)-intercept is zero, meaning the line passes through the origin.

In the equation \( y = \frac{1}{2}x + 2 \) from our exercise, the \( y \)-intercept is \( 2 \). This means that the line crosses the \( y \)-axis at the point (0, 2).

Solving Equations

Solving equations, especially linear equations, is a foundational skill in algebra.
Linear equations can be identified as having terms with only constants and variables raised to the power of one.
To solve equations effectively, follow these general steps:

• Rearrange: Aim to shift all terms involving the variable to one side of the equation and constant terms to the other side. This may require adding or subtracting terms on both sides.
• Isolate the variable: Divide or multiply to get the variable by itself, often by the coefficient of the variable.
• Simplify: Reduce any fractions or complex terms.

In our exercise, rearranging the equation step-by-step allowed us to find the values of the slope and \( y \)-intercept. We transformed \(-4y + 2x + 8=0\) into a more familiar slope-intercept form \( y = \frac{1}{2}x + 2 \), making it easier to understand and solve.