How To Graph The Equations Y = 2x + 4 And Y = -1/6x - 1/3 By Plotting Two Points With Integer Coordinates?

#title: Graphing Linear Equations by Plotting Two Points with Integer Coordinates

In mathematics, a linear equation represents a straight line on a graph. Graphing linear equations is a fundamental skill, and one common method involves plotting two points with integer coordinates. This approach is particularly useful because it provides a clear and straightforward way to visualize the line represented by the equation. In this article, we will delve into the process of graphing two linear equations by plotting two points with integer coordinates. We will explore the steps involved, the underlying principles, and some helpful tips to ensure accuracy and understanding. By mastering this technique, you'll gain a solid foundation for working with linear equations and their graphical representations.

A) ${ y = 2x + 4 }$

Understanding the Equation

To effectively graph the equation ${ y = 2x + 4 }$, we first need to understand its components. This is a linear equation in slope-intercept form, which is generally written as ${ y = mx + b }$. Here,

• ${ y }$ is the dependent variable, representing the vertical coordinate.
• ${ x }$ is the independent variable, representing the horizontal coordinate.
• ${ m }$ is the slope of the line, indicating its steepness and direction.
• ${ b }$ is the y-intercept, the point where the line crosses the y-axis.

In our equation, ${ y = 2x + 4 }$, the slope (${ m }$) is 2, and the y-intercept (${ b }$) is 4. This means that for every 1 unit increase in ${ x }$, ${ y }$ increases by 2 units, and the line intersects the y-axis at the point (0, 4). Understanding the slope and y-intercept gives us valuable insights into the behavior of the line and helps in plotting it accurately.

Finding Two Integer Points

To graph a line, we only need two points. The key is to find points with integer coordinates, as these are easier to plot accurately on a graph. To find these points, we can choose two convenient values for ${ x }$ and then calculate the corresponding values for ${ y }$. Let's choose ${ x = 0 }$ and ${ x = -2 }$ as our two values.

1. When ${ x = 0 }$: Substitute ${ x = 0 }$ into the equation: ${ y = 2(0) + 4 = 4 }$ So, the first point is (0, 4). This is also the y-intercept, which we already identified.
2. When ${ x = -2 }$: Substitute ${ x = -2 }$ into the equation: ${ y = 2(-2) + 4 = -4 + 4 = 0 }$ So, the second point is (-2, 0). This is the x-intercept, where the line crosses the x-axis.

By selecting these values for ${ x }$, we've obtained two points with integer coordinates: (0, 4) and (-2, 0). These points are easy to plot on a graph and will allow us to draw the line accurately.

Plotting the Points and Drawing the Line

Now that we have our two points, (0, 4) and (-2, 0), we can plot them on a coordinate plane. The coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). The point (0, 4) is located on the y-axis, 4 units above the origin (0, 0). The point (-2, 0) is located on the x-axis, 2 units to the left of the origin.

1. Plot the first point (0, 4): Locate the point on the graph where ${ x = 0 }$ and ${ y = 4 }$. Mark this point clearly.
2. Plot the second point (-2, 0): Locate the point on the graph where ${ x = -2 }$ and ${ y = 0 }$. Mark this point clearly.
3. Draw a straight line: Using a ruler or straightedge, draw a straight line that passes through both points. Extend the line beyond the points to indicate that it continues infinitely in both directions.

The line you've drawn represents the equation ${ y = 2x + 4 }$. Any point on this line will satisfy the equation, and any point not on the line will not. This graphical representation provides a visual understanding of the relationship between ${ x }$ and ${ y }$ as defined by the equation.

B) ${ y = -\frac{1}{6}x - \frac{1}{3} }$

Understanding the Equation

The second equation we need to graph is ${ y = -\frac{1}{6}x - \frac{1}{3} }$. This is also a linear equation in slope-intercept form, ${ y = mx + b }$, where:

• The slope (${ m }$) is ${ -\frac{1}{6} }$, indicating that the line slopes downward from left to right and is not very steep.
• The y-intercept (${ b }$) is ${ -\frac{1}{3} }$, meaning the line crosses the y-axis at the point ${ (0, -\frac{1}{3}) }$. However, ${ -\frac{1}{3} }$ is not an integer, which makes it less convenient to plot directly. Therefore, we'll focus on finding two points with integer coordinates.

Recognizing the fractional slope and y-intercept is crucial for selecting appropriate ${ x }$ values that will result in integer ${ y }$ values. This will simplify the graphing process and ensure accuracy.

Finding Two Integer Points

To find two points with integer coordinates for this equation, we need to choose ${ x }$ values that will cancel out the fractions in the equation. The denominators in our equation are 6 and 3, so we should choose ${ x }$ values that are multiples of 6. This will eliminate the fractions and give us integer values for ${ y }$. Let's choose ${ x = 0 }$ and ${ x = 6 }$ as our two values.

1. When ${ x = 0 }$: Substitute ${ x = 0 }$ into the equation: ${ y = -\frac{1}{6}(0) - \frac{1}{3} = -\frac{1}{3} }$ This gives us the point ${ (0, -\frac{1}{3}) }$, which we already knew from the y-intercept. While this is a valid point on the line, it does not have integer coordinates, so we'll use it later to check our line but not for plotting.
2. When ${ x = 6 }$: Substitute ${ x = 6 }$ into the equation: ${ y = -\frac{1}{6}(6) - \frac{1}{3} = -1 - \frac{1}{3} = -\frac{4}{3} }$ This also results in a non-integer y-coordinate. We need to adjust our choices for ${ x }$ to find integer points. Let's try ${ x = -2 }$ and ${ x = 4 }$.
3. When ${ x = -2 }$: Substitute ${ x = -2 }$ into the equation: ${ y = -\frac{1}{6}(-2) - \frac{1}{3} = \frac{1}{3} - \frac{1}{3} = 0 }$ So, one point is ${ (-2, 0) }$.
4. When ${ x = 4 }$: Substitute ${ x = 4 }$ into the equation: ${ y = -\frac{1}{6}(4) - \frac{1}{3} = -\frac{2}{3} - \frac{1}{3} = -1 }$ So, another point is ${ (4, -1) }$.

Now we have two points with integer coordinates: ${ (-2, 0) }$ and ${ (4, -1) }$. These points will allow us to graph the line accurately.

Plotting the Points and Drawing the Line

With our two points, ${ (-2, 0) }$ and ${ (4, -1) }$, we can now plot them on the coordinate plane.

1. Plot the first point ${ (-2, 0) }$: Locate the point on the graph where ${ x = -2 }$ and ${ y = 0 }$. This point lies on the x-axis, 2 units to the left of the origin.
2. Plot the second point ${ (4, -1) }$: Locate the point on the graph where ${ x = 4 }$ and ${ y = -1 }$. This point is 4 units to the right of the origin and 1 unit below the x-axis.
3. Draw a straight line: Using a ruler or straightedge, draw a straight line that passes through both points. Extend the line beyond the points to indicate that it continues infinitely in both directions.

The line you've drawn represents the equation ${ y = -\frac{1}{6}x - \frac{1}{3} }$. This graphical representation gives us a visual understanding of the linear relationship defined by the equation. The negative slope is evident in the line's downward direction from left to right, and the line's position relative to the axes reflects the y-intercept and the overall behavior of the equation.

Conclusion

Graphing linear equations by plotting two points with integer coordinates is a fundamental and practical skill in mathematics. By understanding the equation's components, such as the slope and y-intercept, and carefully selecting ${ x }$ values to obtain integer ${ y }$ values, we can accurately represent linear equations graphically. This method not only provides a visual representation of the equation but also enhances our understanding of the relationship between the variables involved. Mastering this technique will serve as a valuable foundation for more advanced mathematical concepts and applications.

In summary, graphing linear equations involves:

1. Understanding the equation and its slope-intercept form.
2. Finding two points with integer coordinates by choosing appropriate ${ x }$ values and calculating the corresponding ${ y }$ values.
3. Plotting these points on the coordinate plane.
4. Drawing a straight line through the points to represent the equation.

By following these steps, you can confidently graph any linear equation and gain a deeper understanding of its properties and behavior. Remember, practice is key to mastering this skill, so continue working through examples and exploring different types of linear equations.