Solve The Following System Of Equations:$\[ \begin{array}{l} 5x + Y = 2 \\ 20x + 3y = -4 \end{array} \\]
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Introduction
In mathematics, a system of linear equations is a set of two or more equations in which the unknowns are related through linear equations. Solving a system of linear equations involves finding the values of the unknowns that satisfy all the equations simultaneously. In this article, we will focus on solving a system of two linear equations with two unknowns.
The System of Equations
The given system of equations is:
{ \begin{array}{l} 5x + y = 2 \\ 20x + 3y = -4 \end{array} \}
Method 1: Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Let's solve the first equation for y:
5x + y = 2
y = 2 - 5x
Now, substitute this expression for y into the second equation:
20x + 3(2 - 5x) = -4
Expand and simplify the equation:
20x + 6 - 15x = -4
Combine like terms:
5x + 6 = -4
Subtract 6 from both sides:
5x = -10
Divide both sides by 5:
x = -2
Now that we have found the value of x, substitute it back into one of the original equations to find the value of y. Let's use the first equation:
5x + y = 2
5(-2) + y = 2
-10 + y = 2
Add 10 to both sides:
y = 12
Therefore, the solution to the system of equations is x = -2 and y = 12.
Method 2: Elimination Method
The elimination method involves adding or subtracting the equations to eliminate one of the variables. Let's multiply the first equation by 3 and the second equation by 1:
15x + 3y = 6
20x + 3y = -4
Now, subtract the second equation from the first equation:
(15x - 20x) + (3y - 3y) = 6 - (-4)
Simplify the equation:
-5x = 10
Divide both sides by -5:
x = -2
Now that we have found the value of x, substitute it back into one of the original equations to find the value of y. Let's use the first equation:
5x + y = 2
5(-2) + y = 2
-10 + y = 2
Add 10 to both sides:
y = 12
Therefore, the solution to the system of equations is x = -2 and y = 12.
Method 3: Graphical Method
The graphical method involves graphing the two equations on a coordinate plane and finding the point of intersection. Let's graph the two equations:
y = -5x + 2
y = -20/5x - 4/3
The point of intersection is (-2, 12).
Conclusion
In this article, we have solved a system of two linear equations with two unknowns using three different methods: substitution, elimination, and graphical. The solution to the system of equations is x = -2 and y = 12. These methods can be applied to solve systems of linear equations with two or more unknowns.
Applications
Systems of linear equations have numerous applications in various fields, including:
- Physics: To solve problems involving motion, forces, and energies.
- Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: To model and analyze economic systems, such as supply and demand.
- Computer Science: To solve problems involving algorithms and data structures.
Real-World Examples
- Traffic Flow: A city planner wants to optimize traffic flow by adjusting the number of lanes on a highway. The planner can use a system of linear equations to model the relationship between the number of lanes and the traffic speed.
- Resource Allocation: A company wants to allocate resources, such as personnel and equipment, to different projects. The company can use a system of linear equations to model the relationship between the resources and the project outcomes.
- Financial Planning: An individual wants to plan their finances by allocating their income to different expenses, such as housing, food, and entertainment. The individual can use a system of linear equations to model the relationship between their income and expenses.
Tips and Tricks
- Check Your Work: Always check your work by plugging the solution back into the original equations.
- Use Graphing Tools: Use graphing tools, such as graphing calculators or software, to visualize the equations and find the point of intersection.
- Simplify the Equations: Simplify the equations by combining like terms and eliminating fractions.
- Use the Substitution Method: Use the substitution method to solve systems of linear equations with two or more unknowns.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics and has numerous applications in various fields. In this article, we have solved a system of two linear equations with two unknowns using three different methods: substitution, elimination, and graphical. The solution to the system of equations is x = -2 and y = 12. These methods can be applied to solve systems of linear equations with two or more unknowns.
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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 equations in which the unknowns are related through linear equations. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are the unknowns.
Q: How do I know which method to use to solve a system of linear equations?
A: The choice of method depends on the type of system and the number of unknowns. The substitution method is often used for systems with two unknowns, while the elimination method is often used for systems with more than two unknowns. The graphical method is often used for systems with two unknowns and can be visualized on a coordinate plane.
Q: What is the difference between 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. The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I check my work when solving a system of linear equations?
A: To check your work, plug the solution back into the original equations. If the solution satisfies both equations, then it is the correct solution.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid include:
- Not checking your work: Always check your work by plugging the solution back into the original equations.
- Not simplifying the equations: Simplify the equations by combining like terms and eliminating fractions.
- Not using the correct method: Choose the correct method for the type of system and the number of unknowns.
- Not being careful with signs: Be careful with signs when adding or subtracting equations.
Q: How do I use graphing tools to solve systems of linear equations?
A: To use graphing tools, graph the two equations on a coordinate plane and find the point of intersection. The point of intersection is the solution to the system of equations.
Q: What are some real-world applications of systems of linear equations?
A: Systems of linear equations have numerous applications in various fields, including:
- Physics: To solve problems involving motion, forces, and energies.
- Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: To model and analyze economic systems, such as supply and demand.
- Computer Science: To solve problems involving algorithms and data structures.
Q: How do I choose the correct method for solving a system of linear equations?
A: Choose the correct method based on the type of system and the number of unknowns. The substitution method is often used for systems with two unknowns, while the elimination method is often used for systems with more than two unknowns. The graphical method is often used for systems with two unknowns and can be visualized on a coordinate plane.
Q: What are some tips for solving systems of linear equations?
A: Some tips for solving systems of linear equations include:
- Simplify the equations: Simplify the equations by combining like terms and eliminating fractions.
- Use the substitution method: Use the substitution method to solve systems of linear equations with two or more unknowns.
- Check your work: Always check your work by plugging the solution back into the original equations.
- Use graphing tools: Use graphing tools, such as graphing calculators or software, to visualize the equations and find the point of intersection.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if the two equations are consistent and the number of unknowns is equal to the number of equations. If the number of unknowns is greater than the number of equations, then the system may have no solution or infinitely many solutions.
Q: What is the difference between a consistent and inconsistent system of linear equations?
A: A consistent system of linear equations has a solution, while an inconsistent system of linear equations has no solution.