What Is The Value Of \[$ Y \$\] In The Solution To The System Of Equations?$\[ \begin{array}{l} \frac{1}{3} X + \frac{1}{4} Y = 1 \\ 2x - 3y = -30 \end{array} \\]A. \[$-8\$\] B. \[$-3\$\] C. 3 D. 8

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Introduction

When solving a system of linear equations, we often need to find the values of the variables that satisfy both equations simultaneously. In this case, we are given two equations with two variables, x and y, and we need to find the value of y in the solution to the system of equations. To do this, we will use the method of substitution or elimination to solve the system of equations and then identify the value of y.

The System of Equations

The system of equations is given by:

{ \begin{array}{l} \frac{1}{3} x + \frac{1}{4} y = 1 \\ 2x - 3y = -30 \end{array} \}

Step 1: Multiply the First Equation by 12 to Eliminate Fractions

To eliminate the fractions in the first equation, we can multiply both sides of the equation by 12. This will give us:

{ \begin{array}{l} 4x + 3y = 12 \\ 2x - 3y = -30 \end{array} \}

Step 2: Add the Two Equations to Eliminate y

Now that we have eliminated the fractions, we can add the two equations to eliminate the variable y. This will give us:

{ \begin{array}{l} (4x + 3y) + (2x - 3y) = 12 + (-30) \\ 6x = -18 \end{array} \}

Step 3: Solve for x

Now that we have eliminated the variable y, we can solve for x by dividing both sides of the equation by 6. This will give us:

{ \begin{array}{l} x = \frac{-18}{6} \\ x = -3 \end{array} \}

Step 4: Substitute x into One of the Original Equations

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use the first equation:

{ \begin{array}{l} \frac{1}{3} x + \frac{1}{4} y = 1 \\ \frac{1}{3} (-3) + \frac{1}{4} y = 1 \\ -1 + \frac{1}{4} y = 1 \end{array} \}

Step 5: Solve for y

Now that we have substituted the value of x into the equation, we can solve for y by adding 1 to both sides of the equation and then multiplying both sides by 4. This will give us:

{ \begin{array}{l} -1 + \frac{1}{4} y = 1 \\ \frac{1}{4} y = 2 \\ y = 8 \end{array} \}

Conclusion

In conclusion, the value of y in the solution to the system of equations is 8.

Final Answer

The final answer is $\boxed{8}$.

Discussion

This problem requires the student to use the method of substitution or elimination to solve a system of linear equations. The student must also be able to identify the value of y in the solution to the system of equations. This problem is a good example of how to use algebraic techniques to solve a system of linear equations.

Tips and Tricks

• When solving a system of linear equations, it is often helpful to eliminate one of the variables by adding or subtracting the equations.
• When eliminating a variable, make sure to multiply both sides of the equation by the same value to avoid introducing fractions.
• When solving for a variable, make sure to isolate the variable on one side of the equation.
• When substituting a value into an equation, make sure to use the correct value and to simplify the equation before solving for the variable.

Common Mistakes

• Failing to eliminate one of the variables by adding or subtracting the equations.
• Introducing fractions when eliminating a variable.
• Failing to isolate the variable on one side of the equation.
• Substituting the wrong value into the equation.

Real-World Applications

This problem has real-world applications in fields such as engineering, economics, and computer science. For example, in engineering, a system of linear equations may be used to model the behavior of a mechanical system, while in economics, a system of linear equations may be used to model the behavior of a market. In computer science, a system of linear equations may be used to solve a problem in machine learning or data analysis.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon
• [3] "Linear Algebra: A Modern Introduction" by David Poole

Note: The references provided are for general information and are not specific to this problem.

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Q: What is the system of equations given in the problem?

A: The system of equations is given by:

{ \begin{array}{l} \frac{1}{3} x + \frac{1}{4} y = 1 \\ 2x - 3y = -30 \end{array} \}

Q: How do we eliminate the fractions in the first equation?

A: To eliminate the fractions in the first equation, we can multiply both sides of the equation by 12. This will give us:

{ \begin{array}{l} 4x + 3y = 12 \\ 2x - 3y = -30 \end{array} \}

Q: How do we add the two equations to eliminate y?

A: Now that we have eliminated the fractions, we can add the two equations to eliminate the variable y. This will give us:

{ \begin{array}{l} (4x + 3y) + (2x - 3y) = 12 + (-30) \\ 6x = -18 \end{array} \}

Q: How do we solve for x?

A: Now that we have eliminated the variable y, we can solve for x by dividing both sides of the equation by 6. This will give us:

{ \begin{array}{l} x = \frac{-18}{6} \\ x = -3 \end{array} \}

Q: How do we substitute x into one of the original equations?

A: Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use the first equation:

{ \begin{array}{l} \frac{1}{3} x + \frac{1}{4} y = 1 \\ \frac{1}{3} (-3) + \frac{1}{4} y = 1 \\ -1 + \frac{1}{4} y = 1 \end{array} \}

Q: How do we solve for y?

A: Now that we have substituted the value of x into the equation, we can solve for y by adding 1 to both sides of the equation and then multiplying both sides by 4. This will give us:

{ \begin{array}{l} -1 + \frac{1}{4} y = 1 \\ \frac{1}{4} y = 2 \\ y = 8 \end{array} \}

Q: What is the value of y in the solution to the system of equations?

A: The value of y in the solution to the system of equations is 8.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid when solving a system of linear equations include:

• Failing to eliminate one of the variables by adding or subtracting the equations.
• Introducing fractions when eliminating a variable.
• Failing to isolate the variable on one side of the equation.
• Substituting the wrong value into the equation.

Q: What are some real-world applications of solving a system of linear equations?

A: Some real-world applications of solving a system of linear equations include:

• Modeling the behavior of a mechanical system in engineering.
• Modeling the behavior of a market in economics.
• Solving a problem in machine learning or data analysis in computer science.

Q: What are some resources for learning more about solving a system of linear equations?

A: Some resources for learning more about solving a system of linear equations include:

• "Linear Algebra and Its Applications" by Gilbert Strang.
• "Introduction to Linear Algebra" by Jim Hefferon.
• "Linear Algebra: A Modern Introduction" by David Poole.

Note: The resources provided are for general information and are not specific to this problem.