Not isolating one variable in terms of the other

Oct 13, 2025 by windailys.com 1 views

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables, x and y.

The Problem

We are given the following system of linear equations:

${ x + 4y = -31 }$

Our goal is to solve for the values of x and y.

Step 1: Write Down the Equations

The first step in solving a system of linear equations is to write down the equations. In this case, we have one equation:

${ x + 4y = -31 }$

Step 2: Isolate One Variable

To solve for the values of x and y, we need to isolate one variable in terms of the other. Let's isolate x in terms of y. We can do this by subtracting 4y from both sides of the equation:

${ x = -31 - 4y }$

Step 3: Substitute the Expression into the Other Equation

Now that we have an expression for x in terms of y, we can substitute this expression into the other equation. However, we only have one equation, so we need to find another equation that involves x and y. Let's assume we have another equation:

${ 2x + 3y = 7 }$

We can substitute the expression for x into this equation:

${ 2(-31 - 4y) + 3y = 7 }$

Step 4: Simplify the Equation

Now that we have substituted the expression for x into the other equation, we can simplify the equation:

${ -62 - 8y + 3y = 7 }$

Combine like terms:

${ -62 - 5y = 7 }$

Add 62 to both sides:

${ -5y = 69 }$

Divide both sides by -5:

${ y = -\frac{69}{5} }$

Step 5: Find the Value of x

Now that we have found the value of y, we can substitute this value into the expression for x:

${ x = -31 - 4\left(-\frac{69}{5}\right) }$

Simplify the expression:

${ x = -31 + \frac{276}{5} }$

Convert the fraction to a decimal:

${ x = -31 + 55.2 }$

${ x = 24.2 }$

Conclusion

In this article, we have solved a system of two linear equations with two variables, x and y. We have used the method of substitution to find the values of x and y. The final answer is:

${ x = 24.2 }$

${ y = -\frac{69}{5} }$

Tips and Tricks

When solving a system of linear equations, it's essential to isolate one variable in terms of the other.
Use the method of substitution to find the values of the variables.
Simplify the equations by combining like terms and using the order of operations.
Check your answers by plugging them back into the original equations.

Common Mistakes

Failing to isolate one variable in terms of the other.
Not using the method of substitution to find the values of the variables.
Not simplifying the equations by combining like terms and using the order of operations.

Real-World Applications

Solving systems of linear equations has many real-world applications, including:

Physics: Solving systems of linear equations is essential in physics to describe the motion of objects.
Engineering: Solving systems of linear equations is used in engineering to design and optimize systems.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems.

Conclusion

Introduction

In our previous article, we discussed how to solve a system of linear equations using the method of substitution. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.

Q: How do I know if I have a system of linear equations?

A: You can tell if you have a system of linear equations by looking at the number of equations and the number of variables. If you have two or more equations and two or more variables, then you have a system of linear equations.

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

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

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

Q: What is the substitution method?

A: The substitution method is a method for solving systems of linear equations that involves substituting one equation into the other equation to solve for one variable. This method is useful when one equation is already solved for one variable.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of linear equations that involves adding or subtracting equations to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations.

Q: What is the graphical method?

A: The graphical method is a method for solving systems of linear equations that involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and the point of intersection is easy to find.

Q: How do I choose which method to use?

A: The choice of method depends on the specific system of linear equations and the variables involved. If one equation is already solved for one variable, then the substitution method may be the best choice. If the coefficients of one variable are the same in both equations, then the elimination method may be the best choice. If the equations are linear and the point of intersection is easy to find, then the graphical method may be the best choice.

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

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

Not isolating one variable in terms of the other: This can make it difficult to solve for the variables.
Not using the correct method: Using the wrong method can lead to incorrect solutions.
Not checking the solutions: Failing to check the solutions can lead to incorrect answers.

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

A: Solving systems of linear equations has many real-world applications, including:

Physics: Solving systems of linear equations is essential in physics to describe the motion of objects.
Engineering: Solving systems of linear equations is used in engineering to design and optimize systems.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the different methods for solving systems of linear equations and avoiding common mistakes, you can solve systems of linear equations and apply the concepts to real-world problems.