Solve The System Of Equations Using Substitution.${ \begin{align} y &= X + 6 \ y &= -2x - 3 \end{align} }$A. { (-3, 3)$}$ B. { (4, -11)$}$ C. { \left(-6, \frac{3}{2}\right)$}$ D. { (1, 7)$}$

Mar 10, 2025 by ADMIN 197 views

**Solving Systems of Equations Using Substitution**

Introduction

Solving systems of equations is a fundamental concept in algebra, and it is essential to understand various methods to solve them. One of the most common methods is substitution, which involves solving one equation for a variable and then substituting that expression into the other equation. In this article, we will explore how to solve systems of equations using substitution.

What is Substitution?

Substitution is a method of solving systems of equations by solving one equation for a variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is linear and the other is quadratic or has a more complex form.

Step-by-Step Guide to Solving Systems of Equations Using Substitution

To solve a system of equations using substitution, follow these steps:

Identify the equations: Write down the two equations that make up the system.
Solve one equation for a variable: Choose one of the equations and solve it for one of the variables. This will give you an expression that you can substitute into the other equation.
Substitute the expression into the other equation: Take the expression you obtained in step 2 and substitute it into the other equation.
Solve for the remaining variable: Simplify the resulting equation and solve for the remaining variable.
Check your solution: Plug the values of the variables back into both original equations to ensure that they are true.

Example 1: Solving a System of Linear Equations

Let's consider the following system of linear equations:

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

To solve this system using substitution, we can follow the steps outlined above.

Step 1: Identify the equations

The two equations are:

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

Step 2: Solve one equation for a variable

Let's solve the first equation for $y$ :

{ y = x + 6 \}

Step 3: Substitute the expression into the other equation

Now, substitute the expression for $y$ into the second equation:

{ x + 6 = -2x - 3 \}

Step 4: Solve for the remaining variable

Simplify the resulting equation and solve for $x$ :

{ 3x = -9 \}

{ x = -3 \}

Step 5: Check your solution

Plug the value of $x$ back into both original equations to ensure that they are true:

{ y = x + 6 \}

{ y = -2x - 3 \}

Substituting $x = -3$ into both equations, we get:

{ y = -3 + 6 = 3 \}

{ y = -2(-3) - 3 = 3 \}

Both equations are true, so the solution is $x = -3$ and $y = 3$ .

Conclusion

Solving systems of equations using substitution is a powerful method that can be used to solve a wide range of problems. By following the steps outlined above, you can solve systems of linear and nonlinear equations with ease. Remember to always check your solution by plugging the values of the variables back into both original equations.

Answer

The correct answer is:

A. ${$ (-3, 3)$}$

Final Thoughts

Introduction

In our previous article, we explored how to solve systems of equations using substitution. This method is particularly useful when one of the equations is linear and the other is quadratic or has a more complex form. In this article, we will answer some frequently asked questions about solving systems of equations using substitution.

Q: What is the first step in solving a system of equations using substitution?

A: The first step in solving a system of equations using substitution is to identify the two equations that make up the system.

Q: How do I choose which equation to solve for a variable?

A: When choosing which equation to solve for a variable, look for the equation that has the variable you want to solve for. If one equation has a variable that is already isolated, it is often easier to solve for that variable first.

Q: What if I have a system of equations with two variables and one equation is quadratic? Can I still use substitution?

A: Yes, you can still use substitution to solve a system of equations with two variables and one equation is quadratic. However, you may need to use algebraic manipulation to isolate the variable in the quadratic equation.

Q: How do I know if I have found the correct solution?

A: To check if you have found the correct solution, plug the values of the variables back into both original equations. If both equations are true, then you have found the correct solution.

Q: What if I get a system of equations with no solution? How do I handle that?

A: If you get a system of equations with no solution, it means that the two equations are inconsistent. This can happen when the two equations represent parallel lines that never intersect. In this case, there is no solution to the system.

Q: Can I use substitution to solve a system of equations with three or more variables?

A: Yes, you can use substitution to solve a system of equations with three or more variables. However, it may be more complicated and require more algebraic manipulation.

Q: What are some common mistakes to avoid when using substitution to solve systems of equations?

A: Some common mistakes to avoid when using substitution to solve systems of equations include:

Not checking the solution by plugging the values of the variables back into both original equations
Not isolating the variable in one of the equations
Not simplifying the resulting equation after substitution
Not considering the possibility of a system with no solution

Conclusion

Solving systems of equations using substitution is a powerful method that can be used to solve a wide range of problems. By understanding the steps involved and avoiding common mistakes, you can become proficient in using substitution to solve systems of equations.

Final Thoughts

Solving systems of equations using substitution is a fundamental concept in algebra, and it is essential to understand various methods to solve them. By mastering this method, you can solve a wide range of problems and become proficient in algebra.

Additional Resources

For more information on solving systems of equations using substitution, check out the following resources:

Khan Academy: Solving Systems of Equations using Substitution
Mathway: Solving Systems of Equations using Substitution
Wolfram Alpha: Solving Systems of Equations using Substitution

Answer Key

Q: What is the first step in solving a system of equations using substitution? A: The first step in solving a system of equations using substitution is to identify the two equations that make up the system.
Q: How do I choose which equation to solve for a variable? A: When choosing which equation to solve for a variable, look for the equation that has the variable you want to solve for.
Q: What if I have a system of equations with two variables and one equation is quadratic? Can I still use substitution? A: Yes, you can still use substitution to solve a system of equations with two variables and one equation is quadratic.
Q: How do I know if I have found the correct solution? A: To check if you have found the correct solution, plug the values of the variables back into both original equations.
Q: What if I get a system of equations with no solution? How do I handle that? A: If you get a system of equations with no solution, it means that the two equations are inconsistent.
Q: Can I use substitution to solve a system of equations with three or more variables? A: Yes, you can use substitution to solve a system of equations with three or more variables.
Q: What are some common mistakes to avoid when using substitution to solve systems of equations? A: Some common mistakes to avoid when using substitution to solve systems of equations include not checking the solution, not isolating the variable, not simplifying the resulting equation, and not considering the possibility of a system with no solution.