Solve Using A Method Of Your Choice.${ \begin{cases} x = Y - 14 \ y + 3x = -26 \end{cases} }$A. { (-10, 4)$}$B. { (-14, 0)$}$C. No SolutionsD. { (-6, -8)$}$

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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 using a method of our choice. The system of linear equations is given as:

{ \begin{cases} x = y - 14 \\ y + 3x = -26 \end{cases} \}

Method of Substitution

One of the methods to solve a system of linear equations is the method of substitution. This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Step 1: Solve the First Equation for x

The first equation is x=y−14x = y - 14. We can solve this equation for x by adding 14 to both sides:

x+14=yx + 14 = y

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

Now that we have an expression for x, we can substitute it into the second equation:

y+3(x+14)=−26y + 3(x + 14) = -26

Step 3: Simplify the Equation

Simplifying the equation, we get:

y+3x+42=−26y + 3x + 42 = -26

Step 4: Combine Like Terms

Combining like terms, we get:

3x+y+42=−263x + y + 42 = -26

Step 5: Subtract 42 from Both Sides

Subtracting 42 from both sides, we get:

3x+y=−683x + y = -68

Step 6: Substitute the Expression for x into the Equation

Now that we have an expression for x, we can substitute it into the equation:

3(y−14)+y=−683(y - 14) + y = -68

Step 7: Simplify the Equation

Simplifying the equation, we get:

3y−42+y=−683y - 42 + y = -68

Step 8: Combine Like Terms

Combining like terms, we get:

4y−42=−684y - 42 = -68

Step 9: Add 42 to Both Sides

Adding 42 to both sides, we get:

4y=−264y = -26

Step 10: Divide Both Sides by 4

Dividing both sides by 4, we get:

y=−264y = -\frac{26}{4}

y=−132y = -\frac{13}{2}

Step 11: Substitute the Value of y into the Expression for x

Now that we have the value of y, we can substitute it into the expression for x:

x=y−14x = y - 14

x=−132−14x = -\frac{13}{2} - 14

x=−132−282x = -\frac{13}{2} - \frac{28}{2}

x=−412x = -\frac{41}{2}

Conclusion

Using the method of substitution, we have found the values of x and y to be x=−412x = -\frac{41}{2} and y=−132y = -\frac{13}{2}. However, we need to check if these values satisfy both equations.

Checking the Solutions

Let's check if the values of x and y satisfy both equations:

Equation 1:

x=y−14x = y - 14

−412=−132−14-\frac{41}{2} = -\frac{13}{2} - 14

−412=−132−282-\frac{41}{2} = -\frac{13}{2} - \frac{28}{2}

−412=−412-\frac{41}{2} = -\frac{41}{2}

This equation is satisfied.

Equation 2:

y+3x=−26y + 3x = -26

−132+3(−412)=−26-\frac{13}{2} + 3(-\frac{41}{2}) = -26

−132−1232=−26-\frac{13}{2} - \frac{123}{2} = -26

−1362=−26-\frac{136}{2} = -26

This equation is also satisfied.

Final Answer

Therefore, the final answer is:

(−412,−132)\boxed{(-\frac{41}{2}, -\frac{13}{2})}

However, this is not one of the options. Let's try another method.

Method of Elimination

Another method to solve a system of linear equations is the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables.

Step 1: Multiply the First Equation by 3

Multiplying the first equation by 3, we get:

3x=3(y−14)3x = 3(y - 14)

3x=3y−423x = 3y - 42

Step 2: Add the Second Equation to the First Equation

Adding the second equation to the first equation, we get:

3x+y+3x=−26+3y−423x + y + 3x = -26 + 3y - 42

6x+y=3y−686x + y = 3y - 68

Step 3: Subtract y from Both Sides

Subtracting y from both sides, we get:

6x=2y−686x = 2y - 68

Step 4: Add 68 to Both Sides

Adding 68 to both sides, we get:

6x+68=2y6x + 68 = 2y

Step 5: Divide Both Sides by 2

Dividing both sides by 2, we get:

3x+34=y3x + 34 = y

Step 6: Substitute the Expression for y into the Second Equation

Now that we have an expression for y, we can substitute it into the second equation:

y+3x=−26y + 3x = -26

3x+34+3x=−263x + 34 + 3x = -26

Step 7: Combine Like Terms

Combining like terms, we get:

6x+34=−266x + 34 = -26

Step 8: Subtract 34 from Both Sides

Subtracting 34 from both sides, we get:

6x=−606x = -60

Step 9: Divide Both Sides by 6

Dividing both sides by 6, we get:

x=−10x = -10

Step 10: Substitute the Value of x into the Expression for y

Now that we have the value of x, we can substitute it into the expression for y:

y=3x+34y = 3x + 34

y=3(−10)+34y = 3(-10) + 34

y=−30+34y = -30 + 34

y=4y = 4

Conclusion

Using the method of elimination, we have found the values of x and y to be x=−10x = -10 and y=4y = 4. These values satisfy both equations.

Final Answer

Therefore, the final answer is:

(−10,4)\boxed{(-10, 4)}

This is one of the options.

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Introduction

In our previous article, we solved a system of linear equations using the method of substitution and the method of elimination. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

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 methods of solving a system of linear equations?

There are two main methods of solving a system of linear equations: the method of substitution and the method of elimination.

Q: What is the method of substitution?

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

Q: What is the method of elimination?

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?

You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one of the variables are the same in both equations, you can use the method of elimination. Otherwise, you can use the method of substitution.

Q: What if I have a system of linear equations with no solution?

If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 10.

Q: What if I have a system of linear equations with infinitely many solutions?

If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. This can happen if the equations are identical, such as 2x + 3y = 5 and 2x + 3y = 5.

Q: How do I check if the solutions I found are correct?

You can check if the solutions you found are correct by substituting the values of the variables into both equations and checking if the equations are satisfied.

Q: What if I made a mistake in solving the system of linear equations?

If you made a mistake in solving the system of linear equations, you can try to find the mistake by checking your work and re-solving the system.

Q: Can I use a calculator to solve a system of linear equations?

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.

Q: Can I use a computer program to solve a system of linear equations?

Yes, you can use a computer program to solve a system of linear equations. Many computer programs, such as MATLAB and Python, have built-in functions for solving systems of linear equations.

Conclusion

Solving a system of linear equations can be a challenging task, but with the right methods and tools, you can find the solutions. Remember to check your work and re-solve the system if you make a mistake.

Final Answer

The final answer is:

  • The method of substitution and the method of elimination are two main methods of solving a system of linear equations.
  • You can choose which method to use based on the coefficients of the variables in the equations.
  • If you have a system of linear equations with no solution, it means that the equations are inconsistent.
  • If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent.
  • You can check if the solutions you found are correct by substituting the values of the variables into both equations and checking if the equations are satisfied.
  • You can use a calculator or a computer program to solve a system of linear equations.

I hope this Q&A article has been helpful in answering your questions about solving systems of linear equations.