Use The Linear Combination Method To Solve This System Of Equations. What Is The Value Of $x$?$ \begin{array}{r} 3x + 7y = 31 \\ -3x - 2y = -1 \end{array} $A. -6 B. $-3 \frac{2}{3}$ C. $3 \frac{2}{3}$ D. 6
Introduction
In mathematics, a system of equations is a set of equations that involve multiple variables. Solving a system of equations means finding the values of the variables that satisfy all the equations in the system. One of the methods used to solve systems of equations is the linear combination method. This method involves combining two or more equations to eliminate one of the variables and solve for the other variable.
The Linear Combination Method
The linear combination method involves multiplying one or more equations by a constant and then adding the resulting equations to eliminate one of the variables. The goal is to create an equation with only one variable, which can then be solved for that variable.
Step 1: Write Down the System of Equations
The given system of equations is:
\begin{array}{r} 3x + 7y = 31 \ -3x - 2y = -1 \end{array}
Step 2: Multiply the Equations by Constants
To eliminate one of the variables, we need to multiply the equations by constants such that the coefficients of the variable to be eliminated are the same. Let's multiply the first equation by 1 and the second equation by 3.
\begin{array}{r} 3x + 7y = 31 \ -9x - 6y = -3 \end{array}
Step 3: Add the Resulting Equations
Now, let's add the resulting equations to eliminate the variable x.
\begin{array}{r} (3x + 7y) + (-9x - 6y) = 31 + (-3) \ -6x + y = 28 \end{array}
Step 4: Solve for the Variable
Now that we have an equation with only one variable, we can solve for that variable. Let's solve for y.
\begin{array}{r} -6x + y = 28 \ y = 28 + 6x \end{array}
Step 5: Substitute the Value of the Variable into One of the Original Equations
Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's substitute y into the first original equation.
\begin{array}{r} 3x + 7y = 31 \ 3x + 7(28 + 6x) = 31 \end{array}
Step 6: Solve for the Variable
Now that we have an equation with only one variable, we can solve for that variable. Let's solve for x.
\begin{array}{r} 3x + 7(28 + 6x) = 31 \ 3x + 196 + 42x = 31 \ 45x = -165 \ x = -\frac{165}{45} \ x = -\frac{11}{3} \ x = -3 \frac{2}{3} \end{array}
Conclusion
Q: What is the linear combination method?
A: The linear combination method is a technique used to solve systems of equations by combining two or more equations to eliminate one of the variables and solve for the other variable.
Q: How do I know which equations to multiply by constants?
A: To eliminate one of the variables, you need to multiply the equations by constants such that the coefficients of the variable to be eliminated are the same.
Q: What if I have more than two equations? Can I still use the linear combination method?
A: Yes, you can still use the linear combination method even if you have more than two equations. You can multiply the equations by constants and add them to eliminate one of the variables.
Q: How do I know which variable to eliminate?
A: You can eliminate any variable that appears in both equations. The goal is to create an equation with only one variable, which can then be solved for that variable.
Q: What if I get a fraction when I solve for the variable?
A: You can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Q: Can I use the linear combination method to solve systems of equations with more than two variables?
A: No, the linear combination method is typically used to solve systems of equations with two variables. If you have a system of equations with more than two variables, you may need to use a different method, such as substitution or elimination.
Q: What are some common mistakes to avoid when using the linear combination method?
A: Some common mistakes to avoid when using the linear combination method include:
- Multiplying the equations by the wrong constants
- Adding the equations incorrectly
- Forgetting to simplify the fraction when solving for the variable
Q: Can I use the linear combination method to solve systems of equations with decimals or fractions?
A: Yes, you can use the linear combination method to solve systems of equations with decimals or fractions. You can multiply the equations by constants and add them to eliminate one of the variables, just like you would with integers.
Q: How do I know if the linear combination method is the best method to use for a particular system of equations?
A: The linear combination method is a good choice when you have a system of equations with two variables and you want to eliminate one of the variables. However, if you have a system of equations with more than two variables or if you want to use a different method, you may need to use a different method, such as substitution or elimination.
Conclusion
In this article, we answered some frequently asked questions about solving systems of equations using the linear combination method. We covered topics such as how to know which equations to multiply by constants, how to eliminate variables, and how to avoid common mistakes. We also discussed the limitations of the linear combination method and when to use it.