What Is The $x$-value Of The Solution To The System Of Equations?${ \begin{array}{l} 5x + 4y = 8 \ 2x - 3y = 17 \end{array} }$A. $-2$ B. $-3$
Introduction to Systems of Equations
Systems of equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of multiple equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables, and we will determine the $x$-value of the solution.
The System of Equations
The given system of equations is:
{ \begin{array}{l} 5x + 4y = 8 \\ 2x - 3y = 17 \end{array} \}
This system consists of two linear equations with two variables, $x$ and $y$. The first equation is $5x + 4y = 8$, and the second equation is $2x - 3y = 17$. Our objective is to find the values of $x$ and $y$ that satisfy both equations.
Method of Elimination
One of the methods to solve a system of equations is the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables. In this case, we can multiply the first equation by 3 and the second equation by 4 to make the coefficients of $y$ in both equations equal.
Step 1: Multiply the First Equation by 3
Multiplying the first equation by 3 gives us:
Step 2: Multiply the Second Equation by 4
Multiplying the second equation by 4 gives us:
Step 3: Add the Two Equations
Now, we can add the two equations to eliminate the variable $y$:
Simplifying the equation, we get:
Step 4: Solve for $x$
To solve for $x$, we can divide both sides of the equation by 23:
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of elimination. We multiplied the first equation by 3 and the second equation by 4 to make the coefficients of $y$ in both equations equal. Then, we added the two equations to eliminate the variable $y$ and solved for $x$. The $x$-value of the solution is $4$.
Final Answer
The final answer is:
Introduction
In our previous article, we solved a system of two linear equations with two variables using the method of elimination. In this article, we will answer some frequently asked questions about systems of equations.
Q: What is a system of equations?
A: A system of equations is a set of multiple equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously.
Q: What are the different methods to solve a system of equations?
A: There are several methods to solve a system of equations, including:
- Method of elimination
- Method of substitution
- Graphical method
- Matrix method
Q: What is the method of elimination?
A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of the variables in the equations are integers.
Q: What is the method of substitution?
A: The method of substitution involves 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.
Q: What is the graphical method?
A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and the solution is a single point.
Q: What is the matrix method?
A: The matrix method involves representing the system of equations as a matrix and using row operations to solve for the variables. This method is useful when the system of equations is large and complex.
Q: How do I determine which method to use?
A: To determine which method to use, you should consider the following factors:
- The number of equations and variables
- The complexity of the equations
- The desired level of accuracy
- The available tools and resources
Q: What are some common mistakes to avoid when solving systems of equations?
A: Some common mistakes to avoid when solving systems of equations include:
- Not checking the solution for consistency
- Not considering the possibility of multiple solutions
- Not using the correct method for the given system
- Not checking for errors in the calculations
Q: How do I check the solution for consistency?
A: To check the solution for consistency, you should substitute the values of the variables into each equation and verify that the equation is true.
Q: What is the importance of systems of equations in real-life applications?
A: Systems of equations have numerous real-life applications, including:
- Physics and engineering
- Economics and finance
- Computer science and programming
- Data analysis and statistics
Q: How do I apply systems of equations in real-life situations?
A: To apply systems of equations in real-life situations, you should:
- Identify the variables and equations involved
- Choose the appropriate method to solve the system
- Use the solution to make informed decisions or predictions
Conclusion
In this article, we answered some frequently asked questions about systems of equations. We discussed the different methods to solve a system of equations, including the method of elimination, method of substitution, graphical method, and matrix method. We also provided tips on how to determine which method to use and how to check the solution for consistency.