Problem 14 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(0.25,\) passes through \((0,4)\)

Step-by-Step Solution

Answer

The equation is \( y = 0.25x + 4 \).

Step 1: Identify the elements needed for the 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. From the problem, we have the slope \( m = 0.25 \).

Step 2: Use the given point to find the y-intercept

Since the line passes through the point \((0, 4)\), we already have the y-intercept. This is because when \( x = 0 \), the value of \( y \) is the y-intercept. Therefore, \( b = 4 \).

Step 3: Write the equation using the found values

Substitute the values of \( m \) and \( b \) in the slope-intercept form equation: \( y = 0.25x + 4 \). This is the equation of the line that satisfies the given conditions.

Key Concepts

Linear Equations

A linear equation represents a straight line when graphed on a coordinate plane. It is an algebraic equation that involves variables with no exponents higher than one. The general form of a linear equation can be represented as **Ax + By = C**. This type of equation highlights the relationship between the x and y variables in a direct, proportional manner.
An important method to express linear equations is the slope-intercept form. This form is especially helpful in identifying key properties of the line, such as the slope and the y-intercept, at a glance. You'll often encounter linear equations when working with data trends, predicting outcomes, or when calculating rates of change.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Represented by the letter **m** in the equation, it tells us how much y changes for a one-unit change in x. A positive slope means the line rises from left to right, while a negative slope means the line falls. A slope of zero represents a horizontal line, showing that y does not change as x changes.
For instance, in our problem, the slope provided is **0.25**, indicating a gentle upward inclination. To calculate the slope between any two points (x₁, y₁) and (x₂, y₂) on a line, use the formula:

• Slope (m) = (y₂ - y₁) / (x₂ - x₁)

Understanding slope is crucial in fields like physics, economics, and statistics, where rate of change and trends are analyzed.

Y-intercept

The y-intercept is a specific point where a line crosses the y-axis on a graph. It tells us the value of y when x is zero. In the slope-intercept form equation **y = mx + b**, the y-intercept is represented by **b**.
In our example, the line passes through the point

• (0, 4)

which is located directly on the y-axis. Therefore, the y-intercept is **4**. This means when x equals zero, the value of y is four. This point helps in plotting graphs and understanding the starting value of a line in different contexts, such as calculating initial investments or starting conditions in scientific models.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a line through them. The slope-intercept form equation, **y = mx + b**, simplifies this process by directly providing the slope and y-intercept.
To graph the equation **y = 0.25x + 4**, start by plotting the y-intercept at (0, 4). From here, use the slope **0.25**, which means for each step right by one unit on the x-axis, move up by 0.25 units on the y-axis. Connect these points with a straight line extending in both directions.
This method helps visually interpret linear relationships, making it easier to understand and communicate information effectively across various applications.