Solve The System Of Linear Equations Shown Below:${ \begin{array}{l} 2a - 3b = 18 \ 5a + 2b = 7 \end{array} }$Fill In The Blanks To Write The Solution In The Form { (a, B)$} : : : { (\square, \square) \}

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Introduction

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Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple linear equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution.

The System of Linear Equations

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The system of linear equations we will be solving is:

{ \begin{array}{l} 2a - 3b = 18 \\ 5a + 2b = 7 \end{array} \}

Our goal is to find the values of $a$ and $b$ that satisfy both equations.

Method of Substitution

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One way to solve this system is by using the method of substitution. This involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the First Equation for $a$

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

$2a - 3b = 18$

Add $3b$ to both sides:

$2a = 18 + 3b$

Divide both sides by 2:

$a = \frac{18 + 3b}{2}$

Step 2: Substitute the Expression for $a$ into the Second Equation

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

$5a + 2b = 7$

Substitute $a = \frac{18 + 3b}{2}$:

$5\left(\frac{18 + 3b}{2}\right) + 2b = 7$

Step 3: Simplify the Equation

Simplify the equation:

$\frac{5(18 + 3b)}{2} + 2b = 7$

Multiply both sides by 2 to eliminate the fraction:

$5(18 + 3b) + 4b = 14$

Expand the equation:

$90 + 15b + 4b = 14$

Combine like terms:

$90 + 19b = 14$

Subtract 90 from both sides:

$19b = -76$

Divide both sides by 19:

$b = -\frac{76}{19}$

Step 4: Find the Value of $a$

Now that we have the value of $b$, substitute it back into the expression for $a$:

$a = \frac{18 + 3b}{2}$

Substitute $b = -\frac{76}{19}$:

$a = \frac{18 + 3\left(-\frac{76}{19}\right)}{2}$

Simplify the expression:

$a = \frac{18 - \frac{228}{19}}{2}$

Multiply both sides by 19 to eliminate the fraction:

$19a = 19(18 - \frac{228}{19})$

Expand the equation:

$19a = 342 - 228$

Combine like terms:

$19a = 114$

Divide both sides by 19:

$a = \frac{114}{19}$

The Solution

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The solution to the system of linear equations is:

$(a, b) = \left(\frac{114}{19}, -\frac{76}{19}\right)$

Conclusion

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Solving a system of linear equations involves finding the values of variables that satisfy multiple linear equations simultaneously. In this article, we used the method of substitution to solve a system of two linear equations with two variables. We found the values of $a$ and $b$ that satisfy both equations and presented the solution in the form $(a, b)$. This method can be applied to more complex systems of linear equations with multiple variables.

Future Directions

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In the future, we can explore other methods for solving systems of linear equations, such as the method of elimination and the use of matrices. We can also apply these methods to more complex systems of linear equations with multiple variables.

References

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• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon

Glossary

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• System of linear equations: A set of linear equations with multiple variables.
• Method of substitution: A method for solving a system of linear equations by solving one equation for one variable and substituting that expression into the other equation.
• Method of elimination: A method for solving a system of linear equations by adding or subtracting the equations to eliminate one variable.
• Matrix: A rectangular array of numbers used to represent a system of linear equations.
  

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Introduction

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Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra. In our previous article, we explored the method of substitution to solve a system of two linear equations with two variables. In this article, we will address some frequently asked questions related to solving systems of linear equations.

Q&A

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Q: What is a system of linear equations?

A: A system of linear equations is a set of linear equations with multiple variables. Each equation is a linear combination of the variables, and the system is a collection of these equations.

Q: What are the different methods for solving systems of linear equations?

A: There are several methods for solving systems of linear equations, including:

• Method of substitution: This involves solving one equation for one variable and substituting that expression into the other equation.
• Method of elimination: This involves adding or subtracting the equations to eliminate one variable.
• Matrix method: This involves using matrices to represent the system of linear equations and solving for the variables.

Q: How do I choose the method of substitution or elimination?

A: The choice of method depends on the specific system of linear equations and the variables involved. If the system has two variables and two equations, the method of substitution or elimination may be suitable. If the system has multiple variables and equations, the matrix method may be more efficient.

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

A: A system of linear equations consists of linear equations, where each equation is a linear combination of the variables. A system of nonlinear equations, on the other hand, consists of nonlinear equations, where each equation is a nonlinear combination of the variables.

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

A: To solve a system of linear equations with multiple variables, you can use the matrix method. This involves representing the system as a matrix and solving for the variables using matrix operations.

Q: What is the significance of solving systems of linear equations?

A: Solving systems of linear equations has numerous applications in various fields, including physics, engineering, economics, and computer science. It is used to model real-world problems, such as optimization, prediction, and decision-making.

Q: Can I use technology to solve systems of linear equations?

A: Yes, you can use technology, such as calculators, computers, or software, to solve systems of linear equations. Many software packages, such as MATLAB or Mathematica, have built-in functions for solving systems of linear equations.

Q: How do I check my solution to a system of linear equations?

A: To check your solution, substitute the values of the variables back into the original equations and verify that they are satisfied. You can also use the matrix method to check your solution.

Conclusion

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Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we addressed some frequently asked questions related to solving systems of linear equations. We hope that this article has provided you with a better understanding of the methods and techniques for solving systems of linear equations.

Future Directions

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In the future, we can explore other methods for solving systems of linear equations, such as the use of numerical methods or the application of linear algebra techniques. We can also apply these methods to more complex systems of linear equations with multiple variables.

References

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• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon

Glossary

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• System of linear equations: A set of linear equations with multiple variables.
• Method of substitution: A method for solving a system of linear equations by solving one equation for one variable and substituting that expression into the other equation.
• Method of elimination: A method for solving a system of linear equations by adding or subtracting the equations to eliminate one variable.
• Matrix method: A method for solving a system of linear equations by representing the system as a matrix and solving for the variables using matrix operations.