Solve The System Of Equations:${ \begin{aligned} 5x + 9y & = -3 \ -4x - 7y & = 3 \end{aligned} }$
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.
The System of Equations
The given system of equations is:
{ \begin{aligned} 5x + 9y & = -3 \\ -4x - 7y & = 3 \end{aligned} \}
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously.
Method 1: Substitution Method
One way to solve this system is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for x
We can solve the first equation for x by isolating x on one side of the equation.
{ 5x + 9y = -3 \}
Subtracting 9y from both sides gives us:
{ 5x = -3 - 9y \}
Dividing both sides by 5 gives us:
{ x = \frac{-3 - 9y}{5} \}
Step 2: Substitute the Expression for x into the Second Equation
Now that we have an expression for x, we can substitute it into the second equation.
{ -4x - 7y = 3 \}
Substituting the expression for x gives us:
{ -4\left(\frac{-3 - 9y}{5}\right) - 7y = 3 \}
Simplifying the expression gives us:
{ \frac{12 + 36y}{5} - 7y = 3 \}
Multiplying both sides by 5 gives us:
{ 12 + 36y - 35y = 15 \}
Combining like terms gives us:
{ 12 + y = 15 \}
Subtracting 12 from both sides gives us:
{ y = 3 \}
Step 3: Find the Value of x
Now that we have the value of y, we can find the value of x by substituting y into one of the original equations.
Substituting y = 3 into the first equation gives us:
{ 5x + 9(3) = -3 \}
Simplifying the expression gives us:
{ 5x + 27 = -3 \}
Subtracting 27 from both sides gives us:
{ 5x = -30 \}
Dividing both sides by 5 gives us:
{ x = -6 \}
Method 2: Elimination Method
Another way to solve this system is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of x or y in both equations are the same.
Multiplying the first equation by 4 and the second equation by 5 gives us:
{ 20x + 36y = -12 \}
{ -20x - 35y = 15 \}
Step 2: Add the Equations
Now that we have the equations with the same coefficients, we can add them to eliminate one variable.
Adding the equations gives us:
{ 36y - 35y = -12 + 15 \}
Combining like terms gives us:
{ y = 3 \}
Step 3: Find the Value of x
Now that we have the value of y, we can find the value of x by substituting y into one of the original equations.
Substituting y = 3 into the first equation gives us:
{ 5x + 9(3) = -3 \}
Simplifying the expression gives us:
{ 5x + 27 = -3 \}
Subtracting 27 from both sides gives us:
{ 5x = -30 \}
Dividing both sides by 5 gives us:
{ x = -6 \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using two different methods: substitution method and elimination method. We have shown that both methods can be used to solve the system and have obtained the same solution: x = -6 and y = 3. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one variable. Both methods are useful tools for solving systems of linear equations and can be applied to more complex systems as well.
Applications
Solving systems of linear equations has numerous applications in various fields, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as motion, forces, and energies.
- Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and to make predictions about economic trends.
- Biology: Systems of linear equations are used to model population dynamics, such as the growth and decline of populations.
Final Thoughts
Solving systems of linear equations is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to solve systems of linear equations, we can model real-world problems, make predictions, and gain insights into complex systems. Whether you are a student, a researcher, or a practitioner, solving systems of linear equations is an essential skill that can be applied to a wide range of problems.
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Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear combination of the variables, and the system is said to be consistent if it has a solution.
Q: How do I know if a system of linear equations has a solution?
A: To determine if a system of linear equations has a solution, you can use the following methods:
- Graphical Method: Graph the equations on a coordinate plane and see if they intersect. If they intersect, the system has a solution.
- Substitution Method: Solve one equation for one variable and substitute that expression into the other equation. If the resulting equation is true, the system has a solution.
- Elimination Method: Add or subtract the equations to eliminate one variable. If the resulting equation is true, the system has a solution.
Q: What are the different methods for solving systems of linear equations?
A: There are several methods for solving systems of linear equations, including:
- Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
- Elimination Method: Add or subtract the equations to eliminate one variable.
- Graphical Method: Graph the equations on a coordinate plane and see if they intersect.
- Matrix Method: Represent the system as a matrix and use row operations to solve it.
Q: How do I choose the best method for solving a system of linear equations?
A: The best method for solving a system of linear equations depends on the specific system and the variables involved. Here are some general guidelines:
- Substitution Method: Use this method when one equation is easily solvable for one variable.
- Elimination Method: Use this method when the coefficients of the variables are easy to manipulate.
- Graphical Method: Use this method when the system has a simple graphical representation.
- Matrix Method: Use this method when the system is large and complex.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Here are some common mistakes to avoid when solving systems of linear equations:
- Not checking for consistency: Make sure the system is consistent before solving it.
- Not using the correct method: Choose the best method for the specific system.
- Not checking for extraneous solutions: Make sure the solution is not extraneous.
- Not verifying the solution: Verify the solution by plugging it back into the original equations.
Q: How do I verify a solution to a system of linear equations?
A: To verify a solution to a system of linear equations, plug the solution back into the original equations and check if they are true. If they are true, the solution is valid. If they are not true, the solution is extraneous.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has numerous real-world applications, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as motion, forces, and energies.
- Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and to make predictions about economic trends.
- Biology: Systems of linear equations are used to model population dynamics, such as the growth and decline of populations.
Q: How do I use technology to solve systems of linear equations?
A: There are several ways to use technology to solve systems of linear equations, including:
- Graphing Calculators: Use a graphing calculator to graph the equations and find the intersection point.
- Computer Algebra Systems: Use a computer algebra system, such as Mathematica or Maple, to solve the system.
- Online Tools: Use online tools, such as Wolfram Alpha or Symbolab, to solve the system.
Q: What are some advanced topics in solving systems of linear equations?
A: Some advanced topics in solving systems of linear equations include:
- Nonlinear Systems: Solve systems of nonlinear equations, which involve nonlinear functions.
- Systems with Multiple Variables: Solve systems with multiple variables, which involve more than two variables.
- Systems with Parameters: Solve systems with parameters, which involve variables that are not necessarily constants.
- Systems with Constraints: Solve systems with constraints, which involve additional conditions that must be satisfied.