Solve This System Of Equations:$\[ \begin{cases} 3x + 4y = 18 \\ 6x - 2y = 6 \end{cases} \\]A. \[$(2, 3)\$\] B. \[$(-2, -3)\$\] C. \[$(-3, -2)\$\] D. \[$(3, 2)\$\]
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. 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.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The system of equations can be represented graphically as a set of lines on a coordinate plane.
The Method of Substitution
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is easily solvable for one variable.
Step 1: Solve One Equation for One Variable
Let's solve the first equation for x:
3x + 4y = 18
Subtract 4y from both sides:
3x = 18 - 4y
Divide both sides by 3:
x = (18 - 4y) / 3
Step 2: Substitute the Expression into the Other Equation
Now, substitute the expression for x into the second equation:
6x - 2y = 6
Substitute x = (18 - 4y) / 3:
6((18 - 4y) / 3) - 2y = 6
Simplify the equation:
2(18 - 4y) - 2y = 6
Expand and simplify:
36 - 8y - 2y = 6
Combine like terms:
36 - 10y = 6
Subtract 36 from both sides:
-10y = -30
Divide both sides by -10:
y = 3
Step 3: Find the Value of the Other Variable
Now that we have the value of y, substitute it back into one of the original equations to find the value of x. We will use the first equation:
3x + 4y = 18
Substitute y = 3:
3x + 4(3) = 18
Simplify:
3x + 12 = 18
Subtract 12 from both sides:
3x = 6
Divide both sides by 3:
x = 2
The Method of Elimination
The method of elimination involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one variable are the same in both equations.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of that variable the same in both equations. We can do this by multiplying the equations by necessary multiples.
First equation:
3x + 4y = 18
Multiply both sides by 2:
6x + 8y = 36
Second equation:
6x - 2y = 6
Multiply both sides by 4:
24x - 8y = 24
Step 2: Add or Subtract the Equations
Now that the coefficients of y are the same in both equations, we can add or subtract the equations to eliminate y.
Add the equations:
(6x + 8y) + (24x - 8y) = 36 + 24
Simplify:
30x = 60
Divide both sides by 30:
x = 2
Step 3: Find the Value of the Other Variable
Now that we have the value of x, substitute it back into one of the original equations to find the value of y. We will use the first equation:
3x + 4y = 18
Substitute x = 2:
3(2) + 4y = 18
Simplify:
6 + 4y = 18
Subtract 6 from both sides:
4y = 12
Divide both sides by 4:
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 that the solution to the system is x = 2 and y = 3. This solution satisfies both equations simultaneously.
Answer
The correct answer is:
A. (2, 3)
Discussion
This problem is a classic example of a system of linear equations. The method of substitution and elimination are two common methods used to solve such systems. The method of substitution involves solving one equation 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.
In this problem, we used the method of substitution to solve the system of equations. We first solved one equation for x and then substituted that expression into the other equation. We then simplified the resulting equation to find the value of y.
The method of elimination is another common method used to solve systems of linear equations. This method involves adding or subtracting the equations to eliminate one of the variables. In this problem, we multiplied the equations by necessary multiples to make the coefficients of y the same in both equations. We then added the equations to eliminate y and found the value of x.
Both methods are useful for solving systems of linear equations, and the choice of method depends on the specific problem and the coefficients of the equations.
Final Answer
The final answer is:
(2, 3)
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. 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.
Q&A: Solving a System 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 in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the two main methods for solving a system of linear equations?
A: The two main methods for solving a system 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 equation 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 between the method of substitution and the method of elimination?
A: You can choose between the two methods 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 not, you can use the method of substitution.
Q: What are some common mistakes to avoid when solving a system of linear equations?
A: Some common mistakes to avoid when solving a system of linear equations include:
- Not checking if the equations are consistent (i.e., if they have a solution)
- Not using the correct method for the given equations
- Not simplifying the equations correctly
- Not checking if the solution satisfies both equations
Q: How do I check if the solution satisfies both equations?
A: To check if the solution satisfies both equations, you can substitute the values of x and y into both equations and check if the resulting statements are true.
Q: What if I have a system of linear equations with more than two variables?
A: If you have a system of linear equations with more than two variables, you can use the method of substitution or the method of elimination to solve for one variable, and then use the resulting equation to solve for the other variables.
Q: Can I use a calculator to solve a system of linear equations?
A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.
Conclusion
Solving a system of linear equations is a fundamental concept in mathematics, and it requires a clear understanding of the methods of substitution and elimination. By following the steps outlined in this article, you can solve a system of linear equations and find the values of the variables that satisfy the equations.
Final Answer
The final answer is:
(2, 3)