Problem 10 - 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: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start with the given equation \( -4y + 2x + 8 = 0 \). First, isolate the \( y \)-term on one side by subtracting \( 2x + 8 \) from both sides: \( -4y = -2x - 8 \).

Step 2: Solve for y

Divide every term in the equation \( -4y = -2x - 8 \) by \(-4\) to solve for \( y \). This gives \( y = \frac{1}{2}x + 2 \).

Step 3: Identify the Slope and y-Intercept

Now that the equation is in slope-intercept form \( y = \frac{1}{2}x + 2 \), identify the slope \( m = \frac{1}{2} \) and the y-intercept \( b = 2 \). These values correspond to the slope and the point where the line crosses the y-axis.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental way to express the equation of a straight line. This form makes it easy to identify two key characteristics: the slope and the y-intercept. The general representation of this form is:

• \( y = mx + b \)

Here, \( m \) represents the slope, showing how steep the line is, and \( b \) signifies the y-intercept, indicating where the line crosses the y-axis.

When tackling a problem requiring you to find the slope and y-intercept, always aim to rewrite the equation in this form. Take each term and adjust the equation until \( y \) stands alone on one side and is equal to a term involving \( x \) plus a constant. In our exercise, converting \(-4y + 2x + 8 = 0\) into slope-intercept form involved straightforward algebraic manipulations.

This step-by-step transformation helps highlight the characteristics of the line, making mathematical computations and graphing much simpler. Understanding the slope-intercept form is crucial for anyone learning about linear equations.

Slope Calculation

Calculating the slope is an exciting part of understanding linear equations. The slope \( m \) indicates how fast or slow the line rises or falls as it travels from left to right across the graph. A positive slope means the line moves upwards, while a negative slope shows a downward trend.

To find the slope from an equation, it's useful to first express it in the slope-intercept form \( y = mx + b \). Once in this form, identifying the slope becomes a simple matter of recognizing the coefficient of \( x \).

For the equation provided in this exercise, once transformed to \( y = \frac{1}{2}x + 2 \), the slope is clear to see as \( \frac{1}{2} \).

• This means that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{2} \), indicating a moderate upward rise.

Grasping how to find and interpret slope helps visualize how changes in one variable affect another.

Y-Intercept Determination

Determining the y-intercept is another crucial skill when analyzing linear equations in algebra. The y-intercept \( b \) indicates the specific point where the line will cross the y-axis.

When you rearrange any linear equation into the slope-intercept form \( y = mx + b \), \( b \) stands alone as a constant. This value gives a clear starting point on the vertical axis. In our example, the original equation \(-4y + 2x + 8 = 0\) becomes \( y = \frac{1}{2}x + 2 \) when rearranged properly.

• Here, \( b = 2 \), showing that the line crosses the y-axis at 2.

Understanding the y-intercept allows you to accurately graph the line and interpret its position relative to the origins at all times. It's a fundamental part of linking the visual component of algebra to the numerical.