Solve The System Of Equations:${ \begin{array}{l} 6x - 3y = 36 \ 5x = 3y + 30 \end{array} }$

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example to demonstrate the steps involved in solving it.

The System of Equations

The given system of equations is:

$$\begin{array}{l} 6x - 3y = 36 \\ 5x = 3y + 30 \end{array}$$

Step 1: Write Down the Given Equations

The first step is to write down the given equations. In this case, we have two equations:

1. $6x - 3y = 36$
2. $5x = 3y + 30$

Step 2: Solve One of the Equations for One Variable

To solve the system of equations, we need to eliminate one of the variables. We can do this by solving one of the equations for one variable. Let's solve the second equation for $x$:

$$5x = 3y + 30$$

$$x = \frac{3y + 30}{5}$$

Step 3: Substitute the Expression into the Other Equation

Now that we have an expression for $x$, we can substitute it into the first equation:

$$6x - 3y = 36$$

$$6\left(\frac{3y + 30}{5}\right) - 3y = 36$$

Step 4: Simplify the Equation

To simplify the equation, we can start by multiplying both sides by 5 to eliminate the fraction:

$$6(3y + 30) - 15y = 180$$

$$18y + 180 - 15y = 180$$

Step 5: Combine Like Terms

Now we can combine like terms:

$$3y + 180 = 180$$

Step 6: Solve for the Variable

To solve for $y$, we can subtract 180 from both sides:

$$3y = 0$$

$$y = 0$$

Step 7: Find the Value of the Other Variable

Now that we have the value of $y$, we can substitute it into one of the original equations to find the value of the other variable. Let's use the second equation:

$$5x = 3y + 30$$

$$5x = 3(0) + 30$$

$$5x = 30$$

$$x = 6$$

Conclusion

In this article, we solved a system of two linear equations with two variables. We used the given system of equations as an example to demonstrate the steps involved in solving it. We wrote down the given equations, solved one of the equations for one variable, substituted the expression into the other equation, simplified the equation, combined like terms, solved for the variable, and found the value of the other variable. The final answer is $x = 6$ and $y = 0$.

Example Use Cases

Solving systems of linear equations has many practical applications in various fields, including:

• Physics: To solve problems involving motion, forces, and energies.
• Engineering: To design and optimize systems, such as electrical circuits, mechanical systems, and structural systems.
• Economics: To model and analyze economic systems, such as supply and demand, and resource allocation.
• Computer Science: To solve problems involving algorithms, data structures, and computer networks.

Tips and Tricks

Here are some tips and tricks to help you solve systems of linear equations:

• Use the substitution method: Substitute the expression for one variable into the other equation to eliminate one of the variables.
• Use the elimination method: Add or subtract the equations to eliminate one of the variables.
• Check your work: Verify that the solution satisfies both equations.
• Use technology: Use calculators or computer software to solve systems of linear equations.

Conclusion

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

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

A: A system of linear equations has a solution if the two equations are consistent, meaning that they have the same solution. If the equations are inconsistent, there is no solution.

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

A: There are two main methods for solving systems of linear equations:

1. Substitution method: Substitute the expression for one variable into the other equation to eliminate one of the variables.
2. Elimination method: Add or subtract the equations to eliminate one of the variables.

Q: How do I choose which method to use?

A: Choose the method that is most convenient for the problem. If one variable is already isolated in one of the equations, use the substitution method. If the coefficients of one variable are the same in both equations, use the elimination method.

Q: What if I have a system of linear equations with three or more variables?

A: To solve a system of linear equations with three or more variables, use the same methods as before, but with more variables. You can also use matrix methods or graphing methods to solve the system.

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

A: Yes, you can use calculators or computer software to solve systems of linear equations. Many calculators and computer programs have built-in functions for solving systems of linear equations.

Q: How do I check my work when solving a system of linear equations?

A: To check your work, substitute the solution back into both original equations to make sure that it satisfies both equations.

Q: What if I get a system of linear equations with no solution or infinitely many solutions?

A: If you get a system of linear equations with no solution, it means that the equations are inconsistent. If you get a system of linear equations with infinitely many solutions, it means that the equations are dependent.

Q: Can I use systems of linear equations to model real-world problems?

A: Yes, systems of linear equations can be used to model many real-world problems, such as:

• Physics: To solve problems involving motion, forces, and energies.
• Engineering: To design and optimize systems, such as electrical circuits, mechanical systems, and structural systems.
• Economics: To model and analyze economic systems, such as supply and demand, and resource allocation.
• Computer Science: To solve problems involving algorithms, data structures, and computer networks.

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 checking your work: Make sure to substitute the solution back into both original equations to make sure that it satisfies both equations.
• Not using the correct method: Choose the method that is most convenient for the problem.
• Not simplifying the equations: Simplify the equations as much as possible to make it easier to solve the system.

Conclusion

Solving systems of linear equations is an essential skill in mathematics and has many practical applications in various fields. By following the steps outlined in this article and avoiding common mistakes, you can solve systems of linear equations with ease. Remember to use the substitution method, elimination method, and check your work to ensure that you get the correct solution.