The System Of Equations Can Be Solved Using Linear Combination To Eliminate One Of The Variables:$[ \begin{array}{l} 2x - Y = -4 \quad \rightarrow \quad 10x - 5y = -20 \ 3x + 5y = 59 \quad \rightarrow \quad \begin{array}{l} 3x + 5y = 59 \ 13x =

by ADMIN 245 views

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. One of the most common methods used to solve systems of equations is the linear combination method, which involves eliminating one of the variables by adding or subtracting the equations. In this article, we will explore the system of equations and provide a step-by-step guide on how to solve it using linear combination.

What is a System of Equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Each equation in the system is a statement that two expressions are equal. For example, consider the following system of equations:

2x - y = -4 3x + 5y = 59

In this system, we have two equations and two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.

The Linear Combination Method

The linear combination method is a technique used to solve systems of equations by eliminating one of the variables. This method involves adding or subtracting the equations to eliminate one of the variables. To do this, we need to multiply one or both of the equations by a constant, and then add or subtract the resulting equations.

Step 1: Multiply the Equations by a Constant

To eliminate one of the variables, we need to multiply one or both of the equations by a constant. Let's multiply the first equation by 5 and the second equation by 2:

5(2x - y) = 5(-4) 2(3x + 5y) = 2(59)

This gives us:

10x - 5y = -20 6x + 10y = 118

Step 2: Add or Subtract the Equations

Now that we have multiplied the equations by a constant, we can add or subtract the resulting equations to eliminate one of the variables. Let's add the two equations:

(10x - 5y) + (6x + 10y) = (-20) + 118

This gives us:

16x + 5y = 98

Step 3: Solve for One Variable

Now that we have eliminated one of the variables, we can solve for the other variable. Let's solve for x:

16x = 98 - 5y

x = (98 - 5y) / 16

Step 4: Substitute the Value of One Variable into the Other Equation

Now that we have solved for one variable, we can substitute the value of that variable into the other equation to solve for the other variable. Let's substitute the value of x into the second equation:

3x + 5y = 59

x = (98 - 5y) / 16

Substituting this value of x into the second equation, we get:

3((98 - 5y) / 16) + 5y = 59

Step 5: Solve for the Other Variable

Now that we have substituted the value of one variable into the other equation, we can solve for the other variable. Let's solve for y:

3((98 - 5y) / 16) + 5y = 59

Multiplying both sides of the equation by 16 to eliminate the fraction, we get:

3(98 - 5y) + 80y = 944

Expanding the equation, we get:

294 - 15y + 80y = 944

Combine like terms:

65y = 650

Divide both sides of the equation by 65:

y = 10

Step 6: Find the Value of the Other Variable

Now that we have solved for one variable, we can find the value of the other variable. Let's find the value of x:

x = (98 - 5y) / 16

Substituting the value of y into the equation, we get:

x = (98 - 5(10)) / 16

x = (98 - 50) / 16

x = 48 / 16

x = 3

Conclusion

In this article, we have explored the system of equations and provided a step-by-step guide on how to solve it using linear combination. We have shown that by multiplying the equations by a constant, adding or subtracting the resulting equations, solving for one variable, substituting the value of one variable into the other equation, and solving for the other variable, we can find the values of the variables that satisfy both equations. We hope that this article has provided a comprehensive guide to solving systems of equations using linear combination.

Example 2: Solving a System of Equations with Three Variables

Consider the following system of equations:

x + 2y - 3z = 7 2x - 3y + 4z = 11 3x + 5y - 2z = 9

To solve this system of equations, we can use the linear combination method. Let's multiply the first equation by 2 and the second equation by 1:

2(x + 2y - 3z) = 2(7) (2x - 3y + 4z) = (11)

This gives us:

2x + 4y - 6z = 14 2x - 3y + 4z = 11

Now, let's subtract the second equation from the first equation:

(2x + 4y - 6z) - (2x - 3y + 4z) = 14 - 11

This gives us:

7y - 10z = 3

Now, let's multiply the third equation by 2:

2(3x + 5y - 2z) = 2(9)

This gives us:

6x + 10y - 4z = 18

Now, let's add the first equation to the third equation:

(x + 2y - 3z) + (6x + 10y - 4z) = 7 + 18

This gives us:

7x + 12y - 7z = 25

Now, let's multiply the second equation by 7:

7(2x - 3y + 4z) = 7(11)

This gives us:

14x - 21y + 28z = 77

Now, let's add the third equation to the second equation:

(2x - 3y + 4z) + (6x + 10y - 4z) = 11 + 9

This gives us:

8x + 7y = 20

Now, let's multiply the first equation by 7:

7(x + 2y - 3z) = 7(7)

This gives us:

7x + 14y - 21z = 49

Now, let's add the second equation to the first equation:

(7x + 14y - 21z) + (14x - 21y + 28z) = 49 + 77

This gives us:

21x + 7z = 126

Now, let's multiply the third equation by 7:

7(7x + 12y - 7z) = 7(25)

This gives us:

49x + 84y - 49z = 175

Now, let's add the second equation to the third equation:

(8x + 7y) + (49x + 84y - 49z) = 20 + 175

This gives us:

57x + 91y - 49z = 195

Now, let's multiply the first equation by 49:

49(7x + 12y - 7z) = 49(25)

This gives us:

343x + 588y - 343z = 1225

Now, let's add the second equation to the first equation:

(21x + 7z) + (343x + 588y - 343z) = 126 + 1225

This gives us:

364x + 588y = 1351

Now, let's multiply the third equation by 21:

21(7x + 12y - 7z) = 21(25)

This gives us:

147x + 252y - 147z = 525

Now, let's add the second equation to the third equation:

(57x + 91y - 49z) + (147x + 252y - 147z) = 195 + 525

This gives us:

204x + 343y = 720

Now, let's multiply the first equation by 343:

343(7x + 12y - 7z) = 343(25)

This gives us:

2401x + 4116y - 2401z = 8575

Now, let's add the second equation to the first equation:

(364x + 588y) + (2401x + 4116y - 2401z) = 1351 + 8575

This gives us:

2765x + 4704y - 2401z = 9926

Now, let's multiply the third equation by 364:

364(7x + 12y - 7z) = 364(25)

This gives us:

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: What is linear combination?

A: Linear combination is a technique used to solve systems of equations by eliminating one of the variables. This method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I know which variable to eliminate?

A: To determine which variable to eliminate, you need to look at the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can eliminate that variable.

Q: What is the first step in solving a system of equations using linear combination?

A: The first step in solving a system of equations using linear combination is to multiply one or both of the equations by a constant to make the coefficients of one variable the same.

Q: How do I multiply the equations by a constant?

A: To multiply the equations by a constant, you need to multiply each term in the equation by the constant. For example, if you want to multiply the first equation by 2, you would multiply each term in the equation by 2.

Q: What is the next step in solving a system of equations using linear combination?

A: The next step in solving a system of equations using linear combination is to add or subtract the resulting equations to eliminate one of the variables.

Q: How do I add or subtract the equations?

A: To add or subtract the equations, you need to add or subtract the corresponding terms in the equations. For example, if you want to add the first equation to the second equation, you would add the corresponding terms in the equations.

Q: What is the final step in solving a system of equations using linear combination?

A: The final step in solving a system of equations using linear combination is to solve for the remaining variable.

Q: How do I solve for the remaining variable?

A: To solve for the remaining variable, you need to isolate the variable on one side of the equation and the constant on the other side of the equation.

Q: What are some common mistakes to avoid when solving systems of equations using linear combination?

A: Some common mistakes to avoid when solving systems of equations using linear combination include:

  • Not multiplying the equations by a constant to make the coefficients of one variable the same
  • Not adding or subtracting the resulting equations to eliminate one of the variables
  • Not solving for the remaining variable
  • Not checking the solution to make sure it satisfies both equations

Q: What are some real-world applications of solving systems of equations using linear combination?

A: Some real-world applications of solving systems of equations using linear combination include:

  • Finding the intersection of two lines in a coordinate plane
  • Determining the cost of producing a product based on the cost of the materials and the cost of labor
  • Finding the maximum or minimum value of a function
  • Solving problems in physics, engineering, and economics

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

A: You can practice solving systems of equations using linear combination by working through examples and exercises in a textbook or online resource. You can also try solving systems of equations using linear combination on your own by creating your own problems and solutions.

Q: What are some online resources for learning about solving systems of equations using linear combination?

A: Some online resources for learning about solving systems of equations using linear combination include:

  • Khan Academy: Solving Systems of Equations
  • Mathway: Solving Systems of Equations
  • Wolfram Alpha: Solving Systems of Equations
  • MIT OpenCourseWare: Linear Algebra and Differential Equations

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

A: Some tips for mastering solving systems of equations using linear combination include:

  • Practice, practice, practice: The more you practice solving systems of equations using linear combination, the more comfortable you will become with the method.
  • Start with simple problems: Begin with simple problems and gradually work your way up to more complex problems.
  • Use visual aids: Use visual aids such as graphs and charts to help you understand the relationships between the variables.
  • Check your work: Always check your work to make sure that the solution satisfies both equations.