Not simplifying the equations before solving for the variables

Oct 13, 2025 by pynkocean.com 97 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving a system of two linear equations with two variables. We will use algebraic methods to find the solution to the system.

The System of Equations

The given system of equations is:

{ \begin{array}{l} -6y = \frac{3}{2}(2 - 6x) \\ -6y = -9x + 3 \end{array} \}

To solve this system, we need to find the values of x and y that satisfy both equations.

Step 1: Simplify the First Equation

The first equation is:

{ -6y = \frac{3}{2}(2 - 6x) \}

We can simplify this equation by multiplying both sides by 2 to eliminate the fraction:

{ -12y = 3(2 - 6x) \}

Expanding the right-hand side, we get:

{ -12y = 6 - 18x \}

Step 2: Simplify the Second Equation

The second equation is:

{ -6y = -9x + 3 \}

We can simplify this equation by multiplying both sides by 2 to make the coefficients of y the same:

{ -12y = -18x + 6 \}

Step 3: Equate the Two Equations

Now that we have simplified both equations, we can equate them to eliminate one variable:

{ -12y = 6 - 18x \}

{ -12y = -18x + 6 \}

Since both equations are equal, we can set them equal to each other:

{ 6 - 18x = -18x + 6 \}

Step 4: Solve for x

Now we can solve for x by isolating the variable:

{ 6 - 18x = -18x + 6 \}

Subtracting -18x from both sides, we get:

{ 6 = 6 \}

This equation is true for all values of x, which means that x is a free variable. However, we can still find the value of x by setting x to a specific value.

Step 5: Solve for y

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 use the first equation:

{ -6y = \frac{3}{2}(2 - 6x) \}

Substituting x = 0, we get:

{ -6y = \frac{3}{2}(2 - 6(0)) \}

Simplifying, we get:

{ -6y = \frac{3}{2}(2) \}

{ -6y = 3 \}

Dividing both sides by -6, we get:

{ y = -\frac{1}{2} \}

Conclusion

In this article, we solved a system of two linear equations with two variables using algebraic methods. We simplified the equations, equated them, and solved for the variables x and y. The solution to the system is x = 0 and y = -1/2.

Real-World Applications

Solving systems of equations has many real-world applications in various fields such as physics, engineering, and economics. For example, in physics, systems of equations can be used to model the motion of objects under the influence of gravity. In engineering, systems of equations can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, systems of equations can be used to model the behavior of economic systems and make predictions about future trends.

Tips and Tricks

When solving systems of equations, it's essential to simplify the equations and eliminate one variable before solving for the other variable. It's also crucial to check the solution by substituting the values of x and y into the original equations to ensure that they satisfy both equations.

Common Mistakes

When solving systems of equations, some common mistakes to avoid include:

Not simplifying the equations before solving for the variables
Not eliminating one variable before solving for the other variable
Not checking the solution by substituting the values of x and y into the original equations

Conclusion

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve systems of equations using algebraic methods. Remember to simplify the equations, equate them, and solve for the variables x and y. With practice and patience, you can become proficient in solving systems of equations and apply this skill to real-world problems.

Q: What is a system of equations?

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

Q: How do I know if a system of equations has a solution?

A: A system of equations has a solution if the equations are consistent, meaning that they do not contradict each other. If the equations are inconsistent, there is no solution.

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 one equation into the other equation to eliminate one variable.
Elimination method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Algebraic method: This method involves using algebraic techniques such as factoring and solving quadratic equations.

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 type of equations and the variables involved. For example, if the equations are linear, the substitution or elimination method may be the best choice. If the equations are quadratic, the algebraic method may be the best choice.

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

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

Not simplifying the equations before solving for the variables
Not eliminating one variable before solving for the other variable
Not checking the solution by substituting the values of x and y into the original equations

Q: How do I check if my solution is correct?

A: To check if your solution is correct, substitute the values of x and y into the original equations and check if they satisfy both equations.

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

A: Solving systems of equations has many real-world applications in various fields such as physics, engineering, and economics. For example, in physics, systems of equations can be used to model the motion of objects under the influence of gravity. In engineering, systems of equations can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, systems of equations can be used to model the behavior of economic systems and make predictions about future trends.

Q: Can I use technology to solve systems of equations?

A: Yes, technology can be used to solve systems of equations. For example, graphing calculators and computer software can be used to graph the equations and find the point of intersection.

Q: What are some tips for solving systems of equations?

A: Some tips for solving systems of equations include:

Simplify the equations before solving for the variables
Eliminate one variable before solving for the other variable
Check the solution by substituting the values of x and y into the original equations

Conclusion

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving systems of equations and apply this skill to real-world problems.