Solve The System Of Equations:${ \begin{array}{l} 3x - 4y = -40 \ 4x + 3y = 5 \end{array} }$
Introduction
Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution.
What are Systems of Linear Equations?
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 equation, which means it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} 3x - 4y = -40 \\ 4x + 3y = 5 \end{array} \}
This system consists of two linear equations with two variables, x and y.
Method of Substitution
One way to solve this system is by using the method of substitution. 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 adding 4y to both sides:
3x = -40 + 4y
Then, we can divide both sides by 3:
x = (-40 + 4y) / 3
Step 2: Substitute the Expression for x into the Second Equation
Now, we can substitute the expression for x into the second equation:
4((-40 + 4y) / 3) + 3y = 5
Step 3: Simplify the Equation
We can simplify the equation by multiplying both sides by 3:
-160 + 16y + 9y = 15
Combine like terms:
-160 + 25y = 15
Add 160 to both sides:
25y = 175
Divide both sides by 25:
y = 7
Step 4: 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. We will use the first equation:
3x - 4y = -40
Substitute y = 7:
3x - 4(7) = -40
Simplify:
3x - 28 = -40
Add 28 to both sides:
3x = -12
Divide both sides by 3:
x = -4
Conclusion
We have solved the system of equations using the method of substitution. The solution is x = -4 and y = 7.
Method of Elimination
Another way to solve this system is by using the method of elimination. This method involves adding or subtracting the equations to eliminate one variable.
Step 1: Multiply the Equations by Necessary Multiples
We can multiply the first equation by 3 and the second equation by 4 to make the coefficients of y opposites:
(3x - 4y = -40) × 3 (4x + 3y = 5) × 4
This gives us:
9x - 12y = -120 16x + 12y = 20
Step 2: Add the Equations
Now, we can add the equations to eliminate the y-variable:
(9x - 12y) + (16x + 12y) = -120 + 20
Combine like terms:
25x = -100
Divide both sides by 25:
x = -4
Step 3: Find the Value of y
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:
3x - 4y = -40
Substitute x = -4:
3(-4) - 4y = -40
Simplify:
-12 - 4y = -40
Add 12 to both sides:
-4y = -28
Divide both sides by -4:
y = 7
Conclusion
We have solved the system of equations using the method of elimination. The solution is x = -4 and y = 7.
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have shown that both methods can be used to find the solution to the system. The solution is x = -4 and y = 7.
Real-World Applications
Systems of linear equations have many real-world applications, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the flow of fluids.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand.
- Computer Science: Systems of linear equations are used in computer graphics and game development.
Tips and Tricks
Here are some tips and tricks for solving systems of linear equations:
- Use the method of substitution or elimination: Both methods can be used to solve systems of linear equations.
- Check your work: Make sure to check your work by plugging the solution back into the original equations.
- Use a graphing calculator: A graphing calculator can be used to visualize the system of equations and find the solution.
Conclusion
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 equation, which means it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
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, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.
Q: What is the difference between the method of substitution and the method of elimination?
A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.
Q: How do I choose which method to use?
A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, then the method of elimination is usually easier to use. If the coefficients of one variable are different in both equations, then the method of substitution is usually easier to use.
Q: What if I get stuck while solving a system of linear equations?
A: If you get stuck while solving a system of linear equations, try the following:
- Check your work: Make sure to check your work by plugging the solution back into the original equations.
- Use a graphing calculator: A graphing calculator can be used to visualize the system of equations and find the solution.
- Ask for help: Don't be afraid to ask for help if you are stuck.
Q: Can systems of linear equations be used to model real-world problems?
A: Yes, systems of linear equations can be used to model real-world problems. For example, systems of linear equations can be used to model the motion of objects, the flow of fluids, and the behavior of economic systems.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid when solving systems of linear equations include:
- Not checking your work: Make sure to check your work by plugging the solution back into the original equations.
- Not using the correct method: Make sure to use the correct method for the system of equations.
- Not simplifying the equations: Make sure to simplify the equations as much as possible before solving them.
Q: How can I practice solving systems of linear equations?
A: You can practice solving systems of linear equations by:
- Solving problems: Try solving problems from a textbook or online resource.
- Using a graphing calculator: Use a graphing calculator to visualize the system of equations and find the solution.
- Working with a partner: Work with a partner to solve systems of linear equations.
Q: What are some advanced topics related to systems of linear equations?
A: Some advanced topics related to systems of linear equations include:
- Systems of nonlinear equations: Systems of nonlinear equations involve equations that are not linear.
- Systems of differential equations: Systems of differential equations involve equations that involve rates of change.
- Linear algebra: Linear algebra is a branch of mathematics that deals with systems of linear equations and other related topics.
Conclusion
Solving systems of linear equations is a crucial skill for students and professionals alike. In this article, we have answered some frequently asked questions about systems of linear equations, including how to choose which method to use, how to avoid common mistakes, and how to practice solving systems of linear equations. We have also discussed some advanced topics related to systems of linear equations.