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Chapter 5: Problem 62

Find the slope and the \(y\) -intercept of the line given by \(2 y=8 x+7 .[3.6]\)

Short Answer

Expert verified
The slope is 4, and the y-intercept is \(\frac{7}{2}\).

Step by step solution

01

- Rewrite the Equation

Start by rewriting the given equation into the slope-intercept form, which is \(y = mx + b\). The given equation is \(2y = 8x + 7\).
02

- Isolate y

Divide each term in the equation by 2 to solve for \(y\). This gives us \(y = 4x + \frac{7}{2}\).
03

- Identify the Slope

In the slope-intercept form \(y = mx + b\), the coefficient of \(x\) is the slope \(m\). Here, \(m = 4\).
04

- Identify the y-intercept

The constant term \(b\) in the slope-intercept form is the y-intercept. Here, \(b = \frac{7}{2}\).

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Key Concepts

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

Solving Linear Equations
To find the slope and y-intercept of a line, we need to solve linear equations. Linear equations are equations where the highest power of the variable is one. These equations can be written in the form of Ax + By = C. For the exercise given, the equation is 2y = 8x + 7. Our goal is to rearrange this equation into the slope-intercept form. This helps us easily identify the slope (m) and the y-intercept (b).
Slope of a Line
The slope of a line measures its steepness and direction. In the slope-intercept form of a linear equation, y = mx + b, the slope is represented by the coefficient 'm'. For the equation 2y = 8x + 7, we first rewrite it as y = 4x + 7/2, making it clear that the slope (m) is 4. Here are the steps:
  • Rewrite the given equation in slope-intercept form
  • Identify the coefficient of x, which is your slope
The slope tells us that for every one unit increase in x, y increases by 4 units.
y-intercept
The y-intercept is the value of y when x is zero. It is represented by the constant term 'b' in the slope-intercept form y = mx + b. For our exercise, when we solved the equation to get y = 4x + 7/2, we found that the y-intercept (b) is 7/2. This means the line crosses the y-axis at 3.5. To find the y-intercept, follow these steps:
  • Rewrite the equation in slope-intercept form
  • Identify the constant term
The y-intercept is crucial as it shows where the line starts vertically on a graph.

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