Solve The System By The Substitution Method.${ \begin{aligned} 3x + 8y & = -2 \ 4x - Y & = 9 \end{aligned} }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Solution Set Is
Introduction
In mathematics, solving systems of equations is a crucial concept that involves finding the values of variables that satisfy multiple equations simultaneously. There are several methods to solve systems of equations, including substitution, elimination, and graphing. In this article, we will focus on the substitution method, which involves solving one equation for one variable and then substituting that expression into the other equation.
What is the Substitution Method?
The substitution method is a technique used to solve systems of equations by substituting the expression for one variable from one equation into the other equation. This method is particularly useful when one of the equations is easily solvable for one variable. The substitution method involves the following steps:
- Solve one equation for one variable.
- Substitute the expression for that variable into the other equation.
- Solve the resulting equation for the other variable.
- Back-substitute the value of the second variable into one of the original equations to find the value of the first variable.
Step-by-Step Solution
Let's consider the following system of equations:
{ \begin{aligned} 3x + 8y & = -2 \\ 4x - y & = 9 \end{aligned} \}
To solve this system using the substitution method, we will follow the steps outlined above.
Step 1: Solve One Equation for One Variable
We can solve the second equation for y:
4x - y = 9
y = 4x - 9
Step 2: Substitute the Expression into the Other Equation
Now, we will substitute the expression for y into the first equation:
3x + 8(4x - 9) = -2
Step 3: Solve the Resulting Equation for the Other Variable
Expanding and simplifying the equation, we get:
3x + 32x - 72 = -2
35x - 72 = -2
35x = 70
x = 2
Step 4: Back-Substitute the Value of the Second Variable
Now that we have found the value of x, we can back-substitute it into one of the original equations to find the value of y. We will use the second equation:
4x - y = 9
4(2) - y = 9
8 - y = 9
-y = 1
y = -1
Conclusion
In this article, we have demonstrated how to solve a system of equations using the substitution method. We have followed the steps outlined above and have found the values of x and y that satisfy the system. The substitution method is a powerful tool for solving systems of equations, and it is particularly useful when one of the equations is easily solvable for one variable.
Example Problems
Here are a few example problems to help you practice solving systems of equations using the substitution method:
Example 1
{ \begin{aligned} 2x + 3y & = 7 \\ x - 2y & = -3 \end{aligned} \}
Example 2
{ \begin{aligned} x + 2y & = 4 \\ 3x - y & = 5 \end{aligned} \}
Example 3
{ \begin{aligned} 4x - 3y & = 2 \\ 2x + 5y & = 11 \end{aligned} \}
Tips and Tricks
Here are a few tips and tricks to help you solve systems of equations using the substitution method:
- Make sure to solve one equation for one variable before substituting it into the other equation.
- Be careful when expanding and simplifying the resulting equation.
- Make sure to back-substitute the value of the second variable into one of the original equations to find the value of the first variable.
Real-World Applications
Solving systems of equations has many real-world applications, including:
- Physics: Solving systems of equations is used to model real-world phenomena, such as the motion of objects under the influence of gravity.
- Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Solving systems of equations is used to model economic systems and make predictions about future trends.
Conclusion
Introduction
In our previous article, we discussed the substitution method for solving systems of equations. In this article, we will answer some frequently asked questions about the substitution method and provide additional examples to help you practice.
Q: What is the substitution method?
A: The substitution method is a technique used to solve systems of equations by substituting the expression for one variable from one equation into the other equation.
Q: When should I use the substitution method?
A: You should use the substitution method when one of the equations is easily solvable for one variable. This method is particularly useful when you have a linear equation that can be easily solved for one variable.
Q: How do I know which equation to solve for one variable?
A: You should choose the equation that is easiest to solve for one variable. If one equation has a coefficient of 1 for one variable, it is usually easier to solve for that variable.
Q: What if I have two equations with the same variable?
A: If you have two equations with the same variable, you can use the substitution method by solving one equation for that variable and then substituting it into the other equation.
Q: Can I use the substitution method with non-linear equations?
A: No, the substitution method is only suitable for linear equations. If you have a non-linear equation, you may need to use a different method, such as the elimination method or graphing.
Q: How do I know if I have solved the system correctly?
A: To check if you have solved the system correctly, you can plug the values of the variables back into both original equations to make sure they are true.
Q: What if I get a system with no solution or infinitely many solutions?
A: If you get a system with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations. If you get a system 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 the substitution method with systems of three or more equations?
A: Yes, you can use the substitution method with systems of three or more equations. However, it may be more complicated and you may need to use a different method, such as the elimination method or graphing.
Example Problems
Here are a few example problems to help you practice solving systems of equations using the substitution method:
Example 1
{ \begin{aligned} 2x + 3y & = 7 \\ x - 2y & = -3 \end{aligned} \}
Example 2
{ \begin{aligned} x + 2y & = 4 \\ 3x - y & = 5 \end{aligned} \}
Example 3
{ \begin{aligned} 4x - 3y & = 2 \\ 2x + 5y & = 11 \end{aligned} \}
Tips and Tricks
Here are a few tips and tricks to help you solve systems of equations using the substitution method:
- Make sure to solve one equation for one variable before substituting it into the other equation.
- Be careful when expanding and simplifying the resulting equation.
- Make sure to back-substitute the value of the second variable into one of the original equations to find the value of the first variable.
Real-World Applications
Solving systems of equations has many real-world applications, including:
- Physics: Solving systems of equations is used to model real-world phenomena, such as the motion of objects under the influence of gravity.
- Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Solving systems of equations is used to model economic systems and make predictions about future trends.
Conclusion
In conclusion, the substitution method is a powerful tool for solving systems of equations. By following the steps outlined above and practicing with example problems, you can become proficient in solving systems of equations using the substitution method.