Problem 18 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=-\frac{4}{5}, b=0 $$

Step-by-Step Solution

Answer

The equation is \( y = -\frac{4}{5}x \).

Step 1: Understanding the Line Equation

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

Step 2: Substitute the Given Slope

We are given the slope \( m = -\frac{4}{5} \). Substitute this value into the slope-intercept form of the line equation: \( y = -\frac{4}{5}x + b \).

Step 3: Substitute the Given Y-Intercept

The given y-intercept is \( b = 0 \). Substitute this value into the equation: \( y = -\frac{4}{5}x + 0 \).

Step 4: Simplify the Equation

To simplify, remove the \( +0 \) from the equation, resulting in \( y = -\frac{4}{5}x \).

Key Concepts

Slope

In mathematics, especially in algebra and calculus, the concept of slope is crucial for understanding how lines behave on a coordinate plane. The slope of a line measures its steepness or incline; it tells us how much the line rises or falls as it moves from left to right across the plane. This is formally defined as the "change in the y-coordinate divided by the change in the x-coordinate." This ratio is often remembered by the formula:

• \( m = \frac{\text{rise}}{\text{run}} \)

In simpler terms, the slope is the "rise over run." It shows how much the line goes up or down for every unit you move to the right. A positive slope means the line is moving upwards, while a negative slope indicates it is heading downwards.

For example, with a slope of \(-\frac{4}{5}\), the line falls 4 units for every 5 units it moves to the right. Slope is a critical component in forming the slope-intercept form of a linear equation.

Y-Intercept

The y-intercept is another vital element in understanding linear equations. It refers to the point where the line crosses the y-axis on a graph. Mathematically, it is expressed as the constant \( b \) in the slope-intercept form of a linear equation:

• "\( b \) is the y-intercept."

The y-intercept reveals the value of \( y \) when \( x \) is zero.

In practical terms, if you can imagine sliding a line vertically across the plane, the y-intercept is where it finally touches the y-axis. In the example from the exercise given above, the y-intercept \( b = 0 \) means the line passes through the origin \((0,0)\). This y-intercept helps in quickly sketching a line because the line's starting point on the axis is readily known.

Slope-Intercept Form

The slope-intercept form is one of the most common equations used to represent a straight line. It's given by the expression:

• \( y = mx + b \)

where \( m \) is the slope and \( b \) is the y-intercept. This form is handy because it clearly shows both the slope and the starting point of the line.

It is particularly useful for graphing because you can quickly determine how to draw the line just by looking at it. For our example:

• The equation simplifies to \( y = -\frac{4}{5}x \), indicating a slope of \(-\frac{4}{5}\) and a y-intercept of 0.

This means the line starts at the origin \((0,0)\) and moves downwards 4 units for every 5 units it moves horizontally to the right. By putting the line in slope-intercept form, we gain a clear visual representation of its characteristics.