Consider The Following System Of Equations.$\[ \begin{align*} -4x - Y &= 0 \\ -5x + 3y &= 8 \end{align*} \\]We Are Solving This System Using The Elimination Method.Multiply Both Sides Of The First Equation By 3. Enter Your Next Step Here:

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. The elimination method is one of the techniques used to solve a system of equations. This method involves adding or subtracting the equations to eliminate one of the variables. In this article, we will use the elimination method to solve a system of two linear equations.

The System of Equations

The given system of equations is:

{ \begin{align*} -4x - y &= 0 \\ -5x + 3y &= 8 \end{align*} \}

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

Step 1: Multiply Both Sides of the First Equation by 3

To eliminate one of the variables, we need to make the coefficients of either $x$ or $y$ the same in both equations. Let's multiply both sides of the first equation by 3.

{ \begin{align*} -4x - y &= 0 \\ \Rightarrow\qquad -12x - 3y &= 0 \end{align*} \}

Now, we have the first equation multiplied by 3.

Step 2: Multiply Both Sides of the Second Equation by 4

To make the coefficients of $x$ the same in both equations, we need to multiply both sides of the second equation by 4.

{ \begin{align*} -5x + 3y &= 8 \\ \Rightarrow\qquad -20x + 12y &= 32 \end{align*} \}

Now, we have the second equation multiplied by 4.

Step 3: Subtract the First Equation from the Second Equation

Now that we have the coefficients of $x$ the same in both equations, we can subtract the first equation from the second equation to eliminate $x$.

{ \begin{align*} -20x + 12y &= 32 \\ -(-12x - 3y) &= 0 \\ \Rightarrow\qquad -20x + 12y &= 32 \\ \qquad\qquad\qquad 12x + 3y &= 0 \end{align*} \}

Subtracting the first equation from the second equation, we get:

{ \begin{align*} -20x + 12y - (12x + 3y) &= 32 - 0 \\ \Rightarrow\qquad -32x + 9y &= 32 \end{align*} \}

Now, we have a new equation with only one variable, $y$.

Step 4: Solve for $y$

To solve for $y$, we can isolate $y$ on one side of the equation.

{ \begin{align*} -32x + 9y &= 32 \\ \Rightarrow\qquad 9y &= 32 + 32x \\ \Rightarrow\qquad y &= \frac{32 + 32x}{9} \end{align*} \}

Now, we have the value of $y$ in terms of $x$.

Step 5: Substitute the Value of $y$ into One of the Original Equations

We can substitute the value of $y$ into one of the original equations to solve for $x$.

Let's substitute the value of $y$ into the first equation.

{ \begin{align*} -4x - y &= 0 \\ \Rightarrow\qquad -4x - \frac{32 + 32x}{9} &= 0 \end{align*} \}

Multiplying both sides of the equation by 9, we get:

{ \begin{align*} -36x - 32 - 32x &= 0 \\ \Rightarrow\qquad -68x - 32 &= 0 \end{align*} \}

Now, we have a new equation with only one variable, $x$.

Step 6: Solve for $x$

To solve for $x$, we can isolate $x$ on one side of the equation.

{ \begin{align*} -68x - 32 &= 0 \\ \Rightarrow\qquad -68x &= 32 \\ \Rightarrow\qquad x &= -\frac{32}{68} \\ \Rightarrow\qquad x &= -\frac{8}{17} \end{align*} \}

Now, we have the value of $x$.

Step 7: Substitute the Value of $x$ into One of the Original Equations

We can substitute the value of $x$ into one of the original equations to solve for $y$.

Let's substitute the value of $x$ into the first equation.

{ \begin{align*} -4x - y &= 0 \\ \Rightarrow\qquad -4\left(-\frac{8}{17}\right) - y &= 0 \end{align*} \}

Simplifying the equation, we get:

{ \begin{align*} \frac{32}{17} - y &= 0 \\ \Rightarrow\qquad y &= \frac{32}{17} \end{align*} \}

Now, we have the value of $y$.

Conclusion

In this article, we used the elimination method to solve a system of two linear equations. We multiplied both sides of the first equation by 3 and both sides of the second equation by 4 to make the coefficients of $x$ the same in both equations. We then subtracted the first equation from the second equation to eliminate $x$. We solved for $y$ and then substituted the value of $y$ into one of the original equations to solve for $x$. Finally, we substituted the value of $x$ into one of the original equations to solve for $y$. The values of $x$ and $y$ are $x = -\frac{8}{17}$ and $y = \frac{32}{17}$.

Final Answer

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Introduction

In our previous article, we used the elimination method to solve a system of two linear equations. In this article, we will answer some of the most frequently asked questions about solving a system of equations using the elimination method.

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I know which variable to eliminate?

A: To determine which variable to eliminate, you need to look at the coefficients of the variables in both equations. If the coefficients of one variable are the same in both equations, you can eliminate that variable.

Q: What if the coefficients of the variables are not the same in both equations?

A: If the coefficients of the variables are not the same in both equations, you need to multiply both sides of one or both equations by a number that will make the coefficients of the variables the same.

Q: How do I multiply both sides of an equation by a number?

A: To multiply both sides of an equation by a number, you need to multiply each term in the equation by that number.

Q: What if I get a negative number when I multiply both sides of an equation by a number?

A: If you get a negative number when you multiply both sides of an equation by a number, you need to change the sign of each term in the equation.

Q: How do I subtract one equation from another?

A: To subtract one equation from another, you need to subtract each term in the second equation from the corresponding term in the first equation.

Q: What if I get a fraction when I subtract one equation from another?

A: If you get a fraction when you subtract one equation from another, you need to simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor.

Q: How do I solve for the value of a variable?

A: To solve for the value of a variable, you need to isolate that variable on one side of the equation.

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

A: If you have a system of three or more equations, you can use the elimination method to solve for the values of the variables. However, you may need to use a combination of the elimination method and other techniques, such as substitution or graphing.

Q: Can I use the elimination method to solve a system of nonlinear equations?

A: No, the elimination method is only used to solve systems of linear equations. If you have a system of nonlinear equations, you will need to use a different technique, such as substitution or graphing.

Conclusion

In this article, we answered some of the most frequently asked questions about solving a system of equations using the elimination method. We hope that this article has been helpful in understanding the elimination method and how to use it to solve systems of equations.

Final Answer

The final answer is \boxed{There is no final answer, as this is a Q&A article.}