Solve The System Of Equations:$\[ \begin{array}{l} 5x - 3y = 9 \\ 4x + Y = 7 \end{array} \\]A. \[$x = 9 + 3y\$\] B. \[$y = 7 - 4x\$\] C. \[$x = 7 + 4y\$\]
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. 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.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} 5x - 3y = 9 \\ 4x + y = 7 \end{array} \}
Method of Substitution
One way to solve this system of equations is by using the method of substitution. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Let's solve the second equation for y:
4x + y = 7
y = 7 - 4x
Now, substitute this expression for y into the first equation:
5x - 3(7 - 4x) = 9
Expand and simplify the equation:
5x - 21 + 12x = 9
Combine like terms:
17x - 21 = 9
Add 21 to both sides:
17x = 30
Divide both sides by 17:
x = 30/17
Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the second equation:
4x + y = 7
4(30/17) + y = 7
Simplify the equation:
120/17 + y = 7
Multiply both sides by 17 to eliminate the fraction:
120 + 17y = 119
Subtract 120 from both sides:
17y = -1
Divide both sides by 17:
y = -1/17
Method of Elimination
Another way to solve this system of equations is by using the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables.
Let's multiply the first equation by 1 and the second equation by 3 to make the coefficients of y opposites:
5x - 3y = 9
12x + 3y = 21
Add the two equations together to eliminate y:
17x = 30
Divide both sides by 17:
x = 30/17
Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the second equation:
4x + y = 7
4(30/17) + y = 7
Simplify the equation:
120/17 + y = 7
Multiply both sides by 17 to eliminate the fraction:
120 + 17y = 119
Subtract 120 from both sides:
17y = -1
Divide both sides by 17:
y = -1/17
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 found the values of x and y to be x = 30/17 and y = -1/17.
Discussion
The method of substitution and elimination are two common methods used to solve systems of linear equations. The method of substitution involves solving one of the equations 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 of the variables.
Solutions
The solutions to the system of equations are:
- A.
x = 9 + 3y(Incorrect) - B.
y = 7 - 4x(Incorrect) - C.
x = 7 + 4y(Incorrect)
The correct solution is x = 30/17 and y = -1/17.
Final Answer
The final answer is x = 30/17 and y = -1/17.
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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 are solved simultaneously to find the values of the variables.
Q: What are the two main methods used to solve systems of linear equations?
A: The two main methods used to solve systems of linear equations are the method of substitution and the method of elimination.
Q: What is the method of substitution?
A: The method of substitution involves solving one of the equations for one variable and then substituting that expression into the other equation.
Q: What is the method of elimination?
A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
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, you can use the method of elimination. If the coefficients of one variable are different in both equations, you can use the method of substitution.
Q: What if I have a system of three or more linear equations?
A: If you have a system of three or more linear equations, you can use the method of substitution or elimination to solve for two variables, and then use the third equation to solve for the third variable.
Q: Can I use a calculator to solve systems of linear equations?
A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.
Q: Can I use systems of linear equations to model real-world problems?
A: Yes, you can use systems of linear equations to model real-world problems. Systems of linear equations can be used to model a wide range of problems, including business, economics, physics, and engineering.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Business: Systems of linear equations can be used to model business problems, such as finding the optimal price and quantity of a product to sell.
- Economics: Systems of linear equations can be used to model economic problems, such as finding the optimal allocation of resources.
- Physics: Systems of linear equations can be used to model physical problems, such as finding the trajectory of a projectile.
- Engineering: Systems of linear equations can be used to model engineering problems, such as finding the optimal design of a bridge.
Q: Can I use systems of linear equations to solve problems with multiple variables?
A: Yes, you can use systems of linear equations to solve problems with multiple variables. Systems of linear equations can be used to model problems with any number of variables.
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 consistency before solving the system.
- Not using the correct method for solving the system.
- Not checking for infinitely many solutions before solving the system.
- Not checking for no solution before solving the system.
Q: Can I use systems of linear equations to solve problems with fractions or decimals?
A: Yes, you can use systems of linear equations to solve problems with fractions or decimals. Systems of linear equations can be used to model problems with any type of number.
Q: What are some common applications of systems of linear equations in real-world problems?
A: Some common applications of systems of linear equations in real-world problems include:
- Business: Systems of linear equations can be used to model business problems, such as finding the optimal price and quantity of a product to sell.
- Economics: Systems of linear equations can be used to model economic problems, such as finding the optimal allocation of resources.
- Physics: Systems of linear equations can be used to model physical problems, such as finding the trajectory of a projectile.
- Engineering: Systems of linear equations can be used to model engineering problems, such as finding the optimal design of a bridge.
Q: Can I use systems of linear equations to solve problems with multiple equations?
A: Yes, you can use systems of linear equations to solve problems with multiple equations. Systems of linear equations can be used to model problems with any number of equations.
Q: What are some common mistakes to avoid when using systems of linear equations to solve problems?
A: Some common mistakes to avoid when using systems of linear equations to solve problems include:
- Not checking for consistency before solving the system.
- Not using the correct method for solving the system.
- Not checking for infinitely many solutions before solving the system.
- Not checking for no solution before solving the system.
Q: Can I use systems of linear equations to solve problems with variables that have different units?
A: Yes, you can use systems of linear equations to solve problems with variables that have different units. Systems of linear equations can be used to model problems with any type of variable.
Q: What are some common applications of systems of linear equations in science and engineering?
A: Some common applications of systems of linear equations in science and engineering include:
- Physics: Systems of linear equations can be used to model physical problems, such as finding the trajectory of a projectile.
- Engineering: Systems of linear equations can be used to model engineering problems, such as finding the optimal design of a bridge.
- Computer Science: Systems of linear equations can be used to model computer science problems, such as finding the shortest path in a graph.
Q: Can I use systems of linear equations to solve problems with variables that have different exponents?
A: Yes, you can use systems of linear equations to solve problems with variables that have different exponents. Systems of linear equations can be used to model problems with any type of variable.
Q: What are some common mistakes to avoid when using systems of linear equations to solve problems with variables that have different exponents?
A: Some common mistakes to avoid when using systems of linear equations to solve problems with variables that have different exponents include:
- Not checking for consistency before solving the system.
- Not using the correct method for solving the system.
- Not checking for infinitely many solutions before solving the system.
- Not checking for no solution before solving the system.
Q: Can I use systems of linear equations to solve problems with variables that have different bases?
A: Yes, you can use systems of linear equations to solve problems with variables that have different bases. Systems of linear equations can be used to model problems with any type of variable.
Q: What are some common applications of systems of linear equations in finance and economics?
A: Some common applications of systems of linear equations in finance and economics include:
- Business: Systems of linear equations can be used to model business problems, such as finding the optimal price and quantity of a product to sell.
- Economics: Systems of linear equations can be used to model economic problems, such as finding the optimal allocation of resources.
Q: Can I use systems of linear equations to solve problems with variables that have different units and exponents?
A: Yes, you can use systems of linear equations to solve problems with variables that have different units and exponents. Systems of linear equations can be used to model problems with any type of variable.
Q: What are some common mistakes to avoid when using systems of linear equations to solve problems with variables that have different units and exponents?
A: Some common mistakes to avoid when using systems of linear equations to solve problems with variables that have different units and exponents include:
- Not checking for consistency before solving the system.
- Not using the correct method for solving the system.
- Not checking for infinitely many solutions before solving the system.
- Not checking for no solution before solving the system.