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Chapter 0: Problem 20

Find the slope and \(y\) -intercept. $$y-4 x=1$$

Short Answer

Expert verified
The slope is 4 and the y-intercept is 1.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Rewrite the given equation to match this form: \( y - 4x = 1 \) Add \( 4x \) to both sides of the equation to isolate \( y \): \( y = 4x + 1 \).
02

Identify the Slope

Compare the rewritten equation \( y = 4x + 1 \) to the slope-intercept form \( y = mx + b \). The coefficient of \( x \) is the slope. In this case, the slope \( m \) is 4.
03

Identify the Y-Intercept

In the slope-intercept form \( y = mx + b \), the constant term \( b \) is the y-intercept. Here, the y-intercept \( b \) is 1.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is an essential concept in linear equations. It is represented as \( y = mx + b \), where:
  • \( y \): the dependent variable (usually considered as the output or result)
  • \( x \): the independent variable (usually considered as the input)
  • \( m \): the slope of the line, showing the rate of change of \( y \) with respect to \( x \)
  • \( b \): the y-intercept, the value where the line crosses the y-axis
In the given problem, we need to convert the equation \( y - 4x = 1 \) into this form. By adding \( 4x \) to both sides of the equation, we get \( y = 4x + 1 \).
The equation is now in slope-intercept form.
Slope Identification
Once we have our equation in the slope-intercept form \( y = mx + b \), the next step is to identify the slope.
The slope \( m \) is the coefficient of the \( x \) term. In this equation, \( y = 4x + 1 \), the coefficient of \( x \) is 4. Hence, the slope \( m \) is 4.
This means for every 1 unit increase in \( x \), the value of \( y \) increases by 4 units. The slope determines the steepness and direction of the line:
  • A positive slope (like 4) means the line rises as it moves from left to right.
  • A negative slope would mean the line falls as it moves from left to right.
Y-Intercept
The y-intercept is another crucial concept in understanding linear equations. It is represented by \( b \) in the slope-intercept form \( y = mx + b \).
The y-intercept is the point where the line crosses the y-axis, which means it occurs when \( x = 0 \). In our equation \( y = 4x + 1 \), the y-intercept \( b \) is 1.
This indicates that when \( x \) is 0, \( y \) equals 1, so the line intersects the y-axis at the point (0, 1). The y-intercept provides a starting value for the line on the graph, giving a clear point to begin plotting.
  • The y-intercept is always the constant term in the equation.
Understanding the y-intercept helps in graphing and visualizing linear relationships.

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