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Chapter 3: Problem 65

In the following exercises, graph each equation. $$ 4 x+y=2 $$

Short Answer

Expert verified
Graph the line passing through \((0, 2)\) and \((1, -2)\).

Step by step solution

01

Rewrite the equation in slope-intercept form

Rearrange the given equation to the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Starting with \(4x + y = 2\), subtract \(4x\) from both sides to get \(y = -4x + 2\).
02

Identify the slope and y-intercept

From the equation \(y = -4x + 2\), identify the slope \(m = -4\) and the y-intercept \(b = 2\).
03

Plot the y-intercept

On the graph, plot the point where the line crosses the y-axis. The y-intercept is \(b = 2\), so plot the point \((0, 2)\).
04

Use the slope to find another point

Use the slope \(-4\), which means \(-4/1\), to find another point. Starting from the y-intercept \((0, 2)\), move down 4 units and right 1 unit to get to the point \((1, -2)\).
05

Draw the line

Draw a straight line through the points \((0, 2)\) and \((1, -2)\). This is the graph of the equation \(4x + y = 2\).

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Key Concepts

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

slope-intercept form
To graph a linear equation effectively, it's helpful to convert it into slope-intercept form. This form is written as: \( y = mx + b \) Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. The slope-intercept form makes it straightforward to identify both the slope and the y-intercept, which are essential for plotting the graph of the equation.### How to Convert to Slope-Intercept Form You typically start with an equation not in this form, like \(4x + y = 2\). To transform it:
  • Isolate the \( y \)-term on one side of the equation.
  • Subtract terms involving \( x \) from both sides.
  • The result is \( y = -4x + 2 \), clearly showing the slope \( -4 \) and y-intercept \( 2 \).
Understanding the slope-intercept form simplifies the graphing process and aids in visualizing the relationship between the variables.
y-intercept
The y-intercept of a line is where the line crosses the y-axis. In the equation \( y = -4x + 2 \), the y-intercept has a specific role. It is represented by the constant term \( 2 \) in this case. ### Identifying the Y-Intercept
  • Look at the equation in slope-intercept form, \( y = mx + b \), and find the value of \( b \).
  • For the equation \( y = -4x + 2 \), \( b \) is 2. This means the line crosses the y-axis at (0, 2).
### Plotting the Y-Intercept
  • Start at the origin (0, 0) on the graph.
  • Move up or down to the intercept value 鈥� in this case, up to \( (0, 2)\).
  • Place a point at this location to mark the y-intercept.
This point is crucial as it provides a starting location for drawing your graph. From here, you can use the slope to find additional points.
slope
The slope of a line measures its steepness. It is denoted by \( m \) in the equation \( y = mx + b \). The slope indicates how much the y-value changes for a unit change in the x-value. A positive slope means the line rises as you move right, while a negative slope means it falls. ### Understanding Slope
  • Slope is calculated as \( \frac{\Delta y}{\Delta x} \), the change in y divided by the change in x.
  • In the equation \( y = -4x + 2 \), the slope \( m \) is -4. This means for every 1 unit you move to the right, you move 4 units down.
### Using Slope to Find Points
  • Start from the y-intercept \( (0, 2) \).
  • Move according to the slope. Here, go down 4 units and 1 unit to the right to find the next point, \( (1, -2) \).
After finding two points, you can draw a straight line through them. This line is the graph of the equation. Understanding slope gives you the tools to create and interpret graphs accurately.

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Most popular questions from this chapter

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