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Chapter 4: Problem 120

In the following exercises, graph each equation. \(y=2 x\)

Short Answer

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

Step by step solution

01

Understand the Equation

The given equation is in the form of a linear equation, which means it represents a straight line. The general form of a linear equation is given by: \[ y = mx + b \]Here, \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the Slope and Y-intercept

In the equation \( y = 2x \), compare it with the general form \( y = mx + b \). It is clear that the slope \( m \) is 2 and the y-intercept \( b \) is 0. This means the line passes through the origin (0,0).
03

Plot the Y-intercept

Start by plotting the y-intercept on the graph. Since the y-intercept \( b \) is 0, place a point at the origin (0,0).
04

Use the Slope to Find Another Point

The slope \( m \) tells us how much y increases when x increases by 1. Since \( m = 2 \), for every 1 unit increase in x, y increases by 2 units. From the origin, move 1 unit to the right (x = 1) and 2 units up (y = 2). Plot this point (1,2) on the graph.
05

Draw the Line

Once you have at least two points (the origin (0,0) and (1,2)), draw a straight line through these points. Extend the line in both directions to complete the graph.

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Key Concepts

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

linear equations
Linear equations are a type of equation that represents a straight line on a graph. They generally have the form: y = mx + b . The letters in this equation stand for:
  • y : The dependent variable.
  • x : The independent variable.
  • m : The slope of the line.
  • b : The y-intercept of the line.
      • This form tells us everything we need to know to draw the line on a coordinate plane. By identifying the slope and the y-intercept, we can graph the equation quickly and easily.
slope
The slope of a line basically tells us how steep the line is. In the equation y = mx + b , the m dictates the steepness. The slope is calculated as the change in y over the change in x, or rise over run. For instance, in the given equation y = 2x, the slope m is 2. This means that for every unit increase in x, y increases by 2 units. To visualize this:
  • From point (0,0), move 1 unit right.
  • Then, move 2 units up.
This shows the change from (0,0) to (1,2), illustrating a slope of 2.
y-intercept
The y-intercept is the point where the line crosses the y-axis. In the equation y = 2x , the y-intercept term is missing because it is zero. When b is zero, our line passes through the origin (0,0). To plot the y-intercept:
  • Find the y-axis on the graph.
  • Place a point at y = 0, which is the origin.
With the y-intercept marked, you have a starting point for plotting the entire line.
coordinate plane
The coordinate plane consists of two perpendicular lines, the x-axis and the y-axis. These axes divide the plane into four quadrants. Here's how you use the coordinate plane to graph linear equations:
  • The x-axis is the horizontal line with values increasing to the right and decreasing to the left.
  • The y-axis is the vertical line with values increasing upwards and decreasing downwards.
To graph the equation y = 2x :
  • Start plotting at the y-intercept, here it is (0,0).
  • Utilize the slope to determine the next point, moving as the slope directs (1 unit right and 2 units up).
  • Mark at least two points and draw a straight line through them.
You can extend this line in both directions to complete the graph.

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