Lesson 8 Homework Practice Solve Systems Of Equations Algebraically Page 51 Fix

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How to Solve Systems of Equations Algebraically: A Step-by-Step Guide for Lesson 8 Homework Practice

A system of equations is a set of two or more equations that have the same variables. Solving a system of equations means finding the values of the variables that make all the equations true. There are different methods to solve systems of equations, such as graphing, substitution, elimination, and matrix methods. In this article, we will focus on how to solve systems of equations algebraically using substitution and elimination methods.

Solving systems of equations algebraically is an important skill that you will need in algebra and other math courses. It can also help you model and solve real-world problems that involve relationships between different quantities. For example, you can use systems of equations to find the optimal price and quantity for a product, or to determine the speed and distance of two moving objects.

In lesson 8 homework practice, you will learn how to solve systems of equations algebraically using substitution and elimination methods. You will also practice solving different types of systems of equations, such as linear, nonlinear, consistent, inconsistent, dependent, and independent. You will also learn how to check your solutions and write them in different forms.

What are Substitution and Elimination Methods?

Substitution and elimination are two common methods to solve systems of equations algebraically. They both involve manipulating the equations to eliminate one or more variables and find the solution.

The substitution method works by solving one equation for one variable in terms of the other variable(s), and then substituting that expression into the other equation(s). This way, you can reduce the system to a single equation with one variable, which you can then solve easily. The substitution method is useful when one equation is already solved for one variable, or when it is easy to do so.

The elimination method works by adding or subtracting the equations to eliminate one or more variables. This way, you can reduce the system to a single equation with one variable, which you can then solve easily. The elimination method is useful when the coefficients of one variable are opposites or multiples in both equations.

How to Solve Systems of Equations Algebraically Using Substitution Method

To solve systems of equations algebraically using substitution method, you need to follow these steps:

Choose one equation and solve it for one variable in terms of the other variable(s).

Substitute the expression you found in step 1 into the other equation(s) and simplify.

Solve the resulting equation for the remaining variable.

Substitute the value you found in step 3 into the expression you found in step 1 and solve for the other variable(s).

Check your solution by plugging it into the original equations and verifying that they are true.

Write your solution in the form (x,y) or (x,y,z), where x, y, and z are the variables.

Let's look at an example of how to use the substitution method to solve a system of equations.

Example: Solve the system of equations using substitution method.

\begin{align*}

y &= 2x + 3 \tag{1} \\

x - y &= -4 \tag{2}

\end{align*}

Solution:

We choose equation (1) and solve it for y in terms of x. We get: \begin{align*}

y &= 2x + 3

\end{align*}

We substitute the expression for y into equation (2) and simplify. We get: \begin{align*}

x - (2x + 3) &= -4 \\

-x - 3 &= -4 \\

-x &= -1 \\

x &= 1

\end{align*}

We solve for x and get: \begin{align*}

x &= 1

\end{align*}

We substitute the value of x into the expression for y and solve for y. We get: \begin{align*}

y &= 2(1) + 3 \\

y &= 5

\end{align*}

We check our solution by plugging it into the original equations and verifying that they are true. We get: \begin{align*}

y &= 2x + 3 \\

5 &= 2(1) + 3 \\

5 &= 5 \quad \checkmark \\

x - y &= -4 \\

1 - 5 &= -4 \\

-4 &= -4 \quad \checkmark

\end{align*}

We write our solution in the form (x,y). We get: \begin{align*}

(x,y) = (1,5)

\end{align*}

The solution is (1,5).

How to Solve Systems of Equations Algebraically Using Elimination Method

To solve systems of equations algebraically using elimination method, you need to follow these steps:

Choose one variable and multiply or divide both equations by a number to make the coefficients of that variable opposites or multiples.

Add or subtract the equations to eliminate that variable and get a single equation with one variable.

Solve the resulting equation for the remaining variable.

Substitute the value you found in step 3 into any of the original equations and solve for the other variable(s).

Check your solution by plugging it into the original equations and verifying that they are true.

Write your solution in the form (x,y) or (x,y,z), where x, y, and z are the variables.

Let's look at an example of how to use the elimination method to solve a system of equations.

Example: Solve the system of equations using elimination method.

\begin{align*}

3x + 2y &= 8 \tag{1} \\

2x - 2y &= -4 \tag{2}

\end{align*}

Solution:

We choose y as the variable to eliminate and multiply equation (1) by -1 to make the coefficients of y opposites. We get: \begin{align*}

-3x - 2y &= -8 \tag{3} \\

2x - 2y &= -4 \tag{4}

\end{align*}

We add equations (3) and (4) to eliminate y and get a single equation with x. We get: \begin{align*}

-3x - 2y + 2x - 2y &= -8 + (-4) \\

-x &= -12 \\

x &= 12

\end{align*}

We solve for x and get: \begin{align*}

x &= 12

\end{align*}

We substitute the value of x into any of the original equations and solve for y. We choose equation (1). We get: \begin{align*}

3x + 2y &= 8 \\

3(12) + 2y &= 8 \\

36 + 2y &= 8 \\

2y &= -28 \\

y &= -14

\end{align*}

We check our solution by plugging it into the original equations and verifying that they are true. We get: \begin{align*}

3x + 2y &= 8 \\

3(12) + 2(-14) &= 8 \\

36 - 28 &= 8 \\

8 &= 8 \quad \checkmark \\

2x - 2y &= -4 \\

2(12) - 2(-14) &= -4 \\

24 + 28 &= -4 \\

-4 &= -4 \quad \checkmark

\end{align*}

We write our solution in the form (x,y). We get: \begin{align*}

(x,y) = (12,-14)

\end{align*}

The solution is (12,-14).

How to Solve Different Types of Systems of Equations Algebraically

Not all systems of equations are the same. Depending on the number and type of equations and variables, systems of equations can be classified into different categories. Here are some common types of systems of equations that you will encounter in lesson 8 homework practice:

Linear systems: These are systems of equations where all the equations are linear, meaning that the variables have a degree of one. For example: \begin{align*}

2x + y &= 5 \\

3x - y &= 1

\end{align*}

Nonlinear systems: These are systems of equations where at least one equation is nonlinear, meaning that the variables have a degree higher than one or involve other functions. For example: \begin{align*}

x^2 + y^2 &= 25 \\

y &= x + 3

\end{align*}

Consistent systems: These are systems of equations that have at least one solution. For example: \begin{align*}

x + y &= 4 \\

x - y &= 2

\end{align*}

Inconsistent systems: These are systems of equations that have no solution. For example: \begin{align*}

x + y &= 4 \\

x + y &= 5

\end{align*}

Dependent systems: These are systems of equations that have infinitely many solutions. For example: \begin{align*}

x + y &= 4 \\

2x + 2y &= 8

\end{align*}

Independent systems: These are systems of equations that have exactly one solution. For example: \begin{align*}

x + y &= 4 \\

x - y &= 2

\end{align*}

To solve different types of systems of equations algebraically, you need to use the appropriate method and check the nature and number of solutions. You can use substitution or elimination methods for linear systems, but you may need to use other methods for nonlinear systems, such as graphing or factoring. You can also use the determinant of a matrix to determine if a system is consistent or inconsistent.

How to Check Your Solutions and Write Them in Different Forms

After solving a system of equations algebraically, you should always check your solution by plugging it into the original equations and verifying that they are true. This way, you can avoid mistakes and ensure that your solution is correct.

You should also write your solution in the correct form depending on the type and number of variables and solutions. Here are some common forms to write your solution:

If the system has two variables and one solution, write your solution in the form (x,y), where x and y are the values of the variables. For example: \begin{align*}

(x,y) = (2,3)

\end{align*}

If the system has three variables and one solution, write your solution in the form (x,y,z), where x, y, and z are the values of the variables. For example: \begin{align*}

(x,y,z) = (1,-2,4)

\end{align*}

If the system has infinitely many solutions, write your solution in terms of a parameter, such as t or s, and use set notation. For example: \begin{align*}

(x,y) = \{(t,t+3) | t \in \mathbb{R}\}

\end{align*}

If the system has no solution, write "no solution" or use the empty set symbol $\emptyset$. For example: \begin{align*}

\text{no solution} \quad \text{or} \quad \emptyset

\end{align*}

Conclusion

In this article, you learned how to solve systems of equations algebraically using substitution and elimination methods. You also learned how to solve different types of systems of equations, such as linear, nonlinear, consistent, inconsistent, dependent, and independent. You also learned how to check your solutions and write them in different forms.

Solving systems of equations algebraically is an important skill that you will need in algebra and other math courses. It can also help you model and solve real-world problems that involve relationships between different quantities. For example, you can use systems of equations to find the optimal price and quantity for a product, or to determine the speed and distance of two moving objects.

In lesson 8 homework practice, you will practice solving systems of equations algebraically using substitution and elimination methods. You will also practice solving different types of systems of equations and writing your solutions in different forms.

We hope this article was helpful and informative for you. If you have any questions or feedback, please feel free to contact us. Thank you for reading and good luck with your homework! 4aad9cdaf3