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Chapter 0: Problem 38

Write the equation in slope-intercept form. Identify the slope and the \(y\) -intercept. $$4=2 x-8 y$$

Short Answer

Expert verified
The slope is \(\frac{1}{4}\) and the y-intercept is \(-\frac{1}{2}\).

Step by step solution

01

Rewrite in Standard Form

Start with the given equation: \(4 = 2x - 8y\). This equation is already close to the standard form \(Ax + By = C\), but we need it in \(y\) form.
02

Rearrange for y

Rearrange the equation to isolate \(y\) on one side. Start by adding \(8y\) to both sides to get: \(4 + 8y = 2x\).
03

Isolate y

Subtract \(4\) from both sides to further isolate \(y\): \(8y = 2x - 4\).
04

Solve for y

Divide each term by 8 to solve for \(y\): \(y = \frac{2}{8}x - \frac{4}{8}\).
05

Simplify the Equation

Simplify the fractions: \(y = \frac{1}{4}x - \frac{1}{2}\). This is the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
06

Identify Slope and y-intercept

From the equation \(y = \frac{1}{4}x - \frac{1}{2}\), identify the slope \(m = \frac{1}{4}\) and the y-intercept \(b = -\frac{1}{2}\).

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Key Concepts

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

Equation Manipulation
To transform an equation into the slope-intercept form, which is expressed as \(y = mx + b\), we often need to perform various equation manipulations. The goal is to isolate \(y\) on one side of the equation. Here, we'll guide you through this **step-by-step process**:
  • First, start with the given equation and decide what needs to be moved to the other side. In our example, \(4 = 2x - 8y\), we need to move all terms except those involving \(y\) to the opposite side.
  • To move terms around, use basic algebraic techniques like adding, subtracting, multiplying, or dividing both sides of the equation equally. This guarantees that the equation remains balanced.
  • When rearranging the equation to isolate \(y\), pay attention to signs and coefficients. It鈥檚 crucial to perform the same operation on every term to maintain equality.
  • Lastly, simplify your results by reducing fractions or combining like terms if needed.
Performing these manipulations correctly sets up equations in a suitable form for further analysis, such as identifying slopes and intercepts.
Slope
The slope of a line in the equation \(y = mx + b\) is represented by \(m\). It indicates how steep the line is, showing the ratio of the rise over the run between two points on the line. Here鈥檚 a better look at **how you can understand slope**:
  • Mathematically, slope is calculated by the change in \(y\) divided by the change in \(x\). Written explicitly as \(m = \frac{\Delta y}{\Delta x}\).
  • This ratio determines the slant of the line: a larger absolute value of \(m\) means a steeper slope.
  • If \(m\) is positive, the line rises as you move from left to right. Conversely, if \(m\) is negative, the line falls.
  • In our exercise, the slope identified is \(\frac{1}{4}\). This means for every 4 units you move horizontally, the line moves 1 unit vertically.
Understanding slope is crucial for interpreting graphs and predicting the behavior of linear relationships in various contexts.
Y-Intercept
In the equation \(y = mx + b\), the \(y\)-intercept \(b\) is the point where the line crosses the \(y\)-axis. Here鈥檚 how it works and why it's important:
  • When \(x\) is zero, the term \(mx\) disappears, and what remains is \(y = b\). This makes \(b\) the \(y\)-coordinate of where the line intersects the \(y\)-axis.
  • The \(y\)-intercept provides a starting point for plotting the line and helps in quickly sketching the graph when combined with the slope.
  • In our given equation, the \(y\)-intercept is \(-\frac{1}{2}\). This tells us the line meets the \(y\)-axis at the point \((0, -\frac{1}{2})\).
Recognizing the \(y\)-intercept allows for easy visualization of linear equations and plays a vital role in graphing and understanding linear functions.

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Most popular questions from this chapter

Rick and Mike are roommates and leave Gainesville on Interstate 75 at the same time to visit their girlfriends for a long weekend. Rick travels north and Mike travels south. If Mike's average speed is 8 mph faster than Rick's, find the speed of each if they are 210 miles apart in 1 hour and 30 minutes.

A speedboat takes 1 hour longer to go 24 miles up a river than to return. If the boat cruises at 10 mph in still water, what is the rate of the current?

Find the equation of the line that passes through the given point and also satisfies the additional piece of information. Express your answer in slope- intercept form, if possible. (3,5)\(;\) parallel to the \(x\) -axis

Determine whether each statement is true or false. If a line has slope equal to zero, describe a line that is perpendicular to it.

Graph the function represented by each side of the equation in the same viewing rectangle and solve for \(x\) $$2 x+6=4 x-2 x+8-2$$
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