Solve The System By Substitution.${ \begin{align*} -x + 3y &= 15 \ y &= 2x \end{align*} }$Provide Your Answer As An Ordered Pair (x, Y).

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of equations by substitution. This method involves solving one equation for one variable and then substituting that expression into the other equation.

What is Substitution?

Substitution is a method of solving a system of equations by 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. The substitution method involves the following steps:

1. Solve one equation for one variable.
2. Substitute the expression into the other equation.
3. Solve the resulting equation for the other variable.
4. Back-substitute the value of the second variable into one of the original equations to find the value of the first variable.

Step 1: Solve One Equation for One Variable

Let's consider the following system of equations:

$$\begin{align*} -x + 3y &= 15 \\ y &= 2x \end{align*}$$

We can solve the second equation for y:

$$y = 2x$$

This equation is already solved for y, so we can substitute this expression into the first equation.

Step 2: Substitute the Expression into the Other Equation

Substitute the expression y = 2x into the first equation:

$$-x + 3(2x) = 15$$

Expand the equation:

$$-x + 6x = 15$$

Combine like terms:

$$5x = 15$$

Step 3: Solve the Resulting Equation for the Other Variable

Divide both sides of the equation by 5:

$$x = \frac{15}{5}$$

Simplify the fraction:

$$x = 3$$

Step 4: Back-Substitute the Value of the Second Variable into One of the Original Equations

Now that we have found the value of x, we can back-substitute this value into one of the original equations to find the value of y. We will use the second equation:

$$y = 2x$$

Substitute x = 3 into the equation:

$$y = 2(3)$$

Simplify the equation:

$$y = 6$$

Conclusion

We have solved the system of equations by substitution. The solution is x = 3 and y = 6. This means that the ordered pair (x, y) is (3, 6).

Why is Substitution Useful?

Substitution is a useful method for solving systems of equations because it allows us to eliminate one variable and solve for the other variable. This method is particularly useful when one of the equations is already solved for one variable. Substitution is also a good method to use when the coefficients of the variables are large or complex.

Real-World Applications

Solving systems of equations by substitution has many real-world applications. For example, in physics, we can use systems of equations to model the motion of objects. In economics, we can use systems of equations to model the behavior of markets. In engineering, we can use systems of equations to design and optimize systems.

Common Mistakes to Avoid

When solving systems of equations by substitution, there are several common mistakes to avoid. These include:

• Not solving one equation for one variable before substituting the expression into the other equation.
• Not back-substituting the value of the second variable into one of the original equations to find the value of the first variable.
• Not checking the solution to make sure it satisfies both equations.

Conclusion

Introduction

In our previous article, we discussed how to solve a system of equations by substitution. This method involves solving one equation for one variable and then substituting that expression into the other equation. In this article, we will answer some frequently asked questions about solving systems of equations by substitution.

Q: What is the first step in solving a system of equations by substitution?

A: The first step in solving a system of equations by substitution is to solve one equation for one variable. This can be done by isolating the variable on one side of the equation.

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

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

Q: What if I have two equations with two variables, but neither equation is easily solvable for one variable?

A: In this case, you can try to eliminate one variable by multiplying both equations by a constant or adding the two equations together. This will give you a new equation with only one variable.

Q: Can I use substitution to solve a system of equations with more than two variables?

A: Yes, you can use substitution to solve a system of equations with more than two variables. However, it may be more difficult to solve and may require more steps.

Q: What if I make a mistake while solving a system of equations by substitution?

A: If you make a mistake while solving a system of equations by substitution, you may end up with an incorrect solution. To avoid this, make sure to check your work carefully and double-check your solution.

Q: Can I use a calculator to solve a system of equations by substitution?

A: Yes, you can use a calculator to solve a system of equations by substitution. However, it is often easier to solve by hand, especially for simple systems.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, make sure that it satisfies both equations. You can do this by plugging the values of the variables back into both equations and checking if they are true.

Q: What if I have a system of equations with no solution?

A: If you have a system of equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory or if one equation is a multiple of the other.

Q: Can I use substitution to solve a system of equations with dependent variables?

A: Yes, you can use substitution to solve a system of equations with dependent variables. However, the solution will be an infinite number of solutions.

Conclusion

Solving a system of equations by substitution is a powerful method for finding the values of the variables. By following the steps outlined in this article and answering the frequently asked questions, you can solve systems of equations by substitution and find the values of the variables.

Common Mistakes to Avoid

When solving systems of equations by substitution, there are several common mistakes to avoid. These include:

• Not solving one equation for one variable before substituting the expression into the other equation.
• Not back-substituting the value of the second variable into one of the original equations to find the value of the first variable.
• Not checking the solution to make sure it satisfies both equations.
• Not using the correct method for solving the system of equations.

Real-World Applications

Solving systems of equations by substitution has many real-world applications. For example, in physics, we can use systems of equations to model the motion of objects. In economics, we can use systems of equations to model the behavior of markets. In engineering, we can use systems of equations to design and optimize systems.

Conclusion

Solving a system of equations by substitution is a powerful method for finding the values of the variables. By following the steps outlined in this article and answering the frequently asked questions, you can solve systems of equations by substitution and find the values of the variables.