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Chapter 2: Problem 30

Graph each equation. \(y=5-x\)

Short Answer

Expert verified
Plot points (0, 5) and (1, 4), then draw a line through them.

Step by step solution

01

Understand the Equation Type

The given equation is linear, as it represents a straight line. In standard form, a line is often expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify Slope and Intercept

In the equation \( y = 5 - x \), we can rewrite it as \( y = -x + 5 \), where \( m = -1 \) is the slope and \( b = 5 \) is the y-intercept. This tells us the line slopes downwards and crosses the y-axis at 5.
03

Plot the Y-Intercept

Start plotting by marking the y-intercept on the graph. For the equation \( y = 5 - x \), the y-intercept is 5. Place a point at (0, 5) on the graph.
04

Use the Slope to Determine Another Point

Use the slope \( m = -1 \) to determine another point. Starting from (0, 5), move 1 unit down (since the slope is -1) and 1 unit to the right, landing at (1, 4). Plot this point.
05

Draw the Line

Connect the two plotted points (0, 5) and (1, 4) with a straight line, extending it in both directions, as the line continues infinitely.

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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 of a linear equation is an incredibly useful format for understanding linear functions and graphing them. In this form, equations look like this:
  • \( y = mx + b \)
Here, \(m\) is the slope, and \(b\) is the y-intercept. This form is popular because it directly tells us how the line behaves on a graph.
The slope \(m\) measures how steep the line is鈥攈ow much \(y\) changes for every one unit that \(x\) increases. A positive slope means the line ascends from left to right, while a negative slope means it descends.
On the other hand, the y-intercept \(b\) is the point where the line crosses the y-axis. This ground-level understanding aids in quickly graphing the function.
Graphing Linear Functions
Graphing linear functions becomes a breeze once you understand the slope-intercept form. Begin by identifying the y-intercept \(b\). Plot it directly on the y-axis, which gives you your starting point on the graph.
Next, use the slope \(m\) to identify another point. If the slope is a fraction, like \(\frac{2}{3}\), it helps you know how to move to the next point. You would go up 2 units if positive (or down if negative) and right 3 units.
  • For example, with a slope of \(-1\), as in our equation \(y = 5 - x\), start at the y-intercept, (0, 5).
  • Move down 1 unit (since the slope is \(-1\)) and right 1 unit, arriving at (1, 4).
Once you have at least two points, draw a line through them. Remember, this line should extend infinitely in both directions.
Y-Intercept
The y-intercept is a significant feature of any linear equation. It provides a straightforward point where the line crosses the y-axis, shown as \( (0, b) \). This intercept tells us exactly where a line starts on the y-axis and is crucial for quickly plotting the first point of the line on a graph.
When dealing with the equation \(y = 5 - x\), the y-intercept \(b\) is 5, meaning the line crosses the y-axis at (0, 5).
Utilizing the y-intercept efficiently, you can tackle more complex equations and confidently start plotting or sketching the graph without delay.
  • It acts as a foundation or anchor point when drawing a graph.
  • Combined with the slope, it gives a clear picture of the line's direction and position on a plane.

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