Solve Each System By Setting Them Equal:${ \begin{array}{l} y = -8x - 18 \ y = 4x + 18 \end{array} }$Write Your Answer As A Point In The Form Of { (x, Y)$}$. If There Is No Solution, Write No Solution.

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 using the method of substitution. We will use the given system of equations:

$$\begin{array}{l} y = -8x - 18 \\ y = 4x + 18 \end{array}$$

Understanding the Method of Substitution

The method of substitution is a technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is already solved for one variable.

Step 1: Solve One Equation for One Variable

Let's solve the first equation for y:

$$y = -8x - 18$$

We can rewrite this equation as:

$$-8x - 18 = y$$

Now, let's solve the second equation for y:

$$y = 4x + 18$$

We can rewrite this equation as:

$$4x + 18 = y$$

Step 2: Set the Two Equations Equal

Since both equations are equal to y, we can set them equal to each other:

$$-8x - 18 = 4x + 18$$

Step 3: Solve for x

Now, let's solve for x:

$$-8x - 4x = 18 + 18$$

Combine like terms:

$$-12x = 36$$

Divide both sides by -12:

$$x = -3$$

Step 4: Find the Value of y

Now that we have the value of x, let's substitute it into one of the original equations to find the value of y. We will use the first equation:

$$y = -8x - 18$$

Substitute x = -3:

$$y = -8(-3) - 18$$

Simplify:

$$y = 24 - 18$$

$$y = 6$$

Conclusion

Therefore, the solution to the system of linear equations is:

$$(x, y) = (-3, 6)$$

Discussion

In this article, we used the method of substitution to solve a system of two linear equations. We solved one equation for one variable and then substituted that expression into the other equation. This method is useful when one of the equations is already solved for one variable. We also discussed the importance of setting the two equations equal to each other and solving for the variable.

Example Problems

1. Solve the system of linear equations:

$$\begin{array}{l} y = 2x + 5 \\ y = -3x - 2 \end{array}$$

2. Solve the system of linear equations:

$$\begin{array}{l} y = x - 3 \\ y = 2x + 1 \end{array}$$

Tips and Tricks

1. Make sure to solve one equation for one variable before substituting it into the other equation.
2. Set the two equations equal to each other and solve for the variable.
3. Check your solution by substituting the values of x and y into both original equations.

Conclusion

Introduction

In our previous article, we discussed the method of substitution for solving systems of linear equations. In this article, we will answer some frequently asked questions about 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 are solved simultaneously to find the values of the variables.

Q: What is the method of substitution?

The method of substitution is a technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation.

Q: How do I know which equation to solve for first?

You can choose either equation to solve for first. However, it's often easier to solve for the variable that appears in both equations.

Q: What if I get a contradiction when I set the two equations equal?

If you get a contradiction, it means that the system of equations has no solution. This is often indicated by the phrase "No Solution".

Q: Can I use the method of substitution with systems of three or more equations?

Yes, you can use the method of substitution with systems of three or more equations. However, it may be more complicated and require more steps.

Q: What if I get a solution, but it doesn't satisfy both original equations?

If you get a solution, but it doesn't satisfy both original equations, then it's not a valid solution. You should go back and check your work to see where you made a mistake.

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

Yes, you can use a graphing calculator to solve systems of linear equations. Graphing calculators can help you visualize the equations and find the solution.

Q: What are some common mistakes to avoid when solving systems of linear equations?

Some common mistakes to avoid when solving systems of linear equations include:

• Not solving one equation for one variable before substituting it into the other equation
• Not setting the two equations equal to each other
• Not checking the solution by substituting the values of x and y into both original equations

Q: Can I use the method of substitution with systems of nonlinear equations?

No, the method of substitution is only used with systems of linear equations. Nonlinear equations require different methods, such as substitution or elimination.

Q: What are some real-world applications of solving systems of linear equations?

Solving systems of linear equations has many real-world applications, including:

• Physics: Solving systems of linear equations can help you model the motion of objects and predict their behavior.
• Engineering: Solving systems of linear equations can help you design and optimize systems, such as electrical circuits or mechanical systems.
• Economics: Solving systems of linear equations can help you model economic systems and make predictions about the behavior of markets.

Conclusion

In conclusion, solving systems of linear equations using the method of substitution is a useful technique that can be applied to a wide range of problems. By following the steps outlined in this article and avoiding common mistakes, you can solve systems of linear equations and find the values of the variables. Remember to check your solution by substituting the values of x and y into both original equations.