What Is The Solution To The Following System Of Equations?${ \begin{array}{l} 4x + 2y = 18 \ x - Y = 3 \end{array} }$A. { (-4, -1)$}$B. { (1, 4)$}$C. { (-1, -4)$}$D. { (4, 1)$}$

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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 involves 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{array}{l} 4x + 2y = 18 \\ x - y = 3 \end{array} \}

Method 1: 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.

Step 1: Solve the Second Equation for x

We can solve the second equation for x by adding y to both sides of the equation:

x=3+yx = 3 + y

Step 2: Substitute the Expression for x into the First Equation

Now, we can substitute the expression for x into the first equation:

4(3+y)+2y=184(3 + y) + 2y = 18

Step 3: Simplify the Equation

Expanding the equation and combining like terms, we get:

12+4y+2y=1812 + 4y + 2y = 18

Combine like terms:

6y+12=186y + 12 = 18

Subtract 12 from both sides:

6y=66y = 6

Divide both sides by 6:

y=1y = 1

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into the expression we found in Step 1:

x=3+yx = 3 + y x=3+1x = 3 + 1 x=4x = 4

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves multiplying the equations by necessary multiples such that the coefficients of one of the variables (in this case, y) are the same in both equations, and then subtracting one equation from the other.

Step 1: Multiply the First Equation by 1 and the Second Equation by 2

To eliminate the variable y, we can multiply the first equation by 1 and the second equation by 2:

4x+2y=184x + 2y = 18 2x−2y=62x - 2y = 6

Step 2: Add the Two Equations

Now, we can add the two equations to eliminate the variable y:

(4x+2y)+(2x−2y)=18+6(4x + 2y) + (2x - 2y) = 18 + 6 6x=246x = 24

Step 3: Solve for x

Divide both sides by 6:

x=4x = 4

Step 4: Find 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 second equation:

x−y=3x - y = 3 4−y=34 - y = 3

Subtract 4 from both sides:

−y=−1-y = -1

Multiply both sides by -1:

y=1y = 1

Conclusion

In this article, we solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. We found that the solution to the system of equations is x = 4 and y = 1.

Answer

The correct answer is:

(4,1)\boxed{(4, 1)}

This solution matches option D in the given choices.

Final Thoughts

Solving a system of linear equations is an essential skill in mathematics and is used in a wide range of applications, including physics, engineering, and economics. By using the substitution method or the elimination method, we can find the values of the variables that satisfy all the equations in the system.

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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 involves finding the values of the variables that satisfy all the equations in the system.

Q: What are the two main methods for solving systems of linear equations?

A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

A: The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves multiplying the equations by necessary multiples such that the coefficients of one of the variables are the same in both equations, and then subtracting one equation from the other.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. If one of the equations is already solved for one variable, the substitution method may be easier to use. If the coefficients of one of the variables are the same in both equations, the elimination method may be easier to use.

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 same methods as before, but you may need to use additional techniques, such as using matrices or graphing.

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, such as the "Solve" function on a graphing calculator.

Q: What if I get stuck or make a mistake while solving a system of linear equations?

A: If you get stuck or make a mistake while solving a system of linear equations, don't worry! Take a step back, review your work, and try again. You can also ask a teacher or tutor for help.

Q: Are systems of linear equations used in real-life applications?

A: Yes, systems of linear equations are used in a wide range of real-life applications, including physics, engineering, economics, and computer science.

Q: Can I use systems of linear equations to solve problems in other areas of mathematics?

A: Yes, systems of linear equations can be used to solve problems in other areas of mathematics, such as algebra, geometry, and calculus.

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 your work
  • Not using the correct method for the specific system of equations
  • Not following the order of operations
  • Not simplifying the equations before solving

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own, using a calculator or graphing tool to check your work.

Q: What are some resources for learning more about solving systems of linear equations?

A: Some resources for learning more about solving systems of linear equations include:

  • Textbooks on algebra and mathematics
  • Online resources, such as Khan Academy and Mathway
  • Video tutorials and lectures on YouTube and other platforms
  • Practice problems and exercises on websites and apps