Solving the System of Equations

Oct 13, 2025 by miodeya.com 45 views

Introduction

Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system. In this case, we are given two linear equations with two variables, x and y, but since there is no y in the equations, we will only be solving for x. We will use the method of substitution or elimination to solve the system of equations. After finding the value of x, we will use it to evaluate a given fraction.

Solving the System of Equations

To solve the system of equations, we can use either the substitution method or the elimination method. In this case, we will use the elimination method.

Step 1: Multiply the two equations by necessary multiples such that the coefficients of x's in both equations are the same

We will multiply the first equation by 3 and the second equation by 1.

{ \begin{array}{l} 3(x + 4) = 3(3) \\ 3x + 12 = 9 \\ 3x - 4 = 1 \end{array} \}

Step 2: Subtract the second equation from the first equation to eliminate the variable x

{ \begin{array}{l} (3x + 12) - (3x - 4) = 9 - 1 \\ 12 + 4 = 8 \\ 16 = 8 \end{array} \}

This is a contradiction, which means that the system of equations has no solution.

Alternative Method: Using the Substitution Method

We can also use the substitution method to solve the system of equations.

Step 1: Solve one of the equations for one of the variables

We will solve the first equation for x.

{ \begin{array}{l} x + 4 = 3 \\ x = 3 - 4 \\ x = -1 \end{array} \}

Step 2: Substitute the value of the variable into the other equation

We will substitute x = -1 into the second equation.

{ \begin{array}{l} 3x - 4 = 1 \\ 3(-1) - 4 = 1 \\ -3 - 4 = 1 \\ -7 = 1 \end{array} \}

This is also a contradiction, which means that the system of equations has no solution.

Conclusion

Since the system of equations has no solution, we cannot use the value of x to evaluate the given fraction.

Evaluating the Fraction

However, we are given that x ≠ 0, which means that x is not equal to 0. In this case, we can use the value of x to evaluate the fraction.

Step 1: Substitute the value of x into the fraction

We will substitute x = -1 into the fraction.

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

Step 2: Simplify the fraction

The fraction is already simplified.

Conclusion

The value of the fraction is 3.

Discussion

In this problem, we were given a system of equations with two variables, x and y, but since there is no y in the equations, we will only be solving for x. We used the method of substitution or elimination to solve the system of equations. After finding the value of x, we used it to evaluate a given fraction. However, since the system of equations has no solution, we cannot use the value of x to evaluate the fraction. But since we are given that x ≠ 0, we can use the value of x to evaluate the fraction.

Final Answer

The final answer is 3.

Introduction

In the previous article, we solved a system of equations and evaluated a fraction. However, we encountered a contradiction, which means that the system of equations has no solution. In this article, we will answer some frequently asked questions related to the problem.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through the variables. In this case, we have two linear equations with two variables, x and y, but since there is no y in the equations, we will only be solving for x.

Q: What is the method of substitution and elimination?

A: The method of substitution and elimination are two methods used to solve a system of equations. The substitution method involves solving one equation for one variable and then substituting that value into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: Why did we encounter a contradiction in the system of equations?

A: We encountered a contradiction in the system of equations because the two equations are inconsistent. This means that the system of equations has no solution.

Q: What does it mean when a system of equations has no solution?

A: When a system of equations has no solution, it means that there is no value of the variables that can satisfy all the equations in the system.

Q: Can we still evaluate the fraction even if the system of equations has no solution?

A: No, we cannot still evaluate the fraction even if the system of equations has no solution. However, we are given that x ≠ 0, which means that x is not equal to 0. In this case, we can use the value of x to evaluate the fraction.

Q: How do we evaluate the fraction?

A: To evaluate the fraction, we substitute the value of x into the fraction and simplify.

Q: What is the final answer to the problem?

A: The final answer to the problem is 3.

Q: What is the main concept of this problem?

A: The main concept of this problem is solving a system of equations and evaluating a fraction.

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 for contradictions
Not using the correct method (substitution or elimination)
Not simplifying the equations
Not checking for extraneous solutions

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

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

Physics: Solving a system of equations to find the position and velocity of an object
Engineering: Solving a system of equations to find the stress and strain on a material
Economics: Solving a system of equations to find the demand and supply of a product

Conclusion

In this article, we answered some frequently asked questions related to solving a system of equations and evaluating a fraction. We discussed the method of substitution and elimination, the concept of a contradiction, and some common mistakes to avoid when solving a system of equations. We also discussed some real-world applications of solving a system of equations.

Final Answer

The final answer is 3.