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Chapter 3: Problem 37

Write an equation in slope-intercept form for the line that satisfies the given conditions. \(x\) -intercept \(=5, y\) -intercept \(=3\)

Short Answer

Expert verified
The equation in slope-intercept form is \( y = -\frac{3}{5}x + 3 \).

Step by step solution

01

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.
02

Identify the Y-intercept

From the problem, the y-intercept is given as 3. This tells us that \( b = 3 \).
03

Use the X-intercept to Find the Slope

The x-intercept is given as 5, which means the point (5, 0) is on the line. We can use this point along with the y-intercept (0, 3) to find the slope. The formula for the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the points (5, 0) and (0, 3) gives \( m = \frac{0 - 3}{5 - 0} = -\frac{3}{5} \).
04

Write the Equation

Now that we have both the slope \( m = -\frac{3}{5} \) and the y-intercept \( b = 3 \), we can write the equation in slope-intercept form as \( y = -\frac{3}{5}x + 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equation
A linear equation is an equation that describes a straight line on a graph. The most common form of a linear equation is called the slope-intercept form, which looks like this:
  • \( y = mx + b \)
Here, \( y \) represents the dependent variable, usually plotted on the vertical axis, while \( x \) is the independent variable, usually on the horizontal axis. The letters \( m \) and \( b \) stand for two critical components of the line:
  • \( m \) is the slope, showing how steep the line is.
  • \( b \) is the y-intercept, indicating the point where the line crosses the y-axis.
Linear equations are fundamental in math because they model relationships where one quantity depends directly on another. So, understanding them is essential in algebra and beyond.
Y-Intercept
The y-intercept is a significant concept when dealing with graphs of linear equations. It's the point where the line crosses the y-axis, and it occurs when the value of \( x \) is zero.Here's what you need to know:
  • In the slope-intercept form \( y = mx + b \), the y-intercept is the \( b \) value.
  • If a line passes through the y-axis at 3, then \( b = 3 \).
Spotting the y-intercept on a graph is straightforward. It's where the line hits the vertical axis, making it simpler to understand how the line behaves when x equals zero. It's also a vital starting point when drawing or thinking about the line's equation.
X-Intercept
The x-intercept is another important aspect of linear equations. It is where the line crosses the x-axis. Unlike the y-intercept, the x-intercept occurs when \( y \) is zero.Some important notes about x-intercepts include:
  • For a line equation \( y = mx + b \), you find the x-intercept by setting \( y = 0 \) and solving for \( x \).
  • In a given problem where the x-intercept is 5, the line crosses the x-axis at (5, 0).
Finding the x-intercept is crucial when solving problems because it helps in plotting the line accurately on a graph and understanding another fixed point the line passes through.
Slope of a Line
The slope of a line is a measure of its steepness and direction. Calculating the slope is essential for understanding how quickly the y-values change compared to changes in x-values.Here's how it works:
  • The slope \( m \) is found using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  • This formula takes two points on the line, say (\( x_1, y_1 \)) and (\( x_2, y_2 \)), and calculates the change in \( y \) divided by the change in \( x \).
  • In our example, using the points (5, 0) and (0, 3), we find \( m = \frac{0 - 3}{5 - 0} = -\frac{3}{5} \).
The slope tells us two things: how steep the line is (numerically) and whether it鈥檚 going upward or downward (by its sign). A positive slope means the line climbs as it moves from left to right, while a negative slope signals a descent.

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