Problem 38
Question
A general linear equation of a line is given. Find the \(x\) -intercept, the \(y\) -intercept, and the slope of the line. $$ x / 2+2 y=4 $$
Step-by-Step Solution
Verified Answer
The line has a slope of -1/4, a y-intercept of 2, and an x-intercept of 8.
1Step 1: Convert to Slope-Intercept Form
The equation is given as \( \frac{x}{2} + 2y = 4 \). To find the slope and intercepts, first convert this equation to the slope-intercept form \( y = mx + b \). Start by solving for \( y \). First, subtract \( \frac{x}{2} \) from both sides: \[ 2y = -\frac{x}{2} + 4 \]Next, divide all terms by 2:\[ y = -\frac{1}{4}x + 2 \]
2Step 2: Identify the Slope
In the slope-intercept form \( y = mx + b \), \( m \) represents the slope. From Step 1, the equation of the line is:\[ y = -\frac{1}{4}x + 2 \]Thus, the slope \( m \) is \(-\frac{1}{4}\).
3Step 3: Determine the y-intercept
In the equation \( y = -\frac{1}{4}x + 2 \), \( b \) is the y-intercept. From the equation, we see:\[ b = 2 \]So, the \( y \)-intercept is 2, meaning the line crosses the y-axis at the point (0, 2).
4Step 4: Find the x-intercept
To find the x-intercept, set \( y = 0 \) in the original equation and solve for \( x \):\[ \frac{x}{2} + 2(0) = 4 \]\[ \frac{x}{2} = 4 \]Multiply both sides by 2:\[ x = 8 \]Thus, the x-intercept is the point (8, 0).
Key Concepts
Understanding the x-interceptExploring the y-interceptWhat is the slope of a line?
Understanding the x-intercept
In a linear equation, the x-intercept is the point where the line crosses the x-axis. At this specific point, the value of y is zero. To find the x-intercept from a given equation, you simply set the y value to zero and solve for x.
In our example, the equation is \( \frac{x}{2} + 2y = 4 \). To find the x-intercept, we substitute \( y = 0 \) into the equation:
\[ \frac{x}{2} + 2(0) = 4 \]
This simplifies to \( \frac{x}{2} = 4 \).
By multiplying both sides by 2, we find that \( x = 8 \). Thus, the x-intercept is at the point \( (8, 0) \).
In our example, the equation is \( \frac{x}{2} + 2y = 4 \). To find the x-intercept, we substitute \( y = 0 \) into the equation:
\[ \frac{x}{2} + 2(0) = 4 \]
This simplifies to \( \frac{x}{2} = 4 \).
By multiplying both sides by 2, we find that \( x = 8 \). Thus, the x-intercept is at the point \( (8, 0) \).
- The x-intercept is helpful in graphing as it provides a clear point through which the line must pass.
- Identifying where the line crosses the axes can simplify understanding the behavior of the line on the coordinate plane.
Exploring the y-intercept
The y-intercept is where a line crosses the y-axis. At this point, the x value is zero. In the slope-intercept form of a linear equation, \( y = mx + b \), \( b \) is the y-intercept.
For the given line \( y = -\frac{1}{4}x + 2 \), the y-intercept \( b \) is 2.
This tells us that the line crosses the y-axis at point \( (0, 2) \).
Remember, when a line crosses the y-axis, it provides a starting point to graph the rest of the line. Since the x-coordinate is zero at the y-intercept, this point serves as a reference to locate the entire line on a graph.
For the given line \( y = -\frac{1}{4}x + 2 \), the y-intercept \( b \) is 2.
This tells us that the line crosses the y-axis at point \( (0, 2) \).
Remember, when a line crosses the y-axis, it provides a starting point to graph the rest of the line. Since the x-coordinate is zero at the y-intercept, this point serves as a reference to locate the entire line on a graph.
- Knowing the y-intercept allows you to quickly plot one point on the graph.
- It's crucial for verifying your calculations are correct when shifting between different equation forms.
What is the slope of a line?
The slope of a line measures its steepness and direction, defined by the rise over the run between any two points on the line. In the slope-intercept form \( y = mx + b \), \( m \) is the slope.
In our example, the equation \( y = -\frac{1}{4}x + 2 \) indicates the slope \( m \) is \(-\frac{1}{4}\).
This negative slope shows the line decreases as it moves from left to right.
In our example, the equation \( y = -\frac{1}{4}x + 2 \) indicates the slope \( m \) is \(-\frac{1}{4}\).
This negative slope shows the line decreases as it moves from left to right.
- A slope of \(-\frac{1}{4}\) means for every four units the line moves horizontally, it will drop one unit vertically.
- The slope is a key factor in determining the angle of a line relative to the x-axis.
- It helps predict how changes in x will affect y, which is useful in countless real-world applications such as calculating rates of change.
Other exercises in this chapter
Problem 38
Use one or more of the basic trigonometric identities to derive the given identity. \(\sin (\theta) \sin (\phi)=\frac{\cos (\theta-\phi)-\cos (\theta+\phi)}{2}\
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Sketch the given region. \(\\{(x, y):|x| \leq 5,|y|>2\\}\)
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Give a recursive definition of the sequence. $$ \left.f_{n}=2^{((-1) n}\right), n=1,2,3, \ldots $$
View solution Problem 38
Sketch the set on a real number line. \(\left\\{t:\left|t^{2}+6 t\right| \leq 10\right\\}\)
View solution