What Does \[$a\$\] Equal In The Solution Of The System Of Equations Below?$\[ \begin{array}{l} 5a + 5y + 2z = -41 \\ y - 3z = 19 \\ 2y + 3z = -34 \end{array} \\]A. \[$0\$\] B. \[$5\$\] C. \[$-3\$\] D.

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Introduction

In this article, we will delve into the world of linear algebra and explore the solution to a system of linear equations. We will examine a specific system of equations and determine the value of $a$ in the solution. This will involve using various techniques such as substitution and elimination to solve the system of equations.

The System of Equations

The system of equations we will be working with is given by:

{ \begin{array}{l} 5a + 5y + 2z = -41 \\ y - 3z = 19 \\ 2y + 3z = -34 \end{array} \}

Using Substitution to Solve the System

To solve this system of equations, we can use the substitution method. We will start by solving the second equation for $y$ in terms of $z$.

$${ y - 3z = 19 \\ y = 19 + 3z }$$

Substituting into the First Equation

Now, we will substitute the expression for $y$ into the first equation.

$${ 5a + 5(19 + 3z) + 2z = -41 \\ 5a + 95 + 15z + 2z = -41 \\ 5a + 95 + 17z = -41 }$$

Simplifying the Equation

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

$${ 5a + 95 + 17z = -41 \\ 5a + 17z = -136 }$$

Substituting into the Third Equation

Now, we will substitute the expression for $y$ into the third equation.

$${ 2(19 + 3z) + 3z = -34 \\ 38 + 6z + 3z = -34 \\ 38 + 9z = -34 }$$

Simplifying the Equation

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

$${ 38 + 9z = -34 \\ 9z = -72 \\ z = -8 }$$

Finding the Value of $y$

Now that we have found the value of $z$, we can substitute it back into the expression for $y$.

$${ y = 19 + 3z \\ y = 19 + 3(-8) \\ y = 19 - 24 \\ y = -5 }$$

Finding the Value of $a$

Now that we have found the values of $y$ and $z$, we can substitute them back into the simplified equation from the first step.

$${ 5a + 17z = -136 \\ 5a + 17(-8) = -136 \\ 5a - 136 = -136 \\ 5a = 0 \\ a = 0 }$$

Conclusion

In this article, we have solved a system of linear equations and found the value of $a$. We used the substitution method to solve the system, and we were able to find the values of $y$ and $z$ as well. The final answer is $a = 0$.

Discussion

The value of $a$ in the solution of the system of equations is $0$. This is because the system of equations is consistent, and there is a unique solution. The value of $a$ is determined by the first equation, and it is equal to $0$ because the other two equations are linearly dependent.

Final Answer

The final answer is $\boxed{0}$.

Introduction

In our previous article, we solved a system of linear equations and found the value of $a$. We used the substitution method to solve the system, and we were able to find the values of $y$ and $z$ as well. 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: A system of linear equations is a set of two or more linear equations that are related to each other. Each equation is a linear equation, which means it is an equation in which the highest power of the variable is 1.

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

A: There are several methods for solving a system of linear equations, including the substitution method, the elimination method, and the graphing method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method for solving a system of linear equations in which one equation is solved for one variable and then that expression is substituted into the other equation. This method is useful when one of the equations is easily solvable for one variable.

Q: What is the elimination method?

A: The elimination method is a method for solving a system of linear equations in which the equations are added or subtracted to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations.

Q: How do I determine which method to use?

A: To determine which method to use, you should first look at the equations and see if one of them is easily solvable for one variable. If so, the substitution method may be the best choice. If not, you may want to try the elimination method.

Q: What is the graphing method?

A: The graphing method is a method for solving a system of linear equations in which the equations are graphed on a coordinate plane and the point of intersection is found. This method is useful when the equations are not easily solvable using the substitution or elimination methods.

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

A: To graph a system of linear equations, you should first graph each equation on a coordinate plane. Then, you should find the point of intersection of the two graphs. This point of intersection is the solution to the system of equations.

Q: What is the point of intersection?

A: The point of intersection is the point where the two graphs meet. This point is the solution to the system of equations.

Q: How do I find the point of intersection?

A: To find the point of intersection, you should first graph each equation on a coordinate plane. Then, you should find the point where the two graphs meet. This point is the solution to the system of equations.

Q: What is the solution to a system of linear equations?

A: The solution to a system of linear equations is the set of values that satisfy all of the equations in the system. This solution is the point of intersection of the graphs of the equations.

Q: How do I check my solution?

A: To check your solution, you should substitute the values of the variables into each equation and see if the equation is true. If it is, then your solution is correct. If not, then you should recheck your work.

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 graphing method, and we have provided examples of how to use each method. We have also discussed how to determine which method to use and how to check your solution.