Solve The System Of Equations Given Below.${ \begin{aligned} y - 15 &= 3x \ -2x + 5y &= -3 \end{aligned} }$A. { (-3, 6)$}$B. { (-7, -6)$}$C. { (-6, -3)$}$D. { (-6, 3)$}$
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 given below is:
{ \begin{aligned} y - 15 &= 3x \\ -2x + 5y &= -3 \end{aligned} \}
Step 1: Solve the First Equation for y
We can solve the first equation for y by adding 15 to both sides of the equation.
{ \begin{aligned} y - 15 + 15 &= 3x + 15 \\ y &= 3x + 15 \end{aligned} \}
Step 2: Substitute the Expression for y into the Second Equation
We can substitute the expression for y into the second equation by replacing y with 3x + 15.
{ \begin{aligned} -2x + 5(3x + 15) &= -3 \\ -2x + 15x + 75 &= -3 \\ 13x + 75 &= -3 \end{aligned} \}
Step 3: Solve for x
We can solve for x by subtracting 75 from both sides of the equation and then dividing both sides by 13.
{ \begin{aligned} 13x + 75 - 75 &= -3 - 75 \\ 13x &= -78 \\ x &= -78/13 \\ x &= -6 \end{aligned} \}
Step 4: Find the Value of y
We can find the value of y by substituting the value of x into the expression for y.
{ \begin{aligned} y &= 3x + 15 \\ y &= 3(-6) + 15 \\ y &= -18 + 15 \\ y &= -3 \end{aligned} \}
The Solution
The solution to the system of equations is x = -6 and y = -3.
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We found the solution to be x = -6 and y = -3.
Answer
The correct answer is C. {(-6, -3)$}$
Why is this the correct answer?
This is the correct answer because we solved the system of equations using the method of substitution and elimination and found the solution to be x = -6 and y = -3. This matches the answer choice C. {(-6, -3)$}$
What are the other answer choices?
The other answer choices are:
A. {(-3, 6)$}$
B. {(-7, -6)$}$
D. {(-6, 3)$}$
Why are these not the correct answers?
These are not the correct answers because we solved the system of equations using the method of substitution and elimination and found the solution to be x = -6 and y = -3. This does not match the answer choices A, B, or D.
What is the method of substitution and elimination?
The method of substitution and elimination is a technique used to solve systems of linear equations. It involves substituting the expression for one variable into the other equation and then solving for the other variable.
What are the advantages of the method of substitution and elimination?
The advantages of the method of substitution and elimination are:
- It is a simple and straightforward method to use.
- It can be used to solve systems of linear equations with two or more variables.
- It can be used to solve systems of linear equations with two or more equations.
What are the disadvantages of the method of substitution and elimination?
The disadvantages of the method of substitution and elimination are:
- It can be time-consuming to use.
- It can be difficult to use if the equations are complex.
- It may not be the best method to use if the system of equations has multiple solutions.
What are some other methods for solving systems of linear equations?
Some other methods for solving systems of linear equations are:
- The method of graphing
- The method of matrices
- The method of determinants
What are the advantages and disadvantages of these methods?
The advantages and disadvantages of these methods are:
- The method of graphing:
- Advantages: It is a visual method that can be used to solve systems of linear equations.
- Disadvantages: It can be difficult to use if the equations are complex.
- The method of matrices:
- Advantages: It is a powerful method that can be used to solve systems of linear equations.
- Disadvantages: It can be difficult to use if the equations are complex.
- The method of determinants:
- Advantages: It is a simple and straightforward method to use.
- Disadvantages: It can be time-consuming to use.
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 are solved simultaneously to find the values of the variables.
Q: How do I solve a system of linear equations?
A: There are several methods to solve a system of linear equations, including the method of substitution and elimination, the method of graphing, the method of matrices, and the method of determinants.
Q: What is the method of substitution and elimination?
A: The method of substitution and elimination is a technique used to solve systems of linear equations. It involves substituting the expression for one variable into the other equation and then solving for the other variable.
Q: What are the advantages of the method of substitution and elimination?
A: The advantages of the method of substitution and elimination are:
- It is a simple and straightforward method to use.
- It can be used to solve systems of linear equations with two or more variables.
- It can be used to solve systems of linear equations with two or more equations.
Q: What are the disadvantages of the method of substitution and elimination?
A: The disadvantages of the method of substitution and elimination are:
- It can be time-consuming to use.
- It can be difficult to use if the equations are complex.
- It may not be the best method to use if the system of equations has multiple solutions.
Q: What is the method of graphing?
A: The method of graphing is a technique used to solve systems of linear equations. It involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: What are the advantages of the method of graphing?
A: The advantages of the method of graphing are:
- It is a visual method that can be used to solve systems of linear equations.
- It can be used to solve systems of linear equations with two or more variables.
- It can be used to solve systems of linear equations with two or more equations.
Q: What are the disadvantages of the method of graphing?
A: The disadvantages of the method of graphing are:
- It can be difficult to use if the equations are complex.
- It may not be the best method to use if the system of equations has multiple solutions.
Q: What is the method of matrices?
A: The method of matrices is a technique used to solve systems of linear equations. It involves representing the system of equations as a matrix and then solving for the variables.
Q: What are the advantages of the method of matrices?
A: The advantages of the method of matrices are:
- It is a powerful method that can be used to solve systems of linear equations.
- It can be used to solve systems of linear equations with two or more variables.
- It can be used to solve systems of linear equations with two or more equations.
Q: What are the disadvantages of the method of matrices?
A: The disadvantages of the method of matrices are:
- It can be difficult to use if the equations are complex.
- It may not be the best method to use if the system of equations has multiple solutions.
Q: What is the method of determinants?
A: The method of determinants is a technique used to solve systems of linear equations. It involves finding the determinant of the matrix representing the system of equations and then solving for the variables.
Q: What are the advantages of the method of determinants?
A: The advantages of the method of determinants are:
- It is a simple and straightforward method to use.
- It can be used to solve systems of linear equations with two or more variables.
- It can be used to solve systems of linear equations with two or more equations.
Q: What are the disadvantages of the method of determinants?
A: The disadvantages of the method of determinants are:
- It can be time-consuming to use.
- It can be difficult to use if the equations are complex.
- It may not be the best method to use if the system of equations has multiple solutions.
Conclusion
In this article, we answered some frequently asked questions about solving systems of linear equations. We discussed the different methods for solving systems of linear equations, including the method of substitution and elimination, the method of graphing, the method of matrices, and the method of determinants. We also discussed the advantages and disadvantages of each method.