Solve The Simultaneous Equations Involving Fractions:${ \begin{align*} \frac{x+1}{3} + \frac{y-1}{2} &= 5, \ \frac{2x+5}{3} - \frac{y+1}{4} &= 3. \end{align*} }$

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Introduction

Simultaneous equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving simultaneous equations involving fractions. These types of equations can be challenging to solve, but with the right approach and techniques, we can find the solutions.

Understanding the Problem

The given problem involves two simultaneous equations with fractions:

{ \begin{align*} \frac{x+1}{3} + \frac{y-1}{2} &= 5, \\ \frac{2x+5}{3} - \frac{y+1}{4} &= 3. \end{align*} \}

Our goal is to find the values of x and y that satisfy both equations.

Step 1: Eliminate the Fractions

To make the equations easier to work with, we can eliminate the fractions by multiplying both sides of each equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 2 is 6, and the LCM of 3 and 4 is 12.

Multiplying the first equation by 6, we get:

{ \begin{align*} 6 \left( \frac{x+1}{3} + \frac{y-1}{2} \right) &= 6(5), \\ 2(x+1) + 3(y-1) &= 30. \end{align*} \}

Multiplying the second equation by 12, we get:

{ \begin{align*} 12 \left( \frac{2x+5}{3} - \frac{y+1}{4} \right) &= 12(3), \\ 8(2x+5) - 3(y+1) &= 36. \end{align*} \}

Step 2: Simplify the Equations

Now that we have eliminated the fractions, we can simplify the equations by distributing the coefficients.

The first equation becomes:

{ \begin{align*} 2(x+1) + 3(y-1) &= 30, \\ 2x + 2 + 3y - 3 &= 30, \\ 2x + 3y - 1 &= 30. \end{align*} \}

The second equation becomes:

{ \begin{align*} 8(2x+5) - 3(y+1) &= 36, \\ 16x + 40 - 3y - 3 &= 36, \\ 16x - 3y + 37 &= 36. \end{align*} \}

Step 3: Solve the System of Equations

We can now solve the system of equations using the method of substitution or elimination. Let's use the elimination method.

First, we can multiply the first equation by 3 and the second equation by 1 to make the coefficients of y's in both equations the same.

Multiplying the first equation by 3, we get:

{ \begin{align*} 3(2x + 3y - 1) &= 3(30), \\ 6x + 9y - 3 &= 90. \end{align*} \}

The second equation remains the same:

{ \begin{align*} 16x - 3y + 37 &= 36. \end{align*} \}

Step 4: Eliminate the Variable y

Now that we have the same coefficients for y's in both equations, we can eliminate the variable y by adding the two equations.

Adding the two equations, we get:

{ \begin{align*} (6x + 9y - 3) + (16x - 3y + 37) &= 90 + 36, \\ 22x + 6y + 34 &= 126. \end{align*} \}

Step 5: Solve for x

Now that we have eliminated the variable y, we can solve for x by isolating x on one side of the equation.

Subtracting 6y and 34 from both sides, we get:

{ \begin{align*} 22x &= 126 - 6y - 34, \\ 22x &= 92 - 6y. \end{align*} \}

Dividing both sides by 22, we get:

{ \begin{align*} x &= \frac{92 - 6y}{22}, \\ x &= \frac{46 - 3y}{11}. \end{align*} \}

Step 6: 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.

Let's substitute x into the first original equation:

{ \begin{align*} \frac{x+1}{3} + \frac{y-1}{2} &= 5, \\ \frac{\frac{46 - 3y}{11} + 1}{3} + \frac{y-1}{2} &= 5. \end{align*} \}

Step 7: Solve for y

Now that we have substituted x into the first original equation, we can solve for y.

Multiplying both sides of the equation by 6 to eliminate the fractions, we get:

{ \begin{align*} 6 \left( \frac{\frac{46 - 3y}{11} + 1}{3} + \frac{y-1}{2} \right) &= 6(5), \\ 2 \left( \frac{46 - 3y}{11} + 1 \right) + 3(y-1) &= 30. \end{align*} \}

Distributing the coefficients, we get:

{ \begin{align*} \frac{2(46 - 3y)}{11} + 2 + 3y - 3 &= 30, \\ \frac{92 - 6y}{11} + 3y - 1 &= 30. \end{align*} \}

Multiplying both sides of the equation by 11 to eliminate the fractions, we get:

{ \begin{align*} 92 - 6y + 33y - 11 &= 330, \\ 27y + 81 &= 330. \end{align*} \}

Subtracting 81 from both sides, we get:

{ \begin{align*} 27y &= 249. \end{align*} \}

Dividing both sides by 27, we get:

{ \begin{align*} y &= \frac{249}{27}, \\ y &= 9.2222... \end{align*} \}

Step 8: Find the Value of x

Now that we have found the value of y, we can find the value of x by substituting y into the equation x = (46 - 3y)/11.

Substituting y = 9.2222... into the equation, we get:

{ \begin{align*} x &= \frac{46 - 3(9.2222...)}{11}, \\ x &= \frac{46 - 27.6667...}{11}, \\ x &= \frac{18.3333...}{11}, \\ x &= 1.6666... \end{align*} \}

Conclusion

In this article, we have solved the simultaneous equations involving fractions using the method of substitution and elimination. We have found the values of x and y that satisfy both equations, and we have demonstrated the importance of eliminating fractions and simplifying equations in solving simultaneous equations.

The final answer is: 1.6666...\boxed{1.6666...}

Introduction

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

Q: What are simultaneous equations?

A: Simultaneous equations are a set of two or more equations that involve the same variables and are to be solved simultaneously. In other words, we need to find the values of the variables that satisfy all the equations at the same time.

Q: Why do we need to eliminate fractions when solving simultaneous equations?

A: When solving simultaneous equations, it is often easier to eliminate fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This makes the equations easier to work with and helps us to avoid errors.

Q: How do we eliminate fractions in simultaneous equations?

A: To eliminate fractions, we can multiply both sides of the equation by the LCM of the denominators. For example, if we have the equation x+13+y−12=5\frac{x+1}{3} + \frac{y-1}{2} = 5, we can multiply both sides by 6 to eliminate the fractions.

Q: What is the least common multiple (LCM) of two numbers?

A: The LCM of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 3 and 2 is 6, and the LCM of 3 and 4 is 12.

Q: How do we simplify equations after eliminating fractions?

A: After eliminating fractions, we can simplify the equations by distributing the coefficients and combining like terms. This makes the equations easier to work with and helps us to avoid errors.

Q: What is the method of substitution in solving simultaneous equations?

A: The method of substitution involves substituting the value of one variable into the other equation to solve for the other variable. For example, if we have the equations x+y=5x + y = 5 and 2x−y=32x - y = 3, we can substitute the value of x into the second equation to solve for y.

Q: What is the method of elimination in solving simultaneous equations?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables. For example, if we have the equations x+y=5x + y = 5 and 2x−y=32x - y = 3, we can add the two equations to eliminate the variable y.

Q: How do we choose which method to use when solving simultaneous equations?

A: We can choose which method to use based on the complexity of the equations and the variables involved. If the equations are simple and the variables are easy to work with, we can use the method of substitution. If the equations are complex and the variables are difficult to work with, we can use the method of elimination.

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

A: Some common mistakes to avoid when solving simultaneous equations include:

  • Not eliminating fractions
  • Not simplifying equations
  • Not choosing the correct method
  • Not checking the solutions

Q: How do we check the solutions when solving simultaneous equations?

A: We can check the solutions by substituting the values of the variables into both equations to make sure they are true. If the values satisfy both equations, then we have found the correct solution.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving simultaneous equations involving fractions. We have covered topics such as eliminating fractions, simplifying equations, and choosing the correct method. We have also discussed common mistakes to avoid and how to check the solutions. By following these tips and techniques, you can become more confident and proficient in solving simultaneous equations involving fractions.

Additional Resources

If you are looking for additional resources to help you with solving simultaneous equations involving fractions, here are a few suggestions:

  • Khan Academy: Solving Simultaneous Equations
  • Mathway: Solving Simultaneous Equations
  • Wolfram Alpha: Solving Simultaneous Equations

These resources provide step-by-step instructions, examples, and practice problems to help you master the skills of solving simultaneous equations involving fractions.