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

by ADMIN 194 views

Introduction

Solving a system of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. A system of equations is a set of two or more equations that contain the same variables. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the value of $y$ in the solution to the system of equations.

The System of Equations

The system of equations we will be working with is:

{ \begin{align*} \frac{1}{3} x + \frac{1}{4} y &= 1 \\ 2x - 3y &= -30 \end{align*} \}

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

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

{ \begin{align*} 4x + 3y &= 12 \end{align*} \}

Step 2: Multiply the Second Equation by 3 to Make the Coefficients of $y$ Opposite

To make the coefficients of $y$ opposite, we can multiply both sides of the second equation by 3. This will give us:

{ \begin{align*} 6x - 9y &= -90 \end{align*} \}

Step 3: Add the Two Equations to Eliminate the Variable $x$

Now that we have the two equations with opposite coefficients of $y$, we can add the two equations to eliminate the variable $x$. This will give us:

{ \begin{align*} (4x + 3y) + (6x - 9y) &= 12 + (-90) \\ 10x - 6y &= -78 \end{align*} \}

Step 4: Solve for $y$

Now that we have the equation $10x - 6y = -78$, we can solve for $y$. We can do this by isolating $y$ on one side of the equation. This will give us:

{ \begin{align*} -6y &= -78 - 10x \\ y &= \frac{-78 - 10x}{-6} \\ y &= \frac{78 + 10x}{6} \end{align*} \}

Step 5: Substitute the Value of $x$ into the Equation

However, we are not given the value of $x$. To find the value of $y$, we need to find the value of $x$ first. We can do this by substituting the value of $x$ into one of the original equations. Let's substitute the value of $x$ into the first equation:

{ \begin{align*} \frac{1}{3} x + \frac{1}{4} y &= 1 \\ \frac{1}{3} x &= 1 - \frac{1}{4} y \\ x &= 3 - \frac{3}{4} y \end{align*} \}

Step 6: Substitute the Value of $x$ into the Equation for $y$

Now that we have the value of $x$, we can substitute it into the equation for $y$:

{ \begin{align*} y &= \frac{78 + 10x}{6} \\ y &= \frac{78 + 10(3 - \frac{3}{4} y)}{6} \\ y &= \frac{78 + 30 - 7.5y}{6} \\ y &= \frac{108 - 7.5y}{6} \\ 6y &= 108 - 7.5y \\ 13.5y &= 108 \\ y &= \frac{108}{13.5} \\ y &= 8 \end{align*} \}

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We found the value of $y$ in the solution to the system of equations to be $8$. This is the final answer.

Discussion

The value of $y$ in the solution to the system of equations is $8$. This means that when $x = 3 - \frac{3}{4} y$, the value of $y$ is $8$. This is a unique solution to the system of equations.

Final Answer

The final answer is: 8\boxed{8}

Introduction

In our previous article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We found the value of $y$ in the solution to the system of equations to be $8$. In this article, we will answer some frequently asked questions about solving a system of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain the same variables. In other words, it is a collection of equations that are related to each other through the variables.

Q: What are the different methods for solving a system of equations?

A: There are several methods for solving a system of equations, including:

  • Substitution method: This method involves substituting the value of one variable into the other equation to solve for the other variable.
  • Elimination method: This method involves adding or subtracting the two equations to eliminate one of the variables.
  • Graphical method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
  • Matrix method: This method involves using matrices to solve the system of equations.

Q: What is the difference between a linear equation and a nonlinear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, $2x + 3y = 5$ is a linear equation. A nonlinear equation, on the other hand, is an equation in which the highest power of the variable is greater than 1. For example, $x^2 + 3y = 5$ is a nonlinear equation.

Q: Can a system of equations have more than one solution?

A: Yes, a system of equations can have more than one solution. This is known as an infinite solution or a dependent system.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution. This is known as an inconsistent system.

Q: How do I know if a system of equations is consistent or inconsistent?

A: To determine if a system of equations is consistent or inconsistent, you can use the following methods:

  • Graphical method: If the two equations intersect, the system is consistent. If the two equations do not intersect, the system is inconsistent.
  • Substitution method: If the value of one variable is a function of the other variable, the system is consistent. If the value of one variable is not a function of the other variable, the system is inconsistent.
  • Elimination method: If the two equations have the same solution, the system is consistent. If the two equations have different solutions, the system is inconsistent.

Q: What is the importance of solving a system of equations?

A: Solving a system of equations is an important concept in mathematics and has numerous applications in various fields such as physics, engineering, economics, and computer science. It is used to model real-world problems and to make predictions about the behavior of complex systems.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions for solving systems of equations, such as the "solve" function.

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

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

  • Not checking the consistency of the system
  • Not using the correct method for solving the system
  • Not checking the solution for accuracy
  • Not considering the possibility of an infinite solution or an inconsistent system

Conclusion

In this article, we answered some frequently asked questions about solving a system of equations. We discussed the different methods for solving a system of equations, the difference between a linear equation and a nonlinear equation, and the importance of solving a system of equations. We also provided some tips for avoiding common mistakes when solving a system of equations.

Final Answer

The final answer is: 8\boxed{8}