Solve The System Of Equations:$\[ \begin{align*} -4x + Y &= 6 \\ -5x - Y &= 21 \end{align*} \\]
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 involves 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.
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. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are the variables.
Example: A System of Two Linear Equations
Let's consider the following system of two linear equations:
{ \begin{align*} -4x + y &= 6 \\ -5x - y &= 21 \end{align*} \}
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.
Method 1: Substitution Method
One way to solve a system of linear equations is to use the substitution method. This 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:
y = 6 + 4x
Now, substitute this expression for y into the second equation:
-5x - (6 + 4x) = 21
Simplify the equation:
-9x - 6 = 21
Add 6 to both sides:
-9x = 27
Divide both sides by -9:
x = -3
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
-4x + y = 6
Substitute x = -3:
-4(-3) + y = 6
Simplify the equation:
12 + y = 6
Subtract 12 from both sides:
y = -6
Therefore, the solution to the system of linear equations is x = -3 and y = -6.
Method 2: Elimination Method
Another way to solve a system of linear equations is to use the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.
Let's add the two equations:
(-4x + y) + (-5x - y) = 6 + 21
Simplify the equation:
-9x = 27
Divide both sides by -9:
x = -3
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
-4x + y = 6
Substitute x = -3:
-4(-3) + y = 6
Simplify the equation:
12 + y = 6
Subtract 12 from both sides:
y = -6
Therefore, the solution to the system of linear equations is x = -3 and y = -6.
Conclusion
Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. We have discussed two methods for solving a system of linear equations: the substitution method and the elimination method. Both methods involve solving one equation for one variable and then substituting that expression into the other equation. By following these steps, we can find the solution to a system of linear equations.
Tips and Tricks
- Make sure to check your work by plugging the values of x and y back into the original equations.
- If you are having trouble solving a system of linear equations, try using a graphing calculator or a computer algebra system to help you visualize the solution.
- Practice, practice, practice! The more you practice solving systems of linear equations, the more comfortable you will become with the methods and techniques.
Real-World Applications
Systems of linear equations have many real-world applications, including:
- Physics: Systems of linear equations are used to model the motion of objects in physics.
- Engineering: Systems of linear equations are used to design and optimize systems in engineering.
- Economics: Systems of linear equations are used to model economic systems and make predictions about economic trends.
Final Thoughts
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 the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are the 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 and only 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 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 choose which method to use?
A: The choice of method depends on the specific system of linear equations. If one equation is already solved for one variable, then the substitution method may be easier to use. If the coefficients of the variables are the same in both equations, then the elimination method may be 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 by plugging the values of x and y back into the original equations.
- Use a graphing calculator or a computer algebra system to help you visualize the solution.
- Break the problem down into smaller steps and try to solve each step individually.
Q: Can I use a system of linear equations to model real-world problems?
A: Yes, systems of linear equations can be used to model real-world problems. For example, a system of linear equations can be used to model the motion of an object in physics, or to design and optimize systems in engineering.
Q: How do I know if a system of linear equations has a unique solution?
A: A system of linear equations has a unique solution if and only if the two equations are consistent and the coefficients of the variables are not the same in both equations.
Q: Can I use a system of linear equations to solve a problem with more than two variables?
A: Yes, systems of linear equations can be used to solve problems with more than two variables. However, the number of equations must be equal to the number of variables, and the coefficients of the variables must be linear.
Q: How do I check my work when solving a system of linear equations?
A: To check your work when solving a system of linear equations, plug the values of x and y back into the original equations. If the values satisfy both equations, then the solution is correct.
Q: Can I use a system of linear equations to solve a problem with fractions or decimals?
A: Yes, systems of linear equations can be used to solve problems with fractions or decimals. However, the coefficients of the variables must be linear, and the fractions or decimals must be simplified.
Q: How do I know if a system of linear equations has a solution that is a fraction or a decimal?
A: If the solution to a system of linear equations is a fraction or a decimal, then the coefficients of the variables must be linear, and the fractions or decimals must be simplified.
Q: Can I use a system of linear equations to solve a problem with negative numbers?
A: Yes, systems of linear equations can be used to solve problems with negative numbers. However, the coefficients of the variables must be linear, and the negative numbers must be handled correctly.
Q: How do I know if a system of linear equations has a solution that is a negative number?
A: If the solution to a system of linear equations is a negative number, then the coefficients of the variables must be linear, and the negative numbers must be handled correctly.
Conclusion
Solving a system 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 learn how to solve a system of linear equations using the substitution method and the elimination method. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to a wide range of problems in mathematics and other fields.