Graph The Equations To Solve The System:$ \begin{align*} y &= \frac{1}{3}x - 5 \ y &= \frac{1}{2}x - 6 \end{align*} }$Click On The Correct Answer A. One Solution: { {0, 7}$ $ B. Solutions: All Numbers On The Line C. One

Introduction

Graphing equations is a powerful tool for solving systems of linear equations. By visualizing the relationships between variables, we can easily identify the points of intersection, which represent the solutions to the system. In this article, we will explore how to graph the equations $y = \frac{1}{3}x - 5$ and $y = \frac{1}{2}x - 6$ to solve the system.

Understanding the Equations

Before we begin graphing, let's take a closer look at the equations we are working with.

Equation 1: $y = \frac{1}{3}x - 5$

This equation represents a line with a slope of $\frac{1}{3}$ and a y-intercept of $-5$. To graph this line, we can start by plotting the y-intercept, which is the point where the line intersects the y-axis. Since the y-intercept is $-5$, we can plot the point $(0, -5)$.

Next, we need to find another point on the line. We can do this by choosing a value for $x$ and solving for $y$. Let's choose $x = 3$. Plugging this value into the equation, we get:

$$y = \frac{1}{3}(3) - 5 = 0 - 5 = -5$$

So, the point $(3, -5)$ is also on the line.

Equation 2: $y = \frac{1}{2}x - 6$

This equation represents a line with a slope of $\frac{1}{2}$ and a y-intercept of $-6$. To graph this line, we can start by plotting the y-intercept, which is the point where the line intersects the y-axis. Since the y-intercept is $-6$, we can plot the point $(0, -6)$.

Next, we need to find another point on the line. We can do this by choosing a value for $x$ and solving for $y$. Let's choose $x = 4$. Plugging this value into the equation, we get:

$$y = \frac{1}{2}(4) - 6 = 2 - 6 = -4$$

So, the point $(4, -4)$ is also on the line.

Graphing the Equations

Now that we have plotted the y-intercepts and found additional points on each line, we can graph the equations.

Graphing Equation 1

To graph the equation $y = \frac{1}{3}x - 5$, we can use the points we plotted earlier: $(0, -5)$ and $(3, -5)$. We can draw a line through these two points to represent the equation.

Graphing Equation 2

To graph the equation $y = \frac{1}{2}x - 6$, we can use the points we plotted earlier: $(0, -6)$ and $(4, -4)$. We can draw a line through these two points to represent the equation.

Finding the Solution

Now that we have graphed the equations, we can find the solution to the system. The solution is the point where the two lines intersect.

Looking at the graph, we can see that the two lines at the point $(6, -1)$. This means that the solution to the system is $x = 6$ and $y = -1$.

Conclusion

Graphing equations is a powerful tool for solving systems of linear equations. By visualizing the relationships between variables, we can easily identify the points of intersection, which represent the solutions to the system. In this article, we graphed the equations $y = \frac{1}{3}x - 5$ and $y = \frac{1}{2}x - 6$ to solve the system and found the solution to be $x = 6$ and $y = -1$.

Answer

Based on the graph, the correct answer is:

• A. One solution: $\{6, -1\}$

Note: The original answer choice A was $\{0, 7\}$, but this is incorrect based on the graph.

Discussion

Graphing equations is a useful tool for solving systems of linear equations. However, it's not the only method. Other methods, such as substitution and elimination, can also be used to solve systems of linear equations.

In this article, we graphed the equations $y = \frac{1}{3}x - 5$ and $y = \frac{1}{2}x - 6$ to solve the system. However, we could have also used substitution or elimination to solve the system.

Substitution involves solving one equation for one variable and then substituting that expression into the other equation. Elimination involves adding or subtracting the equations to eliminate one variable.

Both substitution and elimination can be useful tools for solving systems of linear equations. However, graphing equations can be a more visual and intuitive way to solve systems, especially for simple systems like the one in this article.

Additional Resources

For more information on graphing equations and solving systems of linear equations, check out the following resources:

• Khan Academy: Graphing Linear Equations
• Mathway: Solving Systems of Linear Equations
• Wolfram Alpha: Graphing and Solving Systems of Linear Equations

Q: What is the purpose of graphing equations to solve systems?

A: The purpose of graphing equations to solve systems is to visualize the relationships between variables and identify the points of intersection, which represent the solutions to the system.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to plot the y-intercept and find another point on the line. You can do this by choosing a value for x and solving for y. Then, you can draw a line through the two points to represent the equation.

Q: What is the y-intercept?

A: The y-intercept is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I find the y-intercept of a linear equation?

A: To find the y-intercept of a linear equation, you can set x equal to 0 and solve for y. This will give you the value of y when x is equal to 0.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is the ratio of the change in y to the change in x. It represents the steepness of the line.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you can use the formula: slope = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the point of intersection?

A: The point of intersection is the point where two lines meet. It represents the solution to the system.

Q: How do I find the point of intersection of two lines?

A: To find the point of intersection of two lines, you can graph the two lines and identify the point where they meet.

Q: What are some common mistakes to avoid when graphing equations?

A: Some common mistakes to avoid when graphing equations include:

• Not plotting the y-intercept
• Not finding another point on the line
• Not drawing a line through the two points
• Not identifying the point of intersection

Q: What are some real-world applications of graphing equations to solve systems?

A: Some real-world applications of graphing equations to solve systems include:

• Physics: Graphing equations to solve systems of linear equations can be used to model the motion of objects.
• Engineering: Graphing equations to solve systems of linear equations can be used to design and optimize systems.
• Economics: Graphing equations to solve systems of linear equations can be used to model the behavior of economic systems.

Q: What are some tips for graphing equations to solve systems?

A: Some tips for graphing equations to solve systems include:

• Use a ruler or straightedge to draw the lines
• Plot the y-intercept and find another point on the line
• Draw a line through the two points
• Identify the point of intersection
• Check your work by plugging in values for x and y to make sure the equation is satisfied.