Given The Following System Of Equations, Find The Value Of $x$ In The Solution.$ \begin{array}{l} 5x - 4y = 28 \ 3x + 4y = 44 \end{array} $1. 8 2. 4 3. 12 4. 9

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. 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 to find the value of x in the solution.

The System of Equations

The given system of equations is:

$$\begin{array}{l} 5x - 4y = 28 \\ 3x + 4y = 44 \end{array}$$

Method 1: Substitution Method

The substitution method is a popular method for solving systems of linear equations. In this method, we solve one equation for one variable and then substitute that expression into the other equation.

Step 1: Solve the First Equation for x

We can solve the first equation for x by adding 4y to both sides of the equation:

$$5x = 28 + 4y$$

Next, we can divide both sides of the equation by 5 to solve for x:

$$x = \frac{28 + 4y}{5}$$

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:

$$3\left(\frac{28 + 4y}{5}\right) + 4y = 44$$

Step 3: Simplify the Equation

To simplify the equation, we can multiply both sides of the equation by 5 to eliminate the fraction:

$$3(28 + 4y) + 20y = 220$$

Next, we can distribute the 3 to the terms inside the parentheses:

$$84 + 12y + 20y = 220$$

Step 4: Combine Like Terms

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

$$84 + 32y = 220$$

Step 5: Solve for y

To solve for y, we can subtract 84 from both sides of the equation:

$$32y = 136$$

Next, we can divide both sides of the equation by 32 to solve for y:

$$y = \frac{136}{32}$$

$$y = \frac{17}{4}$$

Step 6: Substitute the Value of y into the Expression for x

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

$$x = \frac{28 + 4\left(\frac{17}{4}\right)}{5}$$

$$x = \frac{28 + 17}{5}$$

$$x = \frac{45}{5}$$

$$x = 9$$

Method 2: Elimination Method

The elimination method is another popular method for solving systems of linear equations. In this method, we add or subtract the equations to eliminate one variable.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate the y-variable, we can multiply the first equation by 4 and the second equation by 1:

$$20x - 16y = 112$$

$$3x + 4y = 44$$

Step 2: Add the Equations

Now that we have multiplied the equations by necessary multiples, we can add the equations to eliminate the y-variable:

$$(20x + 3x) + (-16y + 4y) = 112 + 44$$

$$23x - 12y = 156$$

Step 3: Solve for x

To solve for x, we can add 12y to both sides of the equation:

$$23x = 156 + 12y$$

Next, we can divide both sides of the equation by 23 to solve for x:

$$x = \frac{156 + 12y}{23}$$

Step 4: Substitute the Value of y into the Expression for x

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

$$x = \frac{156 + 12\left(\frac{17}{4}\right)}{23}$$

$$x = \frac{156 + 51}{23}$$

$$x = \frac{207}{23}$$

$$x = 9$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have found that the value of x in the solution is 9.

Discussion

The system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. The substitution method and the elimination method are two popular methods for solving systems of linear equations. 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.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon
• [3] "Solving Systems of Linear Equations" by Math Open Reference

Keywords

• System of linear equations
• Substitution method
• Elimination method
• Linear algebra
• Mathematics
  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 are solved simultaneously to find the values of the variables.

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

A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.

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.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I choose between the substitution method and the elimination method?

A: The choice between the substitution method and the elimination method depends on the specific system of equations. If the coefficients of one variable are the same in both equations, the elimination method is usually easier to use. If the coefficients of one variable are different in both equations, the substitution method is usually easier to use.

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 checking if the equations are consistent (i.e., if they have a solution)
• Not checking if the equations are independent (i.e., if they are not multiples of each other)
• Not following the correct order of operations (e.g., not multiplying both sides of an equation by the same value)
• Not checking if the solution is a valid solution (i.e., if it satisfies both equations)

Q: How do I check if a system of linear equations has a solution?

A: To check if a system of linear equations has a solution, you can use the following steps:

• Check if the equations are consistent (i.e., if they have a solution)
• Check if the equations are independent (i.e., if they are not multiples of each other)
• If the equations are consistent and independent, then the system has a solution.

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

A: Solving systems of linear equations is an important skill in mathematics and has numerous applications in various fields such as physics, engineering, and economics. It is used to model real-world problems and to make predictions and decisions.

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

A: Yes, you can use technology to solve systems of linear equations. Many graphing calculators and computer algebra systems (CAS) have built-in functions for solving systems of linear equations.

Q: What are some common applications of solving systems of linear equations?

A: Some common applications of solving systems of linear equations include:

• Modeling population growth and decline
• Modeling the spread of diseases
• Modeling the behavior of electrical circuits
• Modeling the behavior of mechanical systems
• Solving optimization problems

Q: Can I use solving systems of linear equations to solve optimization problems?

A: Yes, you can use solving systems of linear equations to solve optimization problems. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.

Q: What are some common types of optimization problems?

A: Some common types of optimization problems include:

• Linear programming problems
• Quadratic programming problems
• Nonlinear programming problems

Q: Can I use solving systems of linear equations to solve nonlinear programming problems?

A: No, you cannot use solving systems of linear equations to solve nonlinear programming problems. Nonlinear programming problems involve finding the maximum or minimum value of a nonlinear function subject to certain constraints.

Q: What are some common tools and techniques for solving nonlinear programming problems?

A: Some common tools and techniques for solving nonlinear programming problems include:

• Gradient descent methods
• Newton's method
• Quasi-Newton methods
• Conjugate gradient methods
• Interior-point methods

Q: Can I use solving systems of linear equations to solve quadratic programming problems?

A: Yes, you can use solving systems of linear equations to solve quadratic programming problems. Quadratic programming problems involve finding the maximum or minimum value of a quadratic function subject to certain constraints.

Q: What are some common types of quadratic programming problems?

A: Some common types of quadratic programming problems include:

• Quadratic programming problems with equality constraints
• Quadratic programming problems with inequality constraints
• Quadratic programming problems with both equality and inequality constraints