Which Equation Results From Adding The Equations In This System?$\[ \begin{array}{l} -8x + 8y = 8 \\ 3x - 8y = -18 \end{array} \\]A. \[$5x = -10\$\]B. \[$5x = 26\$\]C. \[$-5x = 26\$\]D. \[$-5x = -10\$\]
Introduction
Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to add the equations in a system to obtain a new equation. We will use a specific example to illustrate the process and provide a step-by-step guide on how to solve it.
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 variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we want to solve for. For example:
{ \begin{array}{l} -8x + 8y = 8 \\ 3x - 8y = -18 \end{array} \}
Adding the Equations in a System
When we add the equations in a system, we combine the like terms to obtain a new equation. The resulting equation is a linear equation that involves the same variables as the original equations. To add the equations, we follow these steps:
- Identify the like terms: The like terms are the terms that involve the same variable. In this case, the like terms are the x-terms and the y-terms.
- Combine the like terms: We add the coefficients of the like terms to obtain the new coefficients.
- Add the constants: We add the constants on the right-hand side of the equations to obtain the new constant.
Example: Adding the Equations in the Given System
Let's use the given system of linear equations as an example:
{ \begin{array}{l} -8x + 8y = 8 \\ 3x - 8y = -18 \end{array} \}
To add the equations, we follow the steps outlined above:
- Identify the like terms: The like terms are the x-terms and the y-terms.
- Combine the like terms: We add the coefficients of the like terms to obtain the new coefficients.
Step 1: Add the x-terms
The x-terms are -8x and 3x. To add them, we combine the coefficients:
-8x + 3x = -5x
Step 2: Add the y-terms
The y-terms are 8y and -8y. To add them, we combine the coefficients:
8y - 8y = 0
Step 3: Add the constants
The constants are 8 and -18. To add them, we combine the constants:
8 + (-18) = -10
The Resulting Equation
The resulting equation is:
-5x = -10
Conclusion
In this article, we explored how to add the equations in a system of linear equations. We used a specific example to illustrate the process and provided a step-by-step guide on how to solve it. By following the steps outlined above, we can add the equations in a system to obtain a new equation that involves the same variables as the original equations.
Answer
The correct answer is:
A.
Q: What is the purpose of adding equations in a system?
A: The purpose of adding equations in a system is to obtain a new equation that involves the same variables as the original equations. This can help to simplify the system and make it easier to solve.
Q: How do I add the equations in a system?
A: To add the equations in a system, you need to follow these steps:
- Identify the like terms: The like terms are the terms that involve the same variable.
- Combine the like terms: You add the coefficients of the like terms to obtain the new coefficients.
- Add the constants: You add the constants on the right-hand side of the equations to obtain the new constant.
Q: What if the equations have different variables?
A: If the equations have different variables, you cannot add them directly. However, you can try to eliminate one of the variables by multiplying the equations by a suitable constant.
Q: Can I add more than two equations in a system?
A: Yes, you can add more than two equations in a system. However, you need to make sure that the equations are consistent and that the variables are the same.
Q: What if the resulting equation is not linear?
A: If the resulting equation is not linear, you may need to use other methods to solve the system, such as substitution or elimination.
Q: Can I use a calculator to add equations in a system?
A: Yes, you can use a calculator to add equations in a system. However, it's always a good idea to check your work by hand to make sure that the calculations are correct.
Q: How do I know if the resulting equation is correct?
A: To check if the resulting equation is correct, you can plug in the values of the variables and see if the equation holds true. You can also use a calculator to check the equation.
Q: Can I use this method to solve systems with more than two equations?
A: Yes, you can use this method to solve systems with more than two equations. However, you need to make sure that the equations are consistent and that the variables are the same.
Q: What if I get a contradictory equation?
A: If you get a contradictory equation, it means that the system has no solution. This can happen if the equations are inconsistent or if the variables are not the same.
Q: Can I use this method to solve systems with fractions or decimals?
A: Yes, you can use this method to solve systems with fractions or decimals. However, you need to make sure that the calculations are correct and that the resulting equation is linear.
Conclusion
In this article, we answered some frequently asked questions about adding equations in a system. We covered topics such as the purpose of adding equations, how to add equations, and how to check if the resulting equation is correct. We also discussed some common pitfalls and how to avoid them. By following the steps outlined in this article, you can add equations in a system and obtain a new equation that involves the same variables as the original equations.