Three Times $x$ Is 13 Less Than $y$. The Sum Of $ X X X [/tex] And Two Times $y$ Is 12. Write Two Equations And Graph To Find The Value Of $y$.Possible Values For $ Y Y Y [/tex]:A.
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Introduction
In this article, we will delve into a mathematical problem involving two variables, $x$ and $y$. We will start by translating the given information into two equations and then use these equations to find the value of $y$. The problem states that three times $x$ is 13 less than $y$, and the sum of $x$ and two times $y$ is 12. We will use algebraic manipulation and graphing techniques to solve for $y$.
Translating the Problem into Equations
Let's start by translating the given information into two equations. The first statement can be written as:
This equation represents the relationship between $x$ and $y$, where three times $x$ is 13 less than $y$.
The second statement can be written as:
This equation represents the relationship between $x$ and $y$, where the sum of $x$ and two times $y$ is 12.
Solving the System of Equations
To solve for $y$, we can use the method of substitution or elimination. Let's use the elimination method to eliminate $x$ from the two equations.
First, we can multiply the first equation by 1 and the second equation by 3 to make the coefficients of $x$ equal:
Now, we can subtract the first equation from the second equation to eliminate $x$:
Simplifying the equation, we get:
Subtracting 13 from both sides, we get:
Dividing both sides by 5, we get:
Graphing the Equations
To visualize the solution, we can graph the two equations on a coordinate plane. The first equation can be written as:
The second equation can be written as:
We can graph these two equations on a coordinate plane to find the point of intersection, which represents the solution to the system of equations.
Conclusion
In this article, we translated the given information into two equations and used the elimination method to solve for $y$. We found that $y = \frac{23}{5}$. We also graphed the two equations on a coordinate plane to visualize the solution. The graph shows that the two equations intersect at the point $\left(\frac{2}{5}, \frac{23}{5}\right)$, which represents the solution to the system of equations.
Possible Values for $y$
Based on the solution, we can conclude that the possible values for $y$ are:
This is the only possible value for $y$ that satisfies the two equations.
Discussion
This problem is a classic example of a system of linear equations. The elimination method is a powerful tool for solving systems of linear equations. By using this method, we can eliminate one variable and solve for the other variable. The graphing technique is also a useful tool for visualizing the solution and understanding the relationship between the two variables.
In conclusion, this article has demonstrated how to translate the given information into two equations, solve the system of equations using the elimination method, and graph the equations to visualize the solution. The possible values for $y$ are $y = \frac{23}{5}$.
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Introduction
In our previous article, we explored a mathematical problem involving two variables, $x$ and $y$. We translated the given information into two equations and used the elimination method to solve for $y$. We also graphed the two equations on a coordinate plane to visualize the solution. In this article, we will answer some frequently asked questions related to this problem.
Q&A
Q: What is the relationship between $x$ and $y$ in this problem?
A: The relationship between $x$ and $y$ is given by two equations:
These equations represent the relationship between $x$ and $y$, where three times $x$ is 13 less than $y$, and the sum of $x$ and two times $y$ is 12.
Q: How do we solve the system of equations?
A: We can use the elimination method to solve the system of equations. By multiplying the first equation by 1 and the second equation by 3, we can make the coefficients of $x$ equal. Then, we can subtract the first equation from the second equation to eliminate $x$.
Q: What is the value of $y$?
A: The value of $y$ is $y = \frac{23}{5}$.
Q: How do we graph the equations?
A: We can graph the two equations on a coordinate plane to visualize the solution. The first equation can be written as:
The second equation can be written as:
We can graph these two equations on a coordinate plane to find the point of intersection, which represents the solution to the system of equations.
Q: What is the point of intersection?
A: The point of intersection is $\left(\frac{2}{5}, \frac{23}{5}\right)$.
Q: What is the possible value for $y$?
A: The possible value for $y$ is $y = \frac{23}{5}$.
Conclusion
In this article, we have answered some frequently asked questions related to the problem of three times $x$ is 13 less than $y$. We have explained the relationship between $x$ and $y$, how to solve the system of equations, and how to graph the equations. We have also found the value of $y$ and the point of intersection.
Discussion
This problem is a classic example of a system of linear equations. The elimination method is a powerful tool for solving systems of linear equations. By using this method, we can eliminate one variable and solve for the other variable. The graphing technique is also a useful tool for visualizing the solution and understanding the relationship between the two variables.
In conclusion, this article has demonstrated how to answer some frequently asked questions related to the problem of three times $x$ is 13 less than $y$. We hope that this article has been helpful in understanding the solution to this problem.
Additional Resources
For more information on solving systems of linear equations, please refer to the following resources:
- Khan Academy: Solving Systems of Linear Equations
- Mathway: Solving Systems of Linear Equations
- Wolfram Alpha: Solving Systems of Linear Equations
We hope that these resources are helpful in understanding the solution to this problem.