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Chapter 7: Problem 10

State the slope and the \(y\) -intercept of the graph of each equation. $$y=2 x-4$$

Short Answer

Expert verified
The slope is 2, and the y-intercept is -4.

Step by step solution

01

Identify the Structure of the Equation

The given equation is in the slope-intercept form, which is \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the \(y\)-intercept.
02

Determine the Slope

In the equation \(y = 2x - 4\), compare it with the slope-intercept form \(y = mx + b\). Notice that \(m = 2\), thus the slope is 2.
03

Determine the Y-Intercept

Continuing from Step 2, compare \(b\) in the equation \(y = 2x - 4\) to the standard form. Here, \(b = -4\). So, the \(y\)-intercept is \(-4\).

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Key Concepts

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

Understanding Linear Equations
Linear equations are a cornerstone of algebra and appear frequently in various subjects. At their core, these equations model relationships with a constant rate of change, manifesting as straight lines when graphed.
The most common form of a linear equation is the slope-intercept form, represented as \(y = mx + b\). Here, \(m\) represents the slope of the line, telling us how steep the line is. \(b\) serves as the y-intercept, the point where the line crosses the y-axis.
Key features are:
  • y: The dependent variable or output.
  • x: The independent variable or input.
  • m: The slope, indicating how much \(y\) changes for a change in \(x\).
  • b: The y-intercept, where the line hits the y-axis.
Mastering these basics allows you to analyze and graph linear relationships efficiently.
Breaking Down Slope Calculation
The slope of a line is pivotal, determining its direction and steepness. It is denoted as \(m\) in the equation \(y = mx + b\), reflecting how much \(y\) changes in response to \(x\) changes.In our example, \(y = 2x - 4\), \(m = 2\). This tells us for every 1 unit increase in \(x\), \(y\) increases by 2 units. Broadly:
  • A positive slope indicates an upward trend, where values increase together.
  • A negative slope means a downward trend, showing an inverse relationship.
  • A slope of zero leads to a horizontal line, showing no change in \(y\) when \(x\) changes.
Visualizing slope helps you predict behaviors and trends in data, understanding both increase and decrease rates.
How to Identify the Y-Intercept
The y-intercept plays a special role in linear equations, marked as \(b\) in \(y = mx + b\). It signifies the point where the line intersects the y-axis, essentially the starting value of \(y\) when \(x\) is zero. In \(y = 2x - 4\), the y-intercept is \(-4\) (as \(b = -4\)). This tells us that the line crosses the y-axis at the point (0, -4). The steps to locate it are straightforward:
  • Set \(x = 0\) in the equation.
  • The resulting \(y\) value is the y-intercept.
Understanding the y-intercept helps in sketching graphs quickly, setting up initial points from which the line extends.

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Most popular questions from this chapter

Graph each equation using the slope and \(y\) -intercept. $$y=-3$$

Subtract. $$-26-(-26)$$

Write each number in scientific notation. $$.1,680,000$$

State the slope and the \(y\) -intercept of the graph of each equation. $$5 x+4 y=20$$

CHALLENGE What is the \(x\) -intercept of the graph of \(y=m x+b ?\) Explain how you know.
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