Solve The System Of Equations:$\[ \begin{array}{l} -2x + 5y = -22 \\ 5x - 5y = 25 \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 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{array}{l} -2x + 5y = -22 \\ 5x - 5y = 25 \end{array} \}
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: Addition and Subtraction
One way to solve this system is to use the method of addition and subtraction. We can add the two equations to eliminate one of the variables.
Step 1: Add the two equations
{ \begin{array}{l} -2x + 5y = -22 \\ 5x - 5y = 25 \end{array} \}
Adding the two equations, we get:
{ \begin{array}{l} -2x + 5y + 5x - 5y = -22 + 25 \\ 3x = 3 \end{array} \}
Step 2: Solve for x
Now that we have eliminated the variable y, we can solve for x.
{ \begin{array}{l} 3x = 3 \\ x = 1 \end{array} \}
Step 3: Substitute x into one of the original equations
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 substitute x = 1 into the first equation:
{ \begin{array}{l} -2(1) + 5y = -22 \\ -2 + 5y = -22 \end{array} \}
Step 4: Solve for y
Now that we have substituted x = 1 into the first equation, we can solve for y.
{ \begin{array}{l} -2 + 5y = -22 \\ 5y = -20 \\ y = -4 \end{array} \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of addition and subtraction. We have found the values of x and y that satisfy both equations, which are x = 1 and y = -4.
Method 2: Substitution
Another way to solve this system is to use the method of substitution. We can solve one of the equations for one of the variables and then substitute that expression into the other equation.
Step 1: Solve one of the equations for x
Let's solve the second equation for x:
{ \begin{array}{l} 5x - 5y = 25 \\ 5x = 25 + 5y \\ x = \frac{25 + 5y}{5} \end{array} \}
Step 2: Substitute x into the other equation
Now that we have solved the second equation for x, we can substitute that expression into the first equation:
{ \begin{array}{l} -2x + 5y = -22 \\ -2\left(\frac{25 + 5y}{5}\right) + 5y = -22 \end{array} \}
Step 3: Solve for y
Now that we have substituted x into the first equation, we can solve for y.
{ \begin{array}{l} -2\left(\frac{25 + 5y}{5}\right) + 5y = -22 \\ \frac{-50 - 10y + 25y}{5} = -22 \\ \frac{15y}{5} = -22 \\ 3y = -22 \\ y = -\frac{22}{3} \end{array} \}
Step 4: Substitute y into one of the original equations
Now that we have found the value of y, we can substitute it into one of the original equations to find the value of x.
Let's substitute y = -\frac{22}{3} into the second equation:
{ \begin{array}{l} 5x - 5y = 25 \\ 5x - 5\left(-\frac{22}{3}\right) = 25 \\ 5x + \frac{110}{3} = 25 \end{array} \}
Step 5: Solve for x
Now that we have substituted y into the second equation, we can solve for x.
{ \begin{array}{l} 5x + \frac{110}{3} = 25 \\ 5x = 25 - \frac{110}{3} \\ 5x = \frac{75 - 110}{3} \\ 5x = -\frac{35}{3} \\ x = -\frac{7}{3} \end{array} \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of substitution. We have found the values of x and y that satisfy both equations, which are x = -\frac{7}{3} and y = -\frac{22}{3}.
Conclusion
Introduction
In our previous article, we discussed how to solve systems of linear equations using two different methods: addition and subtraction, and substitution. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving systems of linear equations.
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 that 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: To determine if a system of linear equations has a solution, you need to check if the two equations are consistent. If the two equations are consistent, then the system has a solution. If the two equations are inconsistent, then the system does not have a solution.
Q: What is the difference between a consistent and an inconsistent system of linear equations?
A: A consistent system of linear equations is one that has a solution. An inconsistent system of linear equations is one that does not have a solution.
Q: How do I solve a system of linear equations using the method of addition and subtraction?
A: To solve a system of linear equations using the method of addition and subtraction, you need to add or subtract the two equations to eliminate one of the variables. Once you have eliminated one of the variables, you can solve for the other variable.
Q: How do I solve a system of linear equations using the method of substitution?
A: To solve a system of linear equations using the method of substitution, you need to solve one of the equations for one of the variables and then substitute that expression into the other equation. Once you have substituted the expression, you can solve for the other variable.
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 if the two equations are consistent before solving the system
- Not following the correct order of operations when solving the system
- Not checking if the solution satisfies both equations
- Not using the correct method to solve the system (e.g. using addition and subtraction when substitution is more appropriate)
Q: How do I check if a solution satisfies both equations?
A: To check if a solution satisfies both equations, you need to substitute the values of the variables into both equations and check if the resulting equations are true.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has many real-world applications, including:
- Finding the intersection point of two lines
- Determining the cost of producing a product
- Calculating the amount of money in a bank account
- Solving problems in physics and engineering
Conclusion
In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving systems of linear equations. We have discussed how to determine if a system of linear equations has a solution, how to solve a system using the method of addition and subtraction, and how to solve a system using the method of substitution. We have also discussed some common mistakes to avoid when solving systems of linear equations and how to check if a solution satisfies both equations.