Find The Solution Of This System Of Equations. Separate The \[$x\$\]- And \[$y\$\]-values With A Comma.$\[ \begin{aligned} x & = 3 + Y \\ 21x + 8y & = -24 \end{aligned} \\]Enter The Correct Answer.

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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, x and y. We will use the given system of equations to find the values of x and y.

The System of Equations

The given system of equations is:

x=3+y21x+8y=−24\begin{aligned} x & = 3 + y \\ 21x + 8y & = -24 \end{aligned}

Step 1: Substitute the Expression for x into the Second Equation

To solve the system of equations, we can substitute the expression for x from the first equation into the second equation. This will give us an equation with only one variable, y.

x=3+y21(3+y)+8y=−24\begin{aligned} x & = 3 + y \\ 21(3 + y) + 8y & = -24 \end{aligned}

Step 2: Simplify the Equation

Now, we can simplify the equation by distributing the 21 to the terms inside the parentheses and combining like terms.

x=3+y63+21y+8y=−24\begin{aligned} x & = 3 + y \\ 63 + 21y + 8y & = -24 \end{aligned}

x=3+y29y+63=−24\begin{aligned} x & = 3 + y \\ 29y + 63 & = -24 \end{aligned}

Step 3: Isolate the Variable y

Next, we can isolate the variable y by subtracting 63 from both sides of the equation and then dividing both sides by 29.

x=3+y29y=−24−63\begin{aligned} x & = 3 + y \\ 29y & = -24 - 63 \end{aligned}

x=3+y29y=−87\begin{aligned} x & = 3 + y \\ 29y & = -87 \end{aligned}

x=3+yy=−87/29\begin{aligned} x & = 3 + y \\ y & = -87 / 29 \end{aligned}

x=3+yy=−3\begin{aligned} x & = 3 + y \\ y & = -3 \end{aligned}

Step 4: Find the Value of x

Now that we have found the value of y, we can substitute it back into the first equation to find the value of x.

x=3+yx=3+(−3)\begin{aligned} x & = 3 + y \\ x & = 3 + (-3) \end{aligned}

x=3+yx=0\begin{aligned} x & = 3 + y \\ x & = 0 \end{aligned}

Conclusion

In this article, we have solved a system of two linear equations with two variables, x and y. We used the given system of equations to find the values of x and y. The final answer is x = 0, y = -3.

Final Answer

Introduction

In our previous article, we solved a system of two linear equations with two 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 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 system of linear equations has a solution if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that are not linear, meaning that they cannot be written in the form ax + by = c, where a, b, and c are constants.

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

To solve a system of linear equations, you can use the following methods:

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

Q: What is the substitution method?

The substitution method is a method of solving a system of linear equations by substituting the expression for one variable from one equation into the other equation.

Q: What is the elimination method?

The elimination method is a method of solving a system of linear equations by adding or subtracting the two equations to eliminate one variable.

Q: What is the graphical method?

The graphical method is a method of solving a system of linear equations by graphing the two equations on a coordinate plane and finding the point of intersection.

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

A system of linear equations has a unique solution if the two equations are consistent and the lines intersect at a single point. A system of linear equations has no solution if the two equations are inconsistent and the lines do not intersect. A system of linear equations has infinitely many solutions if the two equations are equivalent and the lines coincide.

Q: What is the difference between a dependent system and an independent system?

A dependent system is a system of linear equations that has infinitely many solutions, while an independent system is a system of linear equations that has a unique solution.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the different methods of solving systems of linear equations, including the substitution method, elimination method, and graphical method. We have also discussed the different types of solutions that a system of linear equations can have, including unique solutions, no solutions, and infinitely many solutions.

Final Answer

The final answer is that solving systems of linear equations is an important topic in mathematics that has many real-world applications. By understanding the different methods of solving systems of linear equations, you can solve a wide range of problems in mathematics, science, and engineering.