What Is The Solution To This System Of Equations?${ \begin{array}{l} \frac{1}{4} X + 1 \frac{1}{2} Y = \frac{5}{8} \ \frac{3}{4} X - 1 \frac{1}{2} Y = 3 \frac{3}{8} \end{array} }$A. { \left(4,-\frac{1}{4}\right)$}$B.

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Introduction

Solving a system of equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore the solution to a system of equations involving fractions. We will use the method of substitution and elimination to find the solution.

The System of Equations

The given system of equations is:

{ \begin{array}{l} \frac{1}{4} x + 1 \frac{1}{2} y = \frac{5}{8} \\ \frac{3}{4} x - 1 \frac{1}{2} y = 3 \frac{3}{8} \end{array} \}

To simplify the equations, we can convert the mixed numbers to improper fractions:

{ \begin{array}{l} \frac{1}{4} x + \frac{3}{2} y = \frac{5}{8} \\ \frac{3}{4} x - \frac{3}{2} y = \frac{27}{8} \end{array} \}

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of one of the variables (either x or y) are the same in both equations. Let's multiply the first equation by 2 and the second equation by 1.

{ \begin{array}{l} \frac{1}{2} x + 3y = \frac{5}{4} \\ \frac{3}{4} x - \frac{3}{2} y = \frac{27}{8} \end{array} \}

Step 2: Eliminate One of the Variables

Now, we can eliminate one of the variables by adding or subtracting the equations. Let's eliminate the variable x by subtracting the second equation from the first equation.

{ \begin{array}{l} \frac{1}{2} x + 3y - \left(\frac{3}{4} x - \frac{3}{2} y\right) = \frac{5}{4} - \frac{27}{8} \\ \frac{1}{2} x + 3y - \frac{3}{4} x + \frac{3}{2} y = \frac{5}{4} - \frac{27}{8} \end{array} \}

Simplifying the equation, we get:

{ \begin{array}{l} -\frac{1}{4} x + 6y = -\frac{11}{8} \end{array} \}

Step 3: Solve for One of the Variables

Now, we can solve for one of the variables by isolating it on one side of the equation. Let's solve for y.

{ \begin{array}{l} 6y = -\frac{11}{8} + \frac{1}{4} x \\ y = -\frac{11}{48} + \frac{1}{24} x \end{array} \}

Step 4: Substitute the Expression for One Variable into the Other Equation

Now, we can substitute the expression for y into the other equation to solve for x.

{ \begin{array}{l} \frac{1}{4} x + \frac{3}{2} \left(-\frac{11}{48} + \frac{1}{24} x\right) = \frac{5}{8} \\ \frac{1}{4} x - \frac{11}{32} + \frac{3}{16} x = \frac{5}{8} \end{array} \}

Simplifying the equation, we get:

{ \begin{array}{l} \frac{1}{4} x + \frac{3}{16} x = \frac{5}{8} + \frac{11}{32} \\ \frac{5}{16} x = \frac{20}{32} + \frac{11}{32} \\ \frac{5}{16} x = \frac{31}{32} \end{array} \}

Step 5: Solve for the Other Variable

Now, we can solve for x by multiplying both sides of the equation by the reciprocal of the coefficient of x.

{ \begin{array}{l} x = \frac{31}{32} \times \frac{16}{5} \\ x = \frac{31}{5} \end{array} \}

Step 6: Find the Value of the Other Variable

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations.

{ \begin{array}{l} \frac{1}{4} \left(\frac{31}{5}\right) + \frac{3}{2} y = \frac{5}{8} \\ \frac{31}{20} + \frac{3}{2} y = \frac{5}{8} \\ \frac{3}{2} y = \frac{5}{8} - \frac{31}{20} \\ \frac{3}{2} y = \frac{25}{40} - \frac{62}{40} \\ \frac{3}{2} y = -\frac{37}{40} \\ y = -\frac{37}{80} \end{array} \}

Conclusion

In this article, we solved a system of equations involving fractions using the method of substitution and elimination. We first multiplied the equations by necessary multiples to eliminate one of the variables. Then, we solved for one of the variables and substituted the expression into the other equation to solve for the other variable. Finally, we found the value of the other variable by substituting the value of the first variable into one of the original equations.

The Final Answer

The solution to the system of equations is:

{ \begin{array}{l} x = \frac{31}{5} \\ y = -\frac{37}{80} \end{array} \}

This solution satisfies both equations in the system.

Answer Choice

The correct answer is:

Introduction

In the previous article, we solved a system of equations involving fractions using the method of substitution and elimination. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.

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

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

  • Substitution method: This method involves substituting the expression for one variable into the other equation to solve for the other variable.
  • Elimination method: This method involves eliminating one of the variables by adding or subtracting the equations.
  • Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
  • Matrices method: This method involves using matrices to represent the system of equations and solving for the variables.

Q: What are the advantages and disadvantages of each method?

A: Here are the advantages and disadvantages of each method:

  • Substitution method:
    • Advantages: Easy to understand and apply, especially for simple systems of equations.
    • Disadvantages: Can be time-consuming and difficult to apply for complex systems of equations.
  • Elimination method:
    • Advantages: Can be used to solve systems of equations with multiple variables, and can be faster than the substitution method.
    • Disadvantages: Can be difficult to apply for systems of equations with fractions or decimals.
  • Graphical method:
    • Advantages: Can be used to visualize the system of equations and find the point of intersection.
    • Disadvantages: Can be difficult to apply for complex systems of equations, and may not be accurate for systems of equations with multiple variables.
  • Matrices method:
    • Advantages: Can be used to solve systems of equations with multiple variables, and can be faster than the substitution and elimination methods.
    • Disadvantages: Can be difficult to understand and apply, especially for beginners.

Q: How do I choose the best method for solving a system of equations?

A: The best method for solving a system of equations depends on the complexity of the system and the variables involved. Here are some tips for choosing the best method:

  • Simple systems of equations: Use the substitution or elimination method.
  • Complex systems of equations: Use the matrices method or the graphical method.
  • Systems of equations with fractions or decimals: Use the elimination method or the matrices method.
  • Systems of equations with multiple variables: Use the matrices method or the graphical method.

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

A: Here are some common mistakes to avoid when solving systems of equations:

  • Not checking the solution: Make sure to check the solution by substituting the values of the variables into the original equations.
  • Not using the correct method: Choose the best method for the system of equations, and make sure to apply it correctly.
  • Not simplifying the equations: Simplify the equations before solving them to avoid confusion and errors.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions, especially when using the graphical method.

Conclusion

Solving systems of equations is a fundamental concept in mathematics, and there are several methods for solving them. By understanding the different methods and choosing the best method for the system of equations, you can solve systems of equations with ease. Remember to check the solution, simplify the equations, and avoid common mistakes to ensure accurate results.