Find The Solution Of The System Of Equations.${ \begin{array}{c} -10x - Y = 27 \ -4x - Y = 9 \end{array} } ( A N S W E R A T T E M P T 1 O U T O F 2 ) (Answer Attempt 1 Out Of 2) ( A N S W Er A Tt E M Pt 1 O U T O F 2 ) { (\square, \square)\$} Submit Answer

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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 means 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:

−10x−y=27−4x−y=9{ \begin{array}{c} -10x - y = 27 \\ -4x - y = 9 \end{array} }

Method 1: Substitution Method

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

Step 1: Solve the Second Equation for y

Let's solve the second equation for y:

−4x−y=9{ -4x - y = 9 }

−y=9+4x{ -y = 9 + 4x }

y=−9−4x{ y = -9 - 4x }

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

Now, let's substitute the expression for y into the first equation:

−10x−y=27{ -10x - y = 27 }

−10x−(−9−4x)=27{ -10x - (-9 - 4x) = 27 }

−10x+9+4x=27{ -10x + 9 + 4x = 27 }

−6x+9=27{ -6x + 9 = 27 }

Step 3: Solve for x

Now, let's solve for x:

−6x+9=27{ -6x + 9 = 27 }

−6x=18{ -6x = 18 }

x=−3{ x = -3 }

Step 4: Find the Value of y

Now that we have the value of x, let's find the value of y:

y=−9−4x{ y = -9 - 4x }

y=−9−4(−3){ y = -9 - 4(-3) }

y=−9+12{ y = -9 + 12 }

y=3{ y = 3 }

Method 2: Elimination Method

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

Step 1: Multiply the Two Equations by Necessary Multiples

Let's multiply the two equations by necessary multiples to make the coefficients of y's in both equations equal:

−10x−y=27−4x−y=9{ \begin{array}{c} -10x - y = 27 \\ -4x - y = 9 \end{array} }

−10x−y=2710(−4x−y)=10(9){ \begin{array}{c} -10x - y = 27 \\ 10(-4x - y) = 10(9) \end{array} }

−10x−y=27−40x−10y=90{ \begin{array}{c} -10x - y = 27 \\ -40x - 10y = 90 \end{array} }

Step 2: Add the Two Equations

Now, let's add the two equations:

(−10x−y)+(−40x−10y)=27+90{ (-10x - y) + (-40x - 10y) = 27 + 90 }

−50x−11y=117{ -50x - 11y = 117 }

Step 3: Solve for x

Now, let's solve for x:

−50x−11y=117{ -50x - 11y = 117 }

−50x=117+11y{ -50x = 117 + 11y }

x=−117+11y50{ x = -\frac{117 + 11y}{50} }

Step 4: Substitute the Value of x into One of the Original Equations

Now that we have the value of x, let's substitute it into one of the original equations to find the value of y:

−10x−y=27{ -10x - y = 27 }

−10(−117+11y50)−y=27{ -10(-\frac{117 + 11y}{50}) - y = 27 }

117+11y5−y=27{ \frac{117 + 11y}{5} - y = 27 }

117+11y−5y=135{ 117 + 11y - 5y = 135 }

117+6y=135{ 117 + 6y = 135 }

6y=18{ 6y = 18 }

y=3{ y = 3 }

Conclusion

In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have found that the value of x is -3 and the value of y is 3.

Final Answer

The final answer is (−3,3)\boxed{(-3, 3)}.

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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 means 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 equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable.

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 elimination method. If the coefficients of one variable are different in both equations, you can use the substitution method.

Q: What if I have a system of linear equations with three or more variables?

A: If you have a system of linear equations with three or more variables, you can use the same methods as before, but you may need to use more steps and more equations.

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.

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! You can try re-reading the equations, re-checking your work, or asking for help from a teacher or tutor.

Q: Are there any other methods for solving systems of linear equations?

A: Yes, there are other methods for solving systems of linear equations, such as the graphing method and the matrix method. However, these methods are more advanced and may not be covered in basic algebra classes.

Q: Can I use systems of linear equations to solve real-world problems?

A: Yes, systems of linear equations can be used to solve real-world problems in many fields, such as physics, engineering, economics, and computer science.

Q: What are some common applications of systems of linear equations?

A: Some common applications of systems of linear equations include:

  • Finding the intersection of two lines
  • Determining the cost of producing a product
  • Calculating the amount of money in a bank account
  • Solving problems in physics and engineering

Q: Can I use systems of linear equations to solve problems with multiple variables?

A: Yes, systems of linear equations can be used to solve problems with multiple variables. In fact, many real-world problems involve multiple variables and can be solved using systems of linear equations.

Q: What are some tips for solving systems of linear equations?

A: Some tips for solving systems of linear equations include:

  • Read the equations carefully and make sure you understand what they are asking for
  • Use a systematic approach to solve the equations
  • Check your work carefully to avoid mistakes
  • Use a calculator or computer program to check your answers
  • Practice, practice, practice! The more you practice solving systems of linear equations, the more comfortable you will become with the methods and techniques.