Solve The Following System Of Equations.$\[ \begin{align*} -3x + 4y &= -13 \\ 5x - 3y &= 18 \end{align*} \\]Find The Values Of \[$x\$\] And \[$y\$\].
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is given by:
{ \begin{align*} -3x + 4y &= -13 \\ 5x - 3y &= 18 \end{align*} \}
Method of Elimination
One of the methods used to solve a system of linear equations is the method of elimination. This method involves adding or subtracting the equations in the system to eliminate one of the variables. In this case, we can multiply the first equation by 5 and the second equation by 3 to make the coefficients of x in both equations equal.
Step 1: Multiply the Equations
We multiply the first equation by 5 and the second equation by 3 to get:
{ \begin{align*} -15x + 20y &= -65 \\ 15x - 9y &= 54 \end{align*} \}
Step 2: Add the Equations
We add the two equations to eliminate the variable x:
{ \begin{align*} -15x + 20y &= -65 \\ 15x - 9y &= 54 \end{align*} \}
{ \begin{align*} ( -15x + 20y ) + ( 15x - 9y ) &= -65 + 54 \\ 11y &= -11 \end{align*} \}
Step 3: Solve for y
We solve for y by dividing both sides of the equation by 11:
{ \begin{align*} 11y &= -11 \\ y &= \frac{-11}{11} \\ y &= -1 \end{align*} \}
Step 4: Substitute y into One of the Original Equations
We substitute y = -1 into one of the original equations to solve for x. We will use the first equation:
{ \begin{align*} -3x + 4y &= -13 \\ -3x + 4(-1) &= -13 \\ -3x - 4 &= -13 \end{align*} \}
Step 5: Solve for x
We solve for x by adding 4 to both sides of the equation and then dividing both sides by -3:
{ \begin{align*} -3x - 4 &= -13 \\ -3x &= -9 \\ x &= \frac{-9}{-3} \\ x &= 3 \end{align*} \}
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of elimination. We multiplied the equations to make the coefficients of x equal, added the equations to eliminate x, solved for y, and then substituted y into one of the original equations to solve for x. The values of x and y that satisfy the system of equations are x = 3 and y = -1.
Example Use Cases
Solving systems of linear equations has many practical applications in various fields such as physics, engineering, economics, and computer science. Some examples of use cases include:
- Physics: Solving systems of linear equations can help physicists model the motion of objects under the influence of forces.
- Engineering: Engineers use systems of linear equations to design and optimize systems such as electrical circuits, mechanical systems, and structural systems.
- Economics: Economists use systems of linear equations to model economic systems and make predictions about economic outcomes.
- Computer Science: Computer scientists use systems of linear equations to solve problems in computer graphics, machine learning, and data analysis.
Tips and Tricks
Here are some tips and tricks to help you solve systems of linear equations:
- Use the method of elimination: The method of elimination is a powerful tool for solving systems of linear equations. It involves adding or subtracting the equations in the system to eliminate one of the variables.
- Use substitution: Substitution is another method for solving systems of linear equations. It involves solving one of the equations for one of the variables and then substituting that expression into the other equation.
- Use matrices: Matrices are a powerful tool for solving systems of linear equations. They can be used to represent the coefficients of the equations and the variables.
- Use technology: Technology such as calculators and computer software can be used to solve systems of linear equations. They can be used to perform calculations and visualize the solutions.
Conclusion
Solving systems of linear equations is an important skill in mathematics and has many practical applications in various fields. The method of elimination is a powerful tool for solving systems of linear equations. By following the steps outlined in this article, you can solve systems of linear equations and apply the skills to real-world problems.
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Introduction
In our previous article, we discussed how to solve a system of linear equations using the method of elimination. In this article, we will answer some frequently asked questions about solving systems of linear equations.
Q: 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 set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.
Q: What are some common methods for solving systems of linear equations?
Some common methods for solving systems of linear equations include:
- Method of elimination: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
- Method of substitution: This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.
- Method of matrices: This method involves using matrices to represent the coefficients of the equations and the variables.
Q: How do I choose which method to use?
The choice of method depends on the specific system of equations and the variables involved. The method of elimination is often the most straightforward method, but it may not always be the most efficient. The method of substitution can be useful when one of the equations is easily solvable for one of the variables. The method of matrices can be useful when the system of equations is large or complex.
Q: What are some common mistakes to avoid when solving systems of linear equations?
Some common mistakes to avoid when solving systems of linear equations include:
- Not checking for extraneous solutions: Make sure to check that the solutions you find satisfy all the equations in the system.
- Not using the correct method: Choose the method that is most suitable for the system of equations.
- Not following the steps carefully: Make sure to follow the steps of the method carefully to avoid errors.
Q: How do I check if a solution is extraneous?
To check if a solution is extraneous, plug the values of the variables into each of the equations in the system and check if the equation is satisfied. If the equation is not satisfied, then the solution is extraneous.
Q: What are some real-world applications of solving systems of linear equations?
Solving systems of linear equations has many real-world applications, including:
- Physics: Solving systems of linear equations can help physicists model the motion of objects under the influence of forces.
- Engineering: Engineers use systems of linear equations to design and optimize systems such as electrical circuits, mechanical systems, and structural systems.
- Economics: Economists use systems of linear equations to model economic systems and make predictions about economic outcomes.
- Computer Science: Computer scientists use systems of linear equations to solve problems in computer graphics, machine learning, and data analysis.
Q: How do I use technology to solve systems of linear equations?
There are many software programs and online tools that can be used to solve systems of linear equations, including:
- Graphing calculators: Graphing calculators can be used to visualize the solutions to a system of linear equations.
- Computer algebra systems: Computer algebra systems such as Mathematica and Maple can be used to solve systems of linear equations.
- Online tools: Online tools such as Wolfram Alpha and Symbolab can be used to solve systems of linear equations.
Conclusion
Solving systems of linear equations is an important skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve systems of linear equations and apply the skills to real-world problems.