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Chapter 3: Problem 2

The graph of the linear equation \(y=m x+b\) has _______ \((0, b)\) and _________ m$

Short Answer

Expert verified
The graph intersects the y-axis at \((0, b)\) and has slope \(m\).

Step by step solution

01

Identify the Intercept

In the linear equation \(y = mx + b\), the term \(b\) indicates the y-intercept. This is the point where the line intersects the y-axis. Therefore, the graph intersects the y-axis at the point \((0, b)\).
02

Understand the Slope

In the equation \(y = mx + b\), the term \(m\) represents the slope of the line. The slope is a measure of how steep the line is and is calculated as the change in y over the change in x \(\left(\frac{\Delta y}{\Delta x}\right)\).

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Key Concepts

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

Y-intercept
The y-intercept is one of the most crucial elements when graphing a linear equation. In the context of a linear equation in the form \(y = mx + b\), the y-intercept is the value of \(y\) when \(x\) is zero. It is represented by the term \(b\) in the equation.

To find the y-intercept, simply look at your linear equation and find the \(b\) value. This point is where the line crosses the y-axis, hence the name "y-intercept."
  • The y-intercept gives you a starting point for drawing the line on the graph.
  • Its coordinates will always be \((0, b)\), where \(b\) is the value of the y-intercept.
  • Different values of \(b\) will move the line up and down along the y-axis.
Understanding the y-intercept helps you visualize where your line will start on the y-axis before considering the slope to complete the graph.
Slope
The slope of a line is a fundamental aspect that describes its angle and steepness. It is identified by the letter \(m\) in the linear equation \(y = mx + b\). The slope essentially tells us how much \(y\) changes for a unit change in \(x\).

The calculation of the slope is done using the formula \(\frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the change in the \(y\)-coordinates, and \(\Delta x\) is the change in the \(x\)-coordinates. This is often referred to as "rise over run."
  • A positive slope means the line inclines and moves upwards from left to right.
  • A negative slope means the line declines and moves downwards from left to right.
  • A slope of zero implies a horizontal line.
  • An undefined slope means a vertical line and it cannot be expressed as a fraction.
Recognizing the slope of a line provides insight into its direction and gradient, allowing accurate prediction of how the line progresses across the graph.
Graph of a line
The graph of a line is the visual representation of the linear equation \(y = mx + b\). Understanding the graph helps visualize how the line behaves and interacts with the coordinate axes.

To draw the graph:
  • Start with plotting the y-intercept, \((0, b)\), on the coordinate plane.
  • Use the slope \(m\) to determine the next points. From the y-intercept, move vertically by the "rise" (\(\Delta y\)) and horizontally by the "run" (\(\Delta x\)) to plot the next point.
  • Connect these points with a straight line extending in both directions.
The graph of a line gives a clear depiction of its slope and y-intercept and helps in understanding the relationship between \(x\) and \(y\). With each line uniquely defined by a different equation, it becomes straightforward to identify various lines based on their graphical plots or equations.

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Most popular questions from this chapter

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((2,4)\) and \((-1,-1)\) \((8,0)\) and \((11,5)\)

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ y=-4 x $$

A clothing store advertises that it maintains an inventory of at least \(\$ 4,400\) worth of men's jackets at all times. At the store, leather jackets cost \(\$ 100\) and nylon jackets cost S88. If \(x\) represents the number of leather jackets in stock and \(y\) represents the number of nylon jackets in stock, the graph of \(100 x+88 y \geq 4,400\) shows the possible ways the jackets can be stocked. Graph the inequality. Then find three possible combinations of leather and nylon jackets so that the store lives up to its advertising claim.

Find the slope of each line. See Examples 4 and \(5 .\) $$3 x=-12$$

If we know two points that a line passes through, we can write its equation. Explain how this is done.
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