Solve The System Of Equations. 3 X − 2 Y = 6 5 X + 4 Y = 32 \begin{array}{l} 3x - 2y = 6 \\ 5x + 4y = 32 \end{array} 3 X − 2 Y = 6 5 X + 4 Y = 32 A. (4, 3) B. (6, 6) C. (8, -2) D. (2, 0)

Mar 10, 2025 by ADMIN 192 views

**Solving a System of Linear Equations**

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple 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 Equations

The given system of equations is:

\begin{array}{l} 3x - 2y = 6 \\ 5x + 4y = 32 \end{array}

We are asked to find the values of $x$ and $y$ that satisfy both equations.

Method of Substitution

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

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

3x - 2y = 6

3x = 6 + 2y

x = \frac{6 + 2y}{3}

Now, substitute this expression for $x$ into the second equation:

5x + 4y = 32

5\left(\frac{6 + 2y}{3}\right) + 4y = 32

\frac{5(6 + 2y)}{3} + 4y = 32

\frac{30 + 10y}{3} + 4y = 32

30 + 10y + 12y = 96

22y = 66

y = 3

Now that we have found the value of $y$ , we can substitute it back into one of the original equations to find the value of $x$ . Let's use the first equation:

3x - 2y = 6

3x - 2(3) = 6

3x - 6 = 6

3x = 12

x = 4

Therefore, the solution to the system of equations is $(4, 3)$ .

Method of Elimination

Another way to solve this system of equations is by using the method of elimination. This involves adding or subtracting the equations to eliminate one variable.

Let's multiply the first equation by 4 and the second equation by 2:

12x - 8y = 24

10x + 8y = 64

Now, add the two equations together:

(12x - 8y) + (10x + 8y) = 24 + 64

22x = 88

x = 4

Now that we have found the value of $x$ , we can substitute it back into one of the original equations to find the value of $y$ . Let's use the first equation:

3x - 2y = 6

3(4) - 2y = 6

12 - 2y = 6

-2y = -6

y = 3

Therefore, the solution to the system of equations is $(4, 3)$ .

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have found that the solution to the system of equations is $(4, 3)$ . This is a fundamental concept in mathematics, and it has many applications in real-world problems.

Discussion

The method of substitution and elimination are two common methods used to solve systems of linear equations. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.

In this article, we have used the method of substitution and elimination to solve a system of two linear equations with two variables. We have found that the solution to the system of equations is $(4, 3)$ .

Practice Problems

Solve the system of equations:

\begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array}

Solve the system of equations:

\begin{array}{l} x + 2y = 7 \\ 3x - 2y = 11 \end{array}

Answer Key

$(5, 2)$
$(4, 3)$

References

"Algebra and Trigonometry" by Michael Sullivan
"Linear Algebra and Its Applications" by Gilbert Strang

About the Author

The author is a mathematics teacher with over 10 years of experience. He has taught algebra, geometry, and calculus to students of all levels. He is passionate about making mathematics accessible and enjoyable for everyone.

Contact Information

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it can be a bit challenging for some students. In this article, we will answer some frequently asked questions (FAQs) about 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. For example:

\begin{array}{l} 3x - 2y = 6 \\ 5x + 4y = 32 \end{array}

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

A: There are several methods to solve a system of linear equations, including the method of substitution and the method of elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the difference between the method of substitution and the method of elimination?

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: How do I choose which method to use?

A: You can choose either method, depending on which one seems easier to you. If you are solving a system of two linear equations with two variables, the method of substitution may be easier. If you are solving a system of two linear equations with two variables and the coefficients of the variables are the same, the method of elimination may be easier.

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

A: If you have a system of three or more linear equations, you can use the method of substitution or the method of elimination to solve it. However, you may need to use a more advanced method, such as the method of matrices or the method of determinants.

Q: What if I have a system of linear equations with fractions?

A: If you have a system of linear equations with fractions, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: What if I have a system of linear equations with decimals?

A: If you have a system of linear equations with decimals, you can multiply both sides of the equation by a power of 10 to eliminate the decimals.

Q: How do I check my answer?

A: To check your answer, you can substitute the values of the variables back into the original equations and make sure that they are true.

Q: What if I get a system of linear equations with no solution?

A: If you get a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

Q: What if I get a system of linear equations with infinitely many solutions?

A: If you get a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.