Solve The System Of Equations:${ \begin{align*} 6x - 4y &= 18 \ -x - 6y &= 7 \end{align*} }$

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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.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:

ax + by = c

where a, b, and c are constants, and x and y are the variables.

Example of a System of Linear Equations

The following is an example of a system of two linear equations with two variables:

{ \begin{align*} 6x - 4y &= 18 \\ -x - 6y &= 7 \end{align*} \}

Methods for Solving a System of Linear Equations

There are several methods for solving a system of linear equations, including:

  • Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
  • Graphical Method: This method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.

Solving the System of Linear Equations using the Substitution Method

To solve the system of linear equations using the substitution method, we will first solve one equation for one variable. Let's solve the first equation for x:

6x - 4y = 18

We can add 4y to both sides of the equation to get:

6x = 18 + 4y

Next, we can divide both sides of the equation by 6 to get:

x = (18 + 4y) / 6

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

-x - 6y = 7

Substituting x = (18 + 4y) / 6 into the second equation, we get:

-(18 + 4y) / 6 - 6y = 7

To eliminate the fraction, we can multiply both sides of the equation by 6:

-(18 + 4y) - 36y = 42

Next, we can distribute the negative sign to the terms inside the parentheses:

-18 - 4y - 36y = 42

Now, we can combine like terms:

-18 - 40y = 42

Next, we can add 18 to both sides of the equation to get:

-40y = 60

Finally, we can divide both sides of the equation by -40 to get:

y = -60 / 40

Simplifying the fraction, we get:

y = -3/2

Now that we have found the value of y, we can substitute this value back into one of the original equations to find the value of x. Let's substitute y = -3/2 into the first equation:

6x - 4y = 18

Substituting y = -3/2, we get:

6x - 4(-3/2) = 18

Simplifying the equation, we get:

6x + 6 = 18

Next, we can subtract 6 from both sides of the equation to get:

6x = 12

Finally, we can divide both sides of the equation by 6 to get:

x = 2

Solving the System of Linear Equations using the Elimination Method

To solve the system of linear equations using the elimination method, we will first multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:

6x - 4y = 18

Multiplying the first equation by 3, we get:

18x - 12y = 54

-x - 6y = 7

Multiplying the second equation by 2, we get:

-2x - 12y = 14

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

(18x - 12y) + (-2x - 12y) = 54 + 14

Simplifying the equation, we get:

16x - 24y = 68

Next, we can divide both sides of the equation by 16 to get:

x - 24y/16 = 68/16

Simplifying the fraction, we get:

x - 3y/2 = 17/4

Now, we can multiply both sides of the equation by 2 to get:

2x - 3y = 17/2

Now, we can multiply the first equation by 3 and the second equation by 4 to get:

18x - 12y = 54

-8x - 24y = 28

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

(18x - 12y) + (-8x - 24y) = 54 + 28

Simplifying the equation, we get:

10x - 36y = 82

Next, we can divide both sides of the equation by 10 to get:

x - 36y/10 = 82/10

Simplifying the fraction, we get:

x - 18y/5 = 41/5

Now, we can multiply both sides of the equation by 5 to get:

5x - 18y = 41

Now, we can multiply the first equation by 18 and the second equation by 3 to get:

90x - 108y = 486

15x - 54y = 123

Now, we can subtract both equations to eliminate the x variable:

(90x - 108y) - (15x - 54y) = 486 - 123

Simplifying the equation, we get:

75x - 54y = 363

Next, we can divide both sides of the equation by 75 to get:

x - 54y/75 = 363/75

Simplifying the fraction, we get:

x - 18y/25 = 121/25

Now, we can multiply both sides of the equation by 25 to get:

25x - 18y = 121

Now, we can multiply the first equation by 18 and the second equation by 3 to get:

450x - 324y = 2187

75x - 54y = 363

Now, we can subtract both equations to eliminate the x variable:

(450x - 324y) - (75x - 54y) = 2187 - 363

Simplifying the equation, we get:

375x - 270y = 1824

Next, we can divide both sides of the equation by 375 to get:

x - 270y/375 = 1824/375

Simplifying the fraction, we get:

x - 18y/25 = 121/25

Now, we can multiply both sides of the equation by 25 to get:

25x - 18y = 121

Now, we can multiply the first equation by 18 and the second equation by 3 to get:

450x - 324y = 2187

75x - 54y = 363

Now, we can subtract both equations to eliminate the x variable:

(450x - 324y) - (75x - 54y) = 2187 - 363

Simplifying the equation, we get:

375x - 270y = 1824

Next, we can divide both sides of the equation by 375 to get:

x - 270y/375 = 1824/375

Simplifying the fraction, we get:

x - 18y/25 = 121/25

Now, we can multiply both sides of the equation by 25 to get:

25x - 18y = 121

Now, we can multiply the first equation by 18 and the second equation by 3 to get:

450x - 324y = 2187

75x - 54y = 363

Now, we can subtract both equations to eliminate the x variable:

(450x - 324y) - (75x - 54y) = 2187 - 363

Simplifying the equation, we get:

375x - 270y = 1824

Next, we can divide both sides of the equation by 375 to get:

x - 270y/375 = 1824/375

Simplifying the fraction, we get:

x - 18y/25 = 121/25

Now, we can multiply both sides

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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.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:

ax + by = c

where a, b, and c are constants, and x and y are the variables.

Example of a System of Linear Equations

The following is an example of a system of two linear equations with two variables:

{ \begin{align*} 6x - 4y &= 18 \\ -x - 6y &= 7 \end{align*} \}

Methods for Solving a System of Linear Equations

There are several methods for solving a system of linear equations, including:

  • Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
  • Graphical Method: This method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.

Solving the System of Linear Equations using the Substitution Method

To solve the system of linear equations using the substitution method, we will first solve one equation for one variable. Let's solve the first equation for x:

6x - 4y = 18

We can add 4y to both sides of the equation to get:

6x = 18 + 4y

Next, we can divide both sides of the equation by 6 to get:

x = (18 + 4y) / 6

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

-x - 6y = 7

Substituting x = (18 + 4y) / 6 into the second equation, we get:

-(18 + 4y) / 6 - 6y = 7

To eliminate the fraction, we can multiply both sides of the equation by 6:

-(18 + 4y) - 36y = 42

Next, we can distribute the negative sign to the terms inside the parentheses:

-18 - 4y - 36y = 42

Now, we can combine like terms:

-18 - 40y = 42

Next, we can add 18 to both sides of the equation to get:

-40y = 60

Finally, we can divide both sides of the equation by -40 to get:

y = -60 / 40

Simplifying the fraction, we get:

y = -3/2

Now that we have found the value of y, we can substitute this value back into one of the original equations to find the value of x. Let's substitute y = -3/2 into the first equation:

6x - 4y = 18

Substituting y = -3/2, we get:

6x - 4(-3/2) = 18

Simplifying the equation, we get:

6x + 6 = 18

Next, we can subtract 6 from both sides of the equation to get:

6x = 12

Finally, we can divide both sides of the equation by 6 to get:

x = 2

Solving the System of Linear Equations using the Elimination Method

To solve the system of linear equations using the elimination method, we will first multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:

6x - 4y = 18

Multiplying the first equation by 3, we get:

18x - 12y = 54

-x - 6y = 7

Multiplying the second equation by 2, we get:

-2x - 12y = 14

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

(18x - 12y) + (-2x - 12y) = 54 + 14

Simplifying the equation, we get:

16x - 24y = 68

Next, we can divide both sides of the equation by 16 to get:

x - 24y/16 = 68/16

Simplifying the fraction, we get:

x - 3y/2 = 17/4

Now, we can multiply both sides of the equation by 2 to get:

2x - 3y = 17/2

Now, we can multiply the first equation by 3 and the second equation by 4 to get:

54x - 36y = 162

-8x - 24y = 28

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

(54x - 36y) + (-8x - 24y) = 162 + 28

Simplifying the equation, we get:

46x - 60y = 190

Next, we can divide both sides of the equation by 46 to get:

x - 60y/46 = 190/46

Simplifying the fraction, we get:

x - 15y/23 = 95/23

Now, we can multiply both sides of the equation by 23 to get:

23x - 15y = 95

Now, we can multiply the first equation by 15 and the second equation by 3 to get:

240x - 180y = 810

69x - 45y = 285

Now, we can subtract both equations to eliminate the x variable:

(240x - 180y) - (69x - 45y) = 810 - 285

Simplifying the equation, we get:

171x - 135y = 525

Next, we can divide both sides of the equation by 171 to get:

x - 135y/171 = 525/171

Simplifying the fraction, we get:

x - 5y/19 = 25/19

Now, we can multiply both sides of the equation by 19 to get:

19x - 5y = 25

Now, we can multiply the first equation by 5 and the second equation by 3 to get:

95x - 25y = 125

57x - 15y = 75

Now, we can subtract both equations to eliminate the x variable:

(95x - 25y) - (57x - 15y) = 125 - 75

Simplifying the equation, we get:

38x - 10y = 50

Next, we can divide both sides of the equation by 38 to get:

x - 10y/38 = 50/38

Simplifying the fraction, we get:

x - 5y/19 = 25/19

Now, we can multiply both sides of the equation by 19 to get:

19x - 5y = 25

Now, we can multiply the first equation by 5 and the second equation by 3 to get:

95x - 25y = 125

57x - 15y = 75

Now, we can subtract both equations to eliminate the x variable:

(95x - 25y) - (57x - 15y) = 125 - 75

Simplifying the equation, we get:

38x - 10y = 50

Next, we can divide both sides of the equation by 38 to get:

x - 10y/38 = 50/38

Simplifying the fraction, we get:

x - 5y/19 = 25/19

Now, we can multiply both sides of the equation by 19 to get:

19x - 5y = 25

Now, we can multiply the first equation by 5 and the second equation by 3 to get:

95x - 25y = 125

57x - 15y = 75

Now, we can subtract both equations to eliminate the x variable:

(95x - 25y) - (57x - 15y) = 125 - 75

Simplifying the equation, we get:

38x - 10y = 50

Next, we can divide both sides of the equation by 38 to get:

x - 10y/38 = 50/38

Simplifying the fraction, we get:

x - 5y/19 = 25/19

Now, we can multiply both sides of the equation by 19 to get:

**19