Solve The System Of Equations:$\[ \begin{cases} x = 9 - 2y \\ 3x + 5y = 20 \end{cases} \\]
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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 means 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.
The System of Equations
The system of equations we will be solving is:
{ \begin{cases} x = 9 - 2y \\ 3x + 5y = 20 \end{cases} \}
This system consists of two linear equations with two variables, x and y. The first equation is a simple linear equation, while the second equation is a linear equation with a constant term.
Substitution Method
One way to solve this system of equations is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Let's start by solving the first equation for x:
{ x = 9 - 2y \}
We can rewrite this equation as:
{ x + 2y = 9 \}
Now, let's substitute this expression for x into the second equation:
{ 3(x + 2y) + 5y = 20 \}
Expanding the left-hand side of the equation, we get:
{ 3x + 6y + 5y = 20 \}
Combine like terms:
{ 3x + 11y = 20 \}
Solving for y
Now, we can solve for y by isolating y on one side of the equation. Subtract 3x from both sides:
{ 11y = 20 - 3x \}
Divide both sides by 11:
{ y = \frac{20 - 3x}{11} \}
Substituting y into the First Equation
Now that we have an expression for y, we can substitute it into the first equation:
{ x = 9 - 2y \}
Substitute y = (20 - 3x)/11:
{ x = 9 - 2\left(\frac{20 - 3x}{11}\right) \}
Simplifying the Equation
To simplify the equation, we can start by multiplying both sides by 11 to eliminate the fraction:
{ 11x = 99 - 2(20 - 3x) \}
Expand the right-hand side of the equation:
{ 11x = 99 - 40 + 6x \}
Combine like terms:
{ 11x = 59 + 6x \}
Solving for x
Now, we can solve for x by isolating x on one side of the equation. Subtract 6x from both sides:
{ 5x = 59 \}
Divide both sides by 5:
{ x = \frac{59}{5} \}
Finding 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. Let's use the first equation:
{ x = 9 - 2y \}
Substitute x = 59/5:
{ \frac{59}{5} = 9 - 2y \}
Solving for y
Now, we can solve for y by isolating y on one side of the equation. Subtract 9 from both sides:
{ \frac{59}{5} - 9 = -2y \}
Simplify the left-hand side of the equation:
{ \frac{59}{5} - \frac{45}{5} = -2y \}
Combine like terms:
{ \frac{14}{5} = -2y \}
Divide both sides by -2:
{ y = -\frac{7}{5} \}
Conclusion
In this article, we solved a system of two linear equations with two variables using the substitution method. We started by solving one of the equations for one variable and then substituted that expression into the other equation. We then solved for the other variable and found the values of both variables that satisfy the system of equations.
Final Answer
The final answer is:
x = 11.8 y = -1.4
Note: The final answer is rounded to two decimal places.
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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 the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.
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: This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
- Elimination method: This method involves adding or subtracting the equations to eliminate one of the variables.
- Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
- Matrix method: This method involves using matrices to represent the system of equations and solving for the variables.
Q: What is the substitution method?
A: The substitution method is a method for solving systems of linear equations that involves solving one of the equations for one variable and then substituting that expression into the other equation.
Q: How do I use the substitution method to solve a system of linear equations?
A: To use the substitution method, follow these steps:
- Solve one of the equations for one variable.
- Substitute that expression into the other equation.
- Solve for the other variable.
- Substitute the value of the other variable back into one of the original equations to find the value of the first variable.
Q: What is the elimination method?
A: The elimination method is a method for solving systems of linear equations that involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I use the elimination method to solve a system of linear equations?
A: To use the elimination method, follow these steps:
- Multiply both equations by necessary multiples such that the coefficients of one of the variables are the same in both equations.
- Add or subtract the equations to eliminate one of the variables.
- Solve for the other variable.
- Substitute the value of the other variable back into one of the original equations to find the value of the first variable.
Q: What is the graphical method?
A: The graphical method is a method for solving systems of linear equations that involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: How do I use the graphical method to solve a system of linear equations?
A: To use the graphical method, follow these steps:
- Graph the equations on a coordinate plane.
- Find the point of intersection of the two lines.
- The point of intersection represents the solution to the system of equations.
Q: What is the matrix method?
A: The matrix method is a method for solving systems of linear equations that involves using matrices to represent the system of equations and solving for the variables.
Q: How do I use the matrix method to solve a system of linear equations?
A: To use the matrix method, follow these steps:
- Represent the system of equations as a matrix.
- Use row operations to transform the matrix into row-echelon form.
- Solve for the variables by back-substitution.
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 for extraneous solutions: Make sure to check that the solution satisfies both equations in the system.
- Not using the correct method: Choose the method that is best suited for the system of equations.
- Not following the steps carefully: Make sure to follow the steps carefully and accurately.
- Not checking for errors: Make sure to check for errors in the calculations.
Q: How do I know which method to use when solving a system of linear equations?
A: The choice of method depends on the specific system of equations and the variables involved. Some methods are more suitable for certain types of systems, so it's essential to choose the method that is best suited for the problem.
Q: Can I use technology to solve systems of linear equations?
A: Yes, technology can be used to solve systems of linear equations. Many graphing calculators and computer algebra systems can solve systems of linear equations using various methods.
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:
- Physics and engineering: Solving systems of linear equations is used to model and solve problems in physics and engineering, such as motion and force.
- Economics: Solving systems of linear equations is used to model and solve problems in economics, such as supply and demand.
- Computer science: Solving systems of linear equations is used in computer science to solve problems in computer graphics and game development.
- Data analysis: Solving systems of linear equations is used in data analysis to model and solve problems in statistics and data science.