Solve The System Of Equations:${ \begin{aligned} 9x - 8y &= -29 \ -4x + 5y &= 23 \end{aligned} }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. There Is One Solution. The Solution Of The

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.

The System of Equations

The given system of equations is:

$$\begin{aligned} 9x - 8y &= -29 \\ -4x + 5y &= 23 \end{aligned}$$

Method 1: Substitution Method

The substitution method is a popular method for solving systems of linear equations. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the First Equation for x

We can solve the first equation for x by adding 8y to both sides of the equation:

$$9x = -29 + 8y$$

Next, we can divide both sides of the equation by 9 to solve for x:

$$x = \frac{-29 + 8y}{9}$$

Step 2: Substitute the Expression for x into the Second Equation

Now that we have an expression for x, we can substitute it into the second equation:

$$-4\left(\frac{-29 + 8y}{9}\right) + 5y = 23$$

Step 3: Simplify the Equation

To simplify the equation, we can start by multiplying both sides of the equation by 9 to eliminate the fraction:

$$-4(-29 + 8y) + 45y = 207$$

Next, we can distribute the -4 to the terms inside the parentheses:

$$116 - 32y + 45y = 207$$

Step 4: Combine Like Terms

Now that we have simplified the equation, we can combine like terms:

$$13y = 91$$

Step 5: Solve for y

Finally, we can solve for y by dividing both sides of the equation by 13:

$$y = \frac{91}{13}$$

Step 6: Find the Value of x

Now that we have the value of y, we can substitute it back into the expression for x that we found in Step 1:

$$x = \frac{-29 + 8\left(\frac{91}{13}\right)}{9}$$

Simplifying the expression, we get:

$$x = \frac{-29 + \frac{728}{13}}{9}$$

$$x = \frac{-377 + 728}{117}$$

$$x = \frac{351}{117}$$

$$x = 3$$

Method 2: Elimination Method

The elimination method is another popular method for solving systems of linear equations. This method involves adding or subtracting the equations to eliminate one variable.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of x or y in both equations are the same.

Multiplying the first equation by 4 and the second equation by 9, we get:

$$36x - 32y = -116$$

$$-36x + 45y = 207$$

Step 2: Add the Equations

Now that we have the equations with the same coefficients, we can add them to eliminate one variable:

$$-32y + 45y = -116 + 207$$

$$13y = 91$$

Step 3: Solve for y

Finally, we can solve for y by dividing both sides of the equation by 13:

$$y = \frac{91}{13}$$

Step 4: Find the Value of x

Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x.

Substituting y into the first equation, we get:

$$9x - 8\left(\frac{91}{13}\right) = -29$$

Simplifying the equation, we get:

$$9x - \frac{728}{13} = -29$$

$$9x = -29 + \frac{728}{13}$$

$$9x = \frac{-377 + 728}{13}$$

$$9x = \frac{351}{13}$$

$$x = \frac{351}{117}$$

$$x = 3$$

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method and the elimination method. We found that the solution to the system is x = 3 and y = 7. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one variable. Both methods are useful for solving systems of linear equations, and the choice of method depends on the specific problem and the student's preference.

Final Answer

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it can be a challenging task for many students. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables using the substitution method and the elimination method. In this article, we will provide a Q&A guide to help students understand the concepts and techniques involved in 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 involve two or more variables. Each equation in the system is a linear equation, which means that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know which method to use to solve a system of linear equations?

A: The choice of method depends on the specific problem and the student's preference. The substitution method is often used when one of the equations is easily solvable for one variable, while the elimination method is often used when the coefficients of the variables in both equations are the same.

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is often used when one of the equations is easily solvable for one variable.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable. This method is often used when the coefficients of the variables in both equations are the same.

Q: How do I solve a system of linear equations using the substitution method?

A: To solve a system of linear equations using the substitution method, follow these steps:

1. Solve one equation for one variable.
2. Substitute the expression for the variable into the other equation.
3. Simplify the equation and solve for the other variable.
4. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: How do I solve a system of linear equations using the elimination method?

A: To solve a system of linear equations using the elimination method, follow these steps:

1. Multiply the equations by necessary multiples such that the coefficients of the variables in both equations are the same.
2. Add or subtract the equations to eliminate one variable.
3. Simplify the equation and solve for the other variable.
4. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

• Not following the order of operations when simplifying equations.
• Not checking the solutions to ensure that they satisfy both equations.
• Not using the correct method for the specific problem.
• Not being careful when multiplying and dividing equations.

Q: How can I practice solving systems of linear equations?

A: There are many ways to practice solving systems of linear equations, including:

• Working through practice problems in a textbook or online resource.
• Using online tools or software to generate random systems of linear equations.
• Creating your own systems of linear equations and solving them.
• Joining a study group or working with a tutor to practice solving systems of linear equations.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it requires practice and patience to master. By following the steps outlined in this Q&A guide, students can develop the skills and confidence they need to solve systems of linear equations. Remember to practice regularly and to be careful when simplifying equations and checking solutions.

Final Tips

• Always follow the order of operations when simplifying equations.
• Check the solutions to ensure that they satisfy both equations.
• Use the correct method for the specific problem.
• Be careful when multiplying and dividing equations.
• Practice regularly to develop your skills and confidence.