Solve The System Of Equations:${ \begin{array}{l} x - 3y = 12 \ -x + 9y = -18 \end{array} }$

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:

$$\begin{array}{l} x - 3y = 12 \\ -x + 9y = -18 \end{array}$$

Understanding the System

To solve this system, we need to understand the concept of linear equations and how they can be represented graphically. 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 x

We can solve the first equation for x by adding 3y to both sides:

$$x = 12 + 3y$$

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

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

$$-(12 + 3y) + 9y = -18$$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$$-12 - 3y + 9y = -18$$

$$6y = -6$$

Step 4: Solve for y

Dividing both sides by 6, we get:

$$y = -1$$

Step 5: Substitute the Value of y into the Expression for x

Now, we can substitute the value of y into the expression for x:

$$x = 12 + 3(-1)$$

$$x = 9$$

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 variable.

Step 1: Multiply the First Equation by 1 and the Second Equation by 1

We can multiply the first equation by 1 and the second equation by 1:

$$x - 3y = 12$$

$$-x + 9y = -18$$

Step 2: Add the Two Equations

Adding the two equations, we get:

$$-3y + 9y = 12 - 18$$

$$6y = -6$$

Step 3: Solve for y

Dividing both sides by 6, we get:

$$y = -1$$

Step 4: Substitute the Value of y into One of the Original Equations

Now, we can substitute the value of y into one of the original equations:

$$x - 3(-1) = 12$$

$$x + 3 = 12$$

Step 5: Solve for x

Subtracting 3 from both sides, we get:

$$x = 9$$

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 shown that both methods can be used to solve the system, and we have provided a step-by-step guide on how to do it. We hope that this article has provided a clear understanding of how to solve systems of linear equations and has helped students and professionals alike to improve their math skills.

Tips and Tricks

• When solving systems of linear equations, it is essential to understand the concept of linear equations and how they can be represented graphically.
• The substitution method and the elimination method are two common methods used to solve systems of linear equations.
• When using the substitution method, it is essential to solve one equation for one variable and then substitute that expression into the other equation.
• When using the elimination method, it is essential to add or subtract the equations to eliminate one variable.
• It is always a good idea to check the solution by substituting the values of x and y into the original equations.

Real-World Applications

Systems of linear equations have numerous 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 behavior of electrical circuits.
• Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
• Computer Science: Systems of linear equations are used in computer science to solve problems, such as linear programming and graph theory.

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 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 know which method to use?

A: You can use either method, but the substitution method is often easier to use when one of the equations is already solved for one variable. The elimination method is often easier to use when the coefficients of the variables are the same in both equations.

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

A: In this case, you can use the substitution method or the elimination method to solve for two variables, and then substitute those values into the remaining equation to solve for the third variable.

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

A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: How do I check my solution?

A: To check your solution, you can substitute the values of the variables into the original equations and see if 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.

Q: Can I use systems of linear equations to model real-world problems?

A: Yes, you can use systems of linear equations to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and economic systems.

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

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

• Physics and Engineering: Systems of linear equations are used to model the motion of objects and the behavior of electrical circuits.
• Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
• Computer Science: Systems of linear equations are used in computer science to solve problems, such as linear programming and graph theory.

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

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own using a calculator or a computer program.

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 the solution: Make sure to check your solution by substituting the values of the variables into the original equations.
• Not using the correct method: Make sure to use the correct method for the type of system you are working with.
• Not simplifying the equations: Make sure to simplify the equations before solving them.

Q: How can I improve my skills in solving systems of linear equations?

A: You can improve your skills in solving systems of linear equations by:

• Practicing regularly: Practice solving systems of linear equations regularly to build your skills and confidence.
• Using online resources: Use online resources, such as video tutorials and practice problems, to help you learn and practice solving systems of linear equations.
• Seeking help: Seek help from a teacher or tutor if you are struggling with a particular concept or problem.