Problem 34 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. \(x\) -intercept \(\frac{1}{3}, y-\) intercept \(-\frac{1}{4}\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{3}{4}x - \frac{1}{4} \).

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 of the line and \( b \) is the y-intercept.

Step 2: Determine the Slope of the Line

To calculate the slope \( m \), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The x-intercept indicates a point (\( \frac{1}{3}, 0 \)) and the y-intercept indicates a point (\( 0, -\frac{1}{4} \)). Thus, \( m = \frac{0 - (-\frac{1}{4})}{\frac{1}{3} - 0} = \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4} \).

Step 3: Write the Slope-Intercept Equation

Using the slope \( m = \frac{3}{4} \) and the y-intercept \( b = -\frac{1}{4} \), substitute into the slope-intercept form: \( y = \frac{3}{4}x - \frac{1}{4} \).

Key Concepts

Linear Equations

A linear equation is a mathematical statement that describes a straight line when it is graphed on a coordinate plane. These equations are named 'linear' because they produce a straight line when plotted. The simplest form of a linear equation is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept.

### Slope and Intercept
- **Slope (\( m \))**: This defines how steep the line is. A larger slope means a steeper line. If the slope is positive, the line ascends from left to right, whereas a negative slope means it descends.- **Y-intercept (\( b \))**: The y-intercept is the point at which the line crosses the y-axis. It indicates the value of \( y \) when \( x \) is zero.

Linear equations are foundational in understanding algebra and are widely used to model real-world situations where one variable depends on another.

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of \( y \) is zero. For example, if we have a line with an x-intercept of \( \frac{1}{3} \), it crosses the x-axis at the point \( (\frac{1}{3}, 0) \).

The x-intercept provides valuable information about the behavior of the line. It tells us the x-value when the output, represented by \( y \), equals zero. In practical terms, this might represent the initial condition in a real-world scenario, such as the starting quantity before any changes.
### Determining the X-InterceptTo find the x-intercept for any linear equation, set \( y = 0 \) and solve for \( x \). For example, in the equation \( y = \frac{3}{4}x - \frac{1}{4} \):

• Set \( y \) to 0: \( 0 = \frac{3}{4}x - \frac{1}{4} \)
• Solving gives \( \frac{3}{4}x = \frac{1}{4} \)
• So, \( x = \frac{1}{3} \)

Understanding the x-intercept can help in predicting behavior of the equation when graphed.

Y-Intercept

The y-intercept is where the line hits the y-axis. It indicates the value of \( y \) when \( x \) is zero. In our context, if the y-intercept is \(-\frac{1}{4}\), it means the line crosses the y-axis at \((0, -\frac{1}{4})\). This point tells us the starting value of \( y \) in the line equation.

### Utilizing the Y-Intercept
- **Starting Point**: In many real-life situations, the y-intercept represents an initial condition or starting point before any other factors come into play.- **Equation Representation**: It directly represents \( b \) in the slope-intercept form \( y = mx + b \). Thus, when given a y-intercept, one can plug it directly into the equation to represent the constant value of \( y \) when \( x \) is zero.

In our exercise, with the y-intercept of \(-\frac{1}{4}\), we can be sure that when plotting, the line will precisely cross the y-axis at this value, offering a clear and accurate starting point for analyzing the line's behavior on a graph.