Solve The System Of Equations Using The Substitution Method.$ \begin{align*} 2x + 3y &= 1 \ y &= X - 8 \end{align*} }$Choose The Correct Solution A. { (0, -8)$ $B. { (2, -6)$}$C. { (4, -4)$} D . \[ D. \[ D . \[ (5,

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. There are several methods to solve a system of equations, including the substitution method and the elimination method. In this article, we will focus on the substitution method, which involves substituting one equation into another to solve for the variables.

What is the Substitution Method?

The substitution method is a technique used to solve a system of equations by substituting one equation into another. This method is useful when one of the equations is already solved for one of the variables. The substitution method involves the following steps:

1. Identify the equation that is already solved for one of the variables.
2. Substitute the expression for the variable into the other equation.
3. Solve the resulting equation for the other variable.
4. Substitute the value of the variable back into one of the original equations to find the value of the other variable.

Step-by-Step Solution

Let's use the given system of equations to demonstrate the substitution method.

{ \begin{align*} 2x + 3y &= 1 \\ y &= x - 8 \end{align*} \}

Step 1: Identify the Equation Already Solved for One of the Variables

In this system of equations, the second equation is already solved for the variable $y$. The equation is $y = x - 8$.

Step 2: Substitute the Expression for the Variable into the Other Equation

Substitute the expression for $y$ into the first equation:

$2x + 3(x - 8) = 1$

Step 3: Solve the Resulting Equation for the Other Variable

Expand and simplify the equation:

$2x + 3x - 24 = 1$

Combine like terms:

$5x - 24 = 1$

Add 24 to both sides:

$5x = 25$

Divide both sides by 5:

$x = 5$

Step 4: Substitute the Value of the Variable Back into One of the Original Equations

Substitute the value of $x$ back into the second equation:

$y = x - 8$

$y = 5 - 8$

$y = -3$

Conclusion

The solution to the system of equations is $(5, -3)$.

Choosing the Correct Solution

Now that we have solved the system of equations, let's compare our solution to the given options:

A. $(0, -8)$ B. $(2, -6)$ C. $(4, -4)$ D. $(5, -3)$

Our solution matches option D, $(5, -3)$.

Conclusion

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Introduction

In our previous article, we used the substitution method to solve a system of equations. In this article, we will answer some frequently asked questions about the substitution method and provide additional examples to help you understand the concept better.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of equations by substituting one equation into another. This method is useful when one of the equations is already solved for one of the variables.

Q: How do I know which equation to substitute into the other?

A: You should substitute the equation that is already solved for one of the variables into the other equation. This will help you to eliminate one of the variables and solve for the other variable.

Q: What if I have two equations with two variables, but neither equation is solved for one of the variables?

A: In this case, you can use the elimination method to solve the system of equations. The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: Can I use the substitution method with any type of system of equations?

A: Yes, you can use the substitution method with any type of system of equations, including linear and nonlinear systems.

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

A: In this case, you can use the substitution method to solve for two variables, and then use the elimination method to solve for the remaining variable.

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

A: To check if your solution is correct, substitute the values of the variables back into both original equations. If the equations are true, then your solution is correct.

Q: What if I get a system of equations with no solution or infinitely many solutions?

A: If you get a system of equations with no solution, it means that the equations are inconsistent and there is no solution. If you get a system of equations with infinitely many solutions, it means that the equations are dependent and there are infinitely many solutions.

Examples

Let's use the following system of equations to demonstrate the substitution method:

{ \begin{align*} x + 2y &= 6 \\ y &= 2x - 3 \end{align*} \}

Step 1: Identify the Equation Already Solved for One of the Variables

In this system of equations, the second equation is already solved for the variable $y$. The equation is $y = 2x - 3$.

Step 2: Substitute the Expression for the Variable into the Other Equation

Substitute the expression for $y$ into the first equation:

$x + 2(2x - 3) = 6$

Step 3: Solve the Resulting Equation for the Other Variable

Expand and simplify the equation:

$x + 4x - 6 = 6$

Combine like terms:

$5x - 6 = 6$

Add 6 to both sides:

$5x = 12$

Divide both sides by 5:

$x = \frac{12}{5}$

Step 4: Substitute the Value of the Variable Back into One of the Original Equations

Substitute the value of $x$ back into the second equation:

$y = 2x - 3$

$y = 2(\frac{12}{5}) - 3$

$y = \frac{24}{5} - 3$

$y = \frac{24}{5} - \frac{15}{5}$

$y = \frac{9}{5}$

Conclusion

The solution to the system of equations is $(\frac{12}{5}, \frac{9}{5})$.

Conclusion

In this article, we answered some frequently asked questions about the substitution method and provided additional examples to help you understand the concept better. We demonstrated the substitution method with a system of equations and showed how to check if the solution is correct. We also discussed what to do if you get a system of equations with no solution or infinitely many solutions.