Solve The System Of Equations:1. $3x = -3 + 6y$2. − 1 3 X + 8 = Y -\frac{1}{3}x + 8 = Y − 3 1 ​ X + 8 = Y

Introduction

In mathematics, a system of equations is a set of two or more 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 equations:

1. $3x = -3 + 6y$
2. $-\frac{1}{3}x + 8 = y$

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

Step 1: Write Down the Equations

The first step is to write down the given equations.

Equation 1: $3x = -3 + 6y$

This equation represents a linear relationship between $x$ and $y$. We can see that $x$ is multiplied by 3, and $y$ is multiplied by 6.

Equation 2: $-\frac{1}{3}x + 8 = y$

This equation also represents a linear relationship between $x$ and $y$. We can see that $x$ is multiplied by $-\frac{1}{3}$, and the constant term is 8.

Step 2: Solve One of the Equations for One Variable

To solve the system of equations, we can start by solving one of the equations for one variable. Let's solve Equation 1 for $x$.

Solving Equation 1 for $x$

We can isolate $x$ by dividing both sides of the equation by 3.

$$\frac{3x}{3} = \frac{-3 + 6y}{3}$$

This simplifies to:

$$x = -1 + 2y$$

Now we have Equation 1 solved for $x$.

Step 3: Substitute the Expression for $x$ into the Other Equation

Next, we can substitute the expression for $x$ into Equation 2.

Substituting the Expression for $x$ into Equation 2

We can substitute $x = -1 + 2y$ into Equation 2:

$$-\frac{1}{3}(-1 + 2y) + 8 = y$$

This simplifies to:

$$\frac{1}{3} - \frac{2}{3}y + 8 = y$$

Step 4: Simplify the Equation

Now we can simplify the equation by combining like terms.

Simplifying the Equation

We can combine the constant terms:

$$\frac{1}{3} + 8 = y - \frac{2}{3}y$$

This simplifies to:

$$\frac{25}{3} = \frac{1}{3}y$$

Step 5: Solve for $y$

Now we can solve for $y$ by multiplying both sides of the equation by 3.

Solving for $y$

$$25 = y$$

Now we have found the value of $y$.

Step 6: Substitute the Value of $y$ into One of the Original Equations

Next, we can substitute the value of $y$ into one of the original equations to find the value of $x$.

Substituting the Value of $y$ into Equation 1

We can substitute $y = 25$ into Equation 1:

$$3x = -3 + 6(25)$$

This simplifies to:

$$3x = -3 + 150$$

Step 7: Solve for $x$

Now we can solve for $x$ by dividing both sides of the equation by 3.

Solving for $x$

$$x = \frac{147}{3}$$

This simplifies to:

$$x = 49$$

Conclusion

In this article, we have solved the system of equations using the given equations:

1. $3x = -3 + 6y$
2. $-\frac{1}{3}x + 8 = y$

We have found the values of $x$ and $y$ that satisfy both equations. The final answer is:

$x = 49$ $y = 25$

Final Answer

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Introduction

In our previous article, we solved the system of equations using the given equations:

1. $3x = -3 + 6y$
2. $-\frac{1}{3}x + 8 = y$

We found the values of $x$ and $y$ that satisfy both equations. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is a system of equations?

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

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

A: To determine if a system of equations has a solution, you can use the following methods:

• Check if the equations are consistent (i.e., they have the same solution).
• Check if the equations are inconsistent (i.e., they have no solution).
• Use the method of substitution or elimination to solve the system of equations.

Q: What is the method of substitution?

A: The method of substitution involves solving one of the equations for one variable and then substituting that expression into the other equation.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I choose which method to use?

A: You can choose which method to use based on the following factors:

• The complexity of the equations.
• The number of variables.
• The type of equations (e.g., linear, quadratic).

Q: What if I get stuck while solving the system of equations?

A: If you get stuck while solving the system of equations, you can try the following:

• Check your work for errors.
• Use a different method (e.g., substitution, elimination).
• Ask for help from a teacher or tutor.

Q: Can I use a calculator to solve the system of equations?

A: Yes, you can use a calculator to solve the system of equations. However, it's always a good idea to check your work by hand to ensure that the solution is correct.

Q: How do I know if the solution is correct?

A: To check if the solution is correct, you can plug the values of $x$ and $y$ back into the original equations and verify that they are true.

Q: What if the solution is not unique?

A: If the solution is not unique, it means that there are multiple solutions to the system of equations. In this case, you can use the method of substitution or elimination to find the other solutions.

Conclusion

In this article, we have provided a Q&A guide to help you understand the solution to the system of equations and answer any questions you may have. We hope that this guide has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is $\boxed{49, 25}$.