Problem 40 Given each set of information, f... [FREE SOLUTION]

Chapter 2: Problem 40

Given each set of information, find a linear equation satisfying the conditions, if possible \(x\) intercept at (-5,0) and \(y\) intercept at (0,4)

Short Answer

Expert verified

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

Step by step solution

Understand the Intercepts

The intercepts given are the points where the line crosses the axes. The \(x\)-intercept is \((-5,0)\), which means that when \(y=0\), \(x=-5\). The \(y\)-intercept is \((0,4)\), which means that when \(x=0\), \(y=4\).

Use Intercepts to Find Slope

The slope \(m\) of a line can be calculated using two points \((x_1, y_1)\) and \((x_2, y_2)\). Here, \((-5,0)\) is \((x_1, y_1)\) and \((0,4)\) is \((x_2, y_2)\). The formula for the slope is \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{0 + 5} = \frac{4}{5}\] Thus, the slope \(m\) is \(\frac{4}{5}\).

Write the Equation Using Slope-Intercept Form

The slope-intercept form of a line's equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. From the intercepts, we know the slope \(m = \frac{4}{5}\) and the \(y\)-intercept \(b=4\). Substituting these values, the equation is \[y = \frac{4}{5}x + 4\]

Verify the Equation

Substitute the intercepts into the equation to check correctness. For \(x\)-intercept \((-5, 0)\), substitute \(-5\) for \(x\): \[0 = \frac{4}{5}(-5) + 4 = -4 + 4 = 0\]For \(y\)-intercept \((0, 4)\), substitute \(0\) for \(x\):\[4 = \frac{4}{5}(0) + 4 = 4\]Both intercept checks hold true with this equation. Thus, the equation is validated.

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Key Concepts

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

The Significance of the X-Intercept

The x-intercept of a line is an important concept in linear equations. It represents the point where the line crosses the x-axis, which means the y-value at this point is zero. Understanding the x-intercept helps in graphing the line on a coordinate plane.
In our example, the x-intercept is at (-5, 0). This tells us that the line cuts through the x-axis at x = -5. Recognizing the x-intercept can also provide a starting point or reference when plotting the line on a graph.

To find the x-intercept, set y to zero and solve for x in the equation of the line.
The x-intercept is always written as (x, 0).

Knowing your intercepts is key when constructing and analyzing linear equations.

Understanding the Y-Intercept

The y-intercept is where the line crosses the y-axis. At this point, the x-value is zero. It's crucial for writing linear equations and identifying graph features.
In our given problem, the y-intercept is (0, 4). This tells us that the line goes through the y-axis at y = 4. This makes it easy to graph the line as it's one of the two key points usually used to define a straight line.

To find the y-intercept, set x to zero in the line's equation.
The y-intercept is noted as (0, y).

Recognizing how the y-intercept works helps in quickly putting together the full graph of the line without needing many points.

Slope Calculation Demystified

Calculating the slope is essential in understanding how steep a line is. The slope, often represented by 'm', can be calculated using two points. Here, it's calculated as the change in y over the change in x between two points on the line.
Using points (-5, 0) and (0, 4) in this example, we determine the slope as follows: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{0 + 5} = \frac{4}{5} \)This means for every 5 units the line moves horizontally, it moves 4 units vertically upward.

A positive slope indicates the line rises as it moves from left to right.
A negative slope indicates it falls as it moves from left to right.

Understanding slope provides insight into the direction and steepness of the line on a graph.

Using the Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is written as \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept. This form simplifies graphing and provides a quick way to write down the equation of a line using known points.
From our example, with a slope \( m = \frac{4}{5} \) and a y-intercept \( b = 4 \), the equation becomes:\[ y = \frac{4}{5}x + 4 \]

This formula directly shows how y increases or decreases based on x.
It's perfect for graphing because you can start plotting from the y-intercept.

The slope-intercept form allows for easy interpretation and straightforward graph plotting, showcasing why it's a favorite among students.

91影视

Given each set of information, find a linear equation satisfying the conditions, if possible \(x\) intercept at (-5,0) and \(y\) intercept at (0,4)

Short Answer

Step by step solution

Understand the Intercepts

Use Intercepts to Find Slope

Write the Equation Using Slope-Intercept Form

Verify the Equation

Key Concepts

The Significance of the X-Intercept

Understanding the Y-Intercept

Slope Calculation Demystified

Using the Slope-Intercept Form

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