Solve The System Of Equations:${ \begin{array}{c} 5x + 4y = -30 \ -y = \frac{1}{2}x + 9 \end{array} }$

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will discuss how to solve a system of linear equations using the method of substitution and elimination.

The System of Equations

The system of equations we will be solving is:

$$\begin{array}{c} 5x + 4y = -30 \\ -y = \frac{1}{2}x + 9 \end{array}$$

Step 1: Simplify the Second Equation

The second equation can be simplified by multiplying both sides by -1 to get:

$$y = -\frac{1}{2}x - 9$$

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

We can substitute the expression for y into the first equation to get:

$$5x + 4(-\frac{1}{2}x - 9) = -30$$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$$5x - 2x - 36 = -30$$

Combine like terms:

$$3x - 36 = -30$$

Add 36 to both sides:

$$3x = 6$$

Divide both sides by 3:

$$x = 2$$

Step 4: Find the Value of y

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

$$y = -\frac{1}{2}(2) - 9$$

Simplify the expression:

$$y = -1 - 9$$

$$y = -10$$

Conclusion

We have solved the system of linear equations and found the values of x and y. The solution is x = 2 and y = -10.

Why is Solving Systems of Linear Equations Important?

Solving systems of linear equations is an important skill in mathematics and has many real-world applications. It is used in a variety of fields, including physics, engineering, economics, and computer science. Some examples of how solving systems of linear equations is used in real-world applications include:

• Physics: Solving systems of linear equations is used to describe the motion of objects in physics. For example, the equations of motion for an object under the influence of gravity can be written as a system of linear equations.
• Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of economic variables.
• Computer Science: Solving systems of linear equations is used in computer graphics and game development to create realistic simulations of real-world phenomena.

Tips and Tricks for Solving Systems of Linear Equations

Here are some tips and tricks for solving systems of linear equations:

• Use the method of substitution: The method of substitution is a powerful tool for solving systems of linear equations. It involves substituting the expression for one variable into the other equation to get a single equation in one variable.
• Use the method of elimination: The method of elimination is another powerful tool for solving systems of linear equations. It involves adding or subtracting the equations to eliminate one of the variables.
• Check your work: It is always a good idea to check your work by plugging the values of the variables back into the original equations to make sure they are satisfied.
• Use technology: There are many computer algebra systems and calculators that can be used to solve systems of linear equations. These tools can be very helpful in solving complex systems of equations.

Conclusion

Introduction

In our previous article, we discussed how to solve a system of linear equations using the method of substitution and elimination. In this article, we will answer some common questions that students often have when it comes to solving systems of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that involve the same set of variables. For example:

$$\begin{array}{c} 2x + 3y = 7 \\ x - 2y = -3 \end{array}$$

Q: How do I know which method to use to solve a system of linear equations?

There are two main methods to solve a system of linear equations: substitution and elimination. The method you choose to use will depend on the form of the equations and the variables involved.

• Substitution method: Use this method when one of the variables is already isolated in one of the equations. For example, if one of the equations is already in the form y = mx + b, you can substitute this expression into the other equation.
• Elimination method: Use this method when the coefficients of one of the variables are the same in both equations. For example, if the coefficients of x are the same in both equations, you can add or subtract the equations to eliminate the variable x.

Q: How do I use the substitution method to solve a system of linear equations?

To use the substitution method, follow these steps:

1. Identify one of the equations that already has one of the variables isolated.
2. Substitute this expression into the other equation.
3. Solve the resulting equation for the other variable.
4. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: How do I use the elimination method to solve a system of linear equations?

To use the elimination method, follow these steps:

1. Identify the coefficients of one of the variables that are the same in both equations.
2. Add or subtract the equations to eliminate the variable.
3. Solve the resulting equation for the other variable.
4. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

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

If you have a system of linear equations with three or more variables, you can use the same methods as before to solve the system. However, you may need to use more advanced techniques, such as matrix operations or graphing, to solve the system.

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

Yes, you can use technology to solve a system of linear equations. Many computer algebra systems and calculators have built-in functions to solve systems of linear equations. These tools can be very helpful in solving complex systems of equations.

Q: How do I check my work when solving a system of linear equations?

To check your work, follow these steps:

1. Plug the values of the variables back into original equations to make sure they are satisfied.
2. Check that the values of the variables satisfy both equations.
3. If the values do not satisfy both equations, go back and recheck your work.

Conclusion

Solving systems of linear equations is an important skill in mathematics that has many real-world applications. By using the substitution and elimination methods, and by checking our work, we can solve systems of linear equations and find the values of the variables. With practice and experience, solving systems of linear equations becomes easier and more intuitive.