Given The System Of Equations:${ \begin{array}{l} 0.8 - 4x = -0.4y \ 6x + 0.4y = 4.2 \end{array} }$Which Of The Following Shows The System With Like Terms Aligned?A. $[ \begin{array}{l} -4x - 0.4y = -0.8 \ 6x + 0.4y =
Introduction
Solving systems of equations is a fundamental concept in mathematics, and it is essential to understand how to align like terms to simplify the process. In this article, we will explore how to align like terms in a system of equations and provide a step-by-step guide on how to solve it.
What are Like Terms?
Like terms are terms that have the same variable(s) raised to the same power. In the context of a system of equations, like terms are terms that have the same variable(s) raised to the same power and are multiplied by the same coefficient.
Aligning Like Terms
To align like terms, we need to rearrange the equations so that the like terms are on the same side of the equation. Let's take the given system of equations as an example:
{ \begin{array}{l} 0.8 - 4x = -0.4y \\ 6x + 0.4y = 4.2 \end{array} \}
To align like terms, we need to move the constant terms to the other side of the equation. We can do this by adding or subtracting the same value from both sides of the equation.
Step 1: Move the Constant Terms
Let's move the constant terms to the other side of the equation by adding 4x to both sides of the first equation and subtracting 0.4y from both sides of the second equation.
{ \begin{array}{l} 0.8 - 4x + 4x = -0.4y + 4x \\ 6x + 0.4y - 0.4y = 4.2 - 0.4y \end{array} \}
This simplifies to:
{ \begin{array}{l} 0.8 = -0.4y + 4x \\ 6x + 0.4y = 4.2 \end{array} \}
Step 2: Align the Like Terms
Now that we have moved the constant terms to the other side of the equation, we can align the like terms by rearranging the equations.
{ \begin{array}{l} 4x + 0.4y = 0.8 \\ 6x + 0.4y = 4.2 \end{array} \}
As we can see, the like terms are now aligned, and we can proceed to solve the system of equations.
Solving the System of Equations
To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.
Step 1: Multiply the Equations
To eliminate one of the variables, we need to multiply the equations by a suitable constant. Let's multiply the first equation by 6 and the second equation by -4.
{ \begin{array}{l} 24x + 2.4y = 4.8 \\ -24x - 1.6y = -16.8 \end{array} \}
Step 2: Add the Equations
Now that we have multiplied the equations, we can add them to eliminate one of the variables.
{ \begin{array}{l} 24x + 2.4y - 24x - 1.6y = 4.8 - 16.8 \\ 0.8y = -12 \end{array} \}
Step 3: Solve for the Variable
Now that we have eliminated one of the variables, we can solve for the other variable.
{ \begin{array}{l} 0.8y = -12 \\ y = -12 / 0.8 \\ y = -15 \end{array} \}
Step 4: Substitute the Value
Now that we have found the value of one of the variables, we can substitute it into one of the original equations to find the value of the other variable.
{ \begin{array}{l} 4x + 0.4y = 0.8 \\ 4x + 0.4(-15) = 0.8 \\ 4x - 6 = 0.8 \\ 4x = 6.8 \\ x = 1.7 \end{array} \}
Conclusion
In this article, we have learned how to align like terms in a system of equations and solve it using the elimination method. We have also seen how to use the method of substitution to solve the system of equations. By following these steps, we can solve any system of equations and find the values of the variables.
Discussion
The system of equations we solved in this article is a simple example of a system of linear equations. However, there are many other types of systems of equations, such as non-linear systems, systems with complex coefficients, and systems with multiple variables. In each of these cases, the method of solving the system of equations will be different.
Final Answer
The final answer is:
{
\begin{array}{l}
x = 1.7 \\
y = -15
\end{array}
\}$<br/>
**Frequently Asked Questions (FAQs) About Solving Systems of Equations**
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A: A system of equations is a set of two or more equations that are related to each other through a common variable or variables. A: The purpose of solving a system of equations is to find the values of the variables that satisfy all the equations in the system. A: There are two main methods of solving a system of equations: the substitution method and the elimination method. A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation to solve for the other variable. A: The elimination method involves adding or subtracting the equations in a way that eliminates one of the variables, allowing us to solve for the other variable. A: The choice of method depends on the specific system of equations and the variables involved. If the equations are simple and easy to work with, the substitution method may be easier. If the equations are more complex, the elimination method may be more effective. A: Like terms are terms that have the same variable(s) raised to the same power and are multiplied by the same coefficient. A: To align like terms, we need to rearrange the equations so that the like terms are on the same side of the equation. We can do this by adding or subtracting the same value from both sides of the equation. A: The final answer to the system of equations we solved in the previous article is: {
\begin{array}{l}
x = 1.7 \\
y = -15
\end{array}
\}
</span></p>
<h2><strong>Q: Can I use a calculator to solve a system of equations?</strong></h2>
<p>A: Yes, you can use a calculator to solve a system of equations. In fact, many calculators have built-in functions for solving systems of equations.</p>
<h2><strong>Q: What are some common mistakes to avoid when solving a system of equations?</strong></h2>
<p>A: Some common mistakes to avoid when solving a system of equations include:</p>
<ul>
<li>Not checking the work for errors</li>
<li>Not using the correct method for the specific system of equations</li>
<li>Not simplifying the equations before solving</li>
<li>Not checking the solutions for consistency with the original equations</li>
</ul>
<h2><strong>Q: Can I solve a system of equations with more than two variables?</strong></h2>
<p>A: Yes, you can solve a system of equations with more than two variables. However, the process is more complex and may require the use of matrices or other advanced techniques.</p>
<h2><strong>Q: What are some real-world applications of solving systems of equations?</strong></h2>
<p>A: Solving systems of equations has many real-world applications, including:</p>
<ul>
<li>Physics and engineering: solving systems of equations is used to model and analyze complex systems, such as electrical circuits and mechanical systems.</li>
<li>Economics: solving systems of equations is used to model and analyze economic systems, such as supply and demand curves.</li>
<li>Computer science: solving systems of equations is used in computer graphics and game development to create realistic simulations and animations.</li>
</ul>
<h2><strong>Conclusion</strong></h2>
<p>Solving systems of equations is a fundamental concept in mathematics, and it has many real-world applications. By understanding the different methods of solving systems of equations and avoiding common mistakes, you can become proficient in solving these types of problems.</p>
Q: What is a system of equations?
Q: What is the purpose of solving a system of equations?
Q: What are the different methods of solving a system of equations?
Q: What is the substitution method?
Q: What is the elimination method?
Q: How do I choose which method to use?
Q: What are like terms?
Q: How do I align like terms in a system of equations?
Q: What is the final answer to the system of equations we solved in the previous article?