Solve The System Of Equations:${ \begin{array}{l} 3x + Y = 16 \ -8x + Y = -6 \end{array} }$

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:

$$\begin{array}{l} 3x + y = 16 \\ -8x + y = -6 \end{array}$$

Understanding the System

To solve this system, we need to understand the concept of linear equations and how they relate to each other. A linear equation is an equation in which the highest power of the variable(s) is 1. In this case, we have two linear equations with two variables, x and y.

Method 1: Substitution Method

One way to solve this system is by using the substitution method. 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 y

We can solve the first equation for y by subtracting 3x from both sides:

$$y = 16 - 3x$$

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

Now, we can substitute the expression for y into the second equation:

$$-8x + (16 - 3x) = -6$$

Step 3: Simplify the Equation

We can simplify the equation by combining like terms:

$$-8x + 16 - 3x = -6$$

$$-11x + 16 = -6$$

Step 4: Solve for x

Now, we can solve for x by subtracting 16 from both sides and then dividing by -11:

$$-11x = -22$$

$$x = \frac{-22}{-11}$$

$$x = 2$$

Step 5: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:

$$3(2) + y = 16$$

$$6 + y = 16$$

$$y = 10$$

Method 2: Elimination Method

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

Step 1: Multiply the Equations by Necessary Multiples

We can multiply the first equation by 8 and the second equation by 3 to make the coefficients of x in both equations equal:

$$24x + 8y = 128$$

$$-24x + 3y = -18$$

Step 2: Add the Equations

Now, we can add the equations to eliminate the variable x:

$$(24x - 24x) + (8y + 3y) = 128 - 18$$

$$11y = 110$$

Step 3: Solve for y

Now, we can solve for y by dividing both sides by 11:

$$y = \frac{110}{11}$$

$$y = 10$$

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations. We will use the first equation:

$$3x + 10 = 16$$

$$3x = 6$$

$$x = 2$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. We have shown that both methods can be used to solve systems of linear equations and have provided a step-by-step guide on how to use each method.

Key Takeaways

• Systems of linear equations are a fundamental concept in mathematics.
• There are two main methods for solving systems of linear equations: the substitution method and the elimination method.
• The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
• The elimination method involves adding or subtracting the equations to eliminate one of the variables.
• Both methods can be used to solve systems of linear equations.

Real-World Applications

Systems of linear equations have many real-world applications, including:

• Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the flow of fluids.
• Economics: Systems of linear equations are used to model economic systems and make predictions about economic trends.
• Computer Science: Systems of linear equations are used in computer science to solve problems in fields such as machine learning and data analysis.

Final Thoughts

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 related to each other. Each equation in the system is a linear equation, which means that the highest power of the variable(s) is 1.

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

A: A system of linear equations has a solution if and only if the two equations are not parallel. If the two equations are parallel, then the system has no solution. If the two equations are not parallel, then the system has a solution.

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

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

Q: Can I use the substitution method or the elimination method to solve a system of linear equations?

A: Yes, you can use either the substitution method or the elimination method to solve a system of linear equations. The choice of method depends on the specific system and the variables involved.

Q: How do I know which method to use?

A: To determine which method to use, you can try the following:

• If one of the equations is already solved for one variable, you can use the substitution method.
• If the coefficients of one of the variables are the same in both equations, you can use the elimination method.
• If neither of the above conditions is true, you can try using the elimination method.

Q: What if I get a contradiction when using the elimination method?

A: If you get a contradiction when using the elimination method, it means that the system has no solution. This can happen if the two equations are parallel.

Q: Can I use a graphing calculator to solve a system of linear equations?

A: Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can be used to graph the two equations and find the point of intersection, which is the solution to the system.

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

A: If you have a system of linear equations with three or more variables, you can use the same methods as before, but you may need to use more advanced techniques, such as matrix operations or Gaussian elimination.

Q: Can I use a computer program to solve a system of linear equations?

A: Yes, you can use a computer program to solve a system of linear equations. Many computer programs, such as MATLAB or Python, have built-in functions for solving systems of linear equations.

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

A: If you have a system of linear equations with complex numbers, you can use the same methods as before, but you may need to use more advanced techniques, such as complex arithmetic or matrix operations.

Conclusion

In this article, we have answered some of the most frequently asked questions about systems of linear equations. We have covered topics such as the definition of a system of linear equations, the difference between the substitution method and the elimination method, and how to determine which method to use. We have also discussed some of the real-world applications of systems of linear equations and provided some tips for solving systems of linear equations.

Key Takeaways

• A system of linear equations is a set of two or more linear equations that are related to each other.
• The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
• The elimination method involves adding or subtracting the equations to eliminate one of the variables.
• You can use either the substitution method or the elimination method to solve a system of linear equations.
• If you get a contradiction when using the elimination method, it means that the system has no solution.

Real-World Applications

Systems of linear equations have many real-world applications, including:

• Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the flow of fluids.
• Economics: Systems of linear equations are used to model economic systems and make predictions about economic trends.
• Computer Science: Systems of linear equations are used in computer science to solve problems in fields such as machine learning and data analysis.

Final Thoughts

Solving systems of linear equations is a crucial skill for students and professionals alike. By understanding the concept of linear equations and how to solve them, we can apply this knowledge to real-world problems and make predictions about complex systems. In this article, we have provided some answers to frequently asked questions about systems of linear equations and have covered some of the real-world applications of this concept. We hope that this article has been helpful in understanding the concept of systems of linear equations.