Solve The System Of Equations.$\[ \begin{array}{c} x = Y - 4 \\ 2x - 5y = 3 \end{array} \\]Which One-variable Linear Equation Represents The System Of Equations?A. \[$2(y-4) - 5y = 3\$\]B. \[$2x - 5(y-4) = 3\$\]C. \[$2x -

Introduction

A system of linear equations is a set of two or more linear equations that involve the same variables. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution to solve the system of equations.

The System of Equations

The given system of equations is:

$$\begin{array}{c} x = y - 4 \\ 2x - 5y = 3 \end{array}$$

Step 1: Substitute the First Equation into the Second Equation

To solve the system of equations, we can substitute the first equation into the second equation. We will substitute $x = y - 4$ into the second equation $2x - 5y = 3$.

$$2(y - 4) - 5y = 3$$

Step 2: Simplify the Equation

Now, we will simplify the equation by combining like terms.

$$2y - 8 - 5y = 3$$

$$-3y - 8 = 3$$

Step 3: Isolate the Variable

Next, we will isolate the variable $y$ by adding 8 to both sides of the equation.

$$-3y = 11$$

Step 4: Solve for the Variable

Finally, we will solve for the variable $y$ by dividing both sides of the equation by -3.

$$y = -\frac{11}{3}$$

Step 5: Find the Value of the Other Variable

Now that we have found the value of $y$, we can substitute it into one of the original equations to find the value of the other variable $x$. We will substitute $y = -\frac{11}{3}$ into the first equation $x = y - 4$.

$$x = -\frac{11}{3} - 4$$

$$x = -\frac{11}{3} - \frac{12}{3}$$

$$x = -\frac{23}{3}$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution. We have found the values of both variables $x$ and $y$.

Which One-Variable Linear Equation Represents the System of Equations?

Now, let's look at the options given in the problem.

A. $2(y-4) - 5y = 3$

B. $2x - 5(y-4) = 3$

C. $2x - 5y = 3$

We have already solved the system of equations and found the values of both variables $x$ and $y$. We can substitute these values into any of the options to see which one represents the system of equations.

Let's substitute $x = -\frac{23}{3}$ and $y = -\frac{11}{3}$ into option A.

$$2(-\frac{11}{3} - 4) - 5(-\frac{11}{3}) = 3$$

$$2(-\frac{11}{3} - \frac{12}{3}) + \frac{55}{3} = 3$$

$$2(-\frac{23}{3}) + \frac{55}{3} = 3$$

$$-\frac{46}{3} + \frac{55}{3} = 3$$

$$\frac{9}{3} = 3$$

$$3 = 3$$

This is true, so option A represents the system of equations.

Final Answer

$$

Introduction

In our previous article, we solved a system of two linear equations with two variables using the method of substitution. We found the values of both variables $x$ and $y$. 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 system of linear equations is a set of two or more linear equations that involve the same variables. For example:

$$\begin{array}{c} x + y = 4 \\ 2x - 3y = 5 \end{array}$$

Q: How do I solve a system of linear equations?

There are several methods to solve a system of linear equations, including:

• Substitution method: Substitute one equation into the other equation to solve for one variable.
• Elimination method: Add or subtract the equations to eliminate one variable.
• Graphical method: Graph the equations on a coordinate plane to find the point of intersection.

Q: What is the substitution method?

The substitution method involves substituting one equation into the other equation to solve for one variable. For example:

$$\begin{array}{c} x + y = 4 \\ 2x - 3y = 5 \end{array}$$

We can substitute $x = 4 - y$ into the second equation:

$$2(4 - y) - 3y = 5$$

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one variable. For example:

$$\begin{array}{c} x + y = 4 \\ 2x - 3y = 5 \end{array}$$

We can add the equations to eliminate the $y$ variable:

$$(x + y) + (2x - 3y) = 4 + 5$$

$$3x - 2y = 9$$

Q: How do I choose which method to use?

The choice of method depends on the specific system of equations. If the equations are easy to substitute, the substitution method may be the best choice. If the equations are easy to add or subtract, the elimination method may be the best choice.

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

If you have a system of three or more linear equations, you can use the same methods as before, but you may need to use additional techniques, such as:

• Gaussian elimination: Use row operations to transform the system into upper triangular form.
• Matrix methods: Use matrices to represent the system and solve for the variables.

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

Yes, you can use technology, such as graphing calculators or computer software, to solve systems of linear equations. These tools can help you visualize the system and find the solution.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the substitution method, the elimination method, and the graphical method, as well as the choice of method and the use of technology. We hope this article has been helpful in understanding how to solve systems of linear equations.

Common Mistakes to Avoid

When solving systems of linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not checking your work: Make sure to check your work by plugging the solution back into the original equations.
• Not using the correct method: Choose the method that is best suited for the system of equations.
• Not simplifying the equations: Simplify the equations as much as possible to make it easier to solve the system.

Final Tips

Solving systems of linear equations can be challenging, but with practice and patience, you can become proficient. Here are some final tips:

• Practice, practice, practice: The more you practice, the better you will become at solving systems of linear equations.
• Use technology: Technology can be a powerful tool in solving systems of linear equations.
• Check your work: Always check your work to make sure you have found the correct solution.

We hope this article has been helpful in understanding how to solve systems of linear equations. If you have any further questions, please don't hesitate to ask.