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Chapter 10: Problem 31

In Exercises 29 to \(34,\) draw the line described. With \(y\) intercept 5 and with \(m=-\frac{3}{4}\)

Short Answer

Expert verified
Line has equation \( y = -\frac{3}{4}x + 5 \) and passes through \( (0, 5) \) and \( (4, 2) \).

Step by step solution

01

Understand the Line Equation

The equation of a line in slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Here, you are given \( m = -\frac{3}{4} \) and \( b = 5 \).
02

Write the Equation of the Line

Substitute the given slope \( m = -\frac{3}{4} \) and \( y \)-intercept \( b = 5 \) into the equation \( y = mx + b \). This gives \( y = -\frac{3}{4}x + 5 \).
03

Plot the Y-Intercept

The \( y \)-intercept is the point where the line crosses the \( y \)-axis. Plot the point \( (0, 5) \) on the graph.
04

Use the Slope to Find Another Point

The slope \( m = -\frac{3}{4} \) means that for every 4 units you move horizontally to the right, you move 3 units down vertically. From the point \( (0, 5) \), move 4 units right to \( (4, 0) \) and 3 units down to \( (4, 2) \). Plot \( (4, 2) \).
05

Draw the Line

Using a ruler, draw a straight line passing through the points \( (0, 5) \) and \( (4, 2) \). This line represents the equation \( y = -\frac{3}{4}x + 5 \).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a way to visually represent solutions to equations on a coordinate plane. Linear equations can be expressed in various forms, but the slope-intercept form is among the simplest and most useful for graphing. This form is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The equation represents a straight line, and to graph it, you need at least two points.
  • Start by identifying the y-intercept, which is the point where the line crosses the y-axis.
  • Use the slope to find another point on the line.
  • Connect these points with a straight line using a ruler or a straight edge.
Breaking down the process into these steps makes it easier to understand how changes in the equation affect the graph. Moving beyond static forms, graphing empowers you to visualize relationships between variables in real-world contexts.
Slope
The slope of a line reflects the angle and direction of the line in a coordinate plane. It is represented by \( m \) in the equation \( y = mx + b \). The slope is calculated as the "rise over run," indicating how much the \( y \) value changes for a given change in the \( x \) value. Here's what you need to know:
  • A positive slope means the line rises as it moves from left to right, indicating an increasing relationship.
  • A negative slope means the line falls as it moves from left to right, indicating a decreasing relationship.
  • A zero slope denotes a horizontal line, showing no change in \( y \) for any \( x \).
  • An undefined slope indicates a vertical line, where \( x \) is constant.
For example, in our case, the slope \( m = -\frac{3}{4} \) indicates the line moves 3 units down for every 4 units it moves to the right. This helps in plotting the line accurately, guiding how you move from the y-intercept to another point on the line.
Y-Intercept
The y-intercept of a line is the point at which the line crosses the y-axis. In the equation \( y = mx + b \), the y-intercept is \( b \). It is always given by the point \((0, b)\) because the x-coordinate is zero at the y-axis.
  • The y-intercept provides a starting point for graphing a line.
  • It represents the value of \( y \) when \( x = 0 \).
  • In contexts involving time or distance, the y-intercept often shows the initial value or starting condition.
Using the y-intercept, like \( b = 5 \) in our example, we can begin the graphing process. Plotting \((0, 5)\) gives you a clear anchor on the plane. From here, you apply the slope to find the line's path, ensuring accurate representation of the linear equation.

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