Chapters

 

Solving Systems of Equations by the Elimination Method

1

Prepare the two equations and multiply by the appropriate numbers in order to eliminate one of the unknown values.

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2

Add the systems and eliminate one of the unknowns.

3

Solve the resulting equation.

4

Substitute the value obtained into one of the initial equations and then solve.

5

The two values obtained are the solution of the system.

\left\{\begin{matrix} 3x - 4y = -6 \\ 2x + 4y = 16 \end{matrix}\right

The easiest method is to remove the y, this way the equations do not have to be prepared. However, by choosing to remove the x, the process can be seen better.

\left\{\begin{matrix} 3x - 4y = -6 \rightarrow \times 2 \rightarrow \\ 2x + 4y = 16 \rightarrow \times (-3) \rightarrow \end{matrix}\right \left\{\begin{matrix} 6x - 8y = -12 \\ -6x - 12y = -48 \end{matrix}\right

Add and solve the equation:

6x - 8y - 6x - 12y = -12 - 48

-20y = -60

y = 3

Replace the value of y in any of the equations, we are replacing in the second equation.

2x + 4y = 16

2x + 4 . 3 = 16

2x = 4

x = 2

Solution:

x = 2, y = 3

 

\left\{\begin{matrix} 2x + 3y = -1 \\ 3x + 4y = 0 \end{matrix}\right

\left\{\begin{matrix} 2x + 3y = -1 \rightarrow \times (3) \rightarrow \\ 3x + 4y = 0 \rightarrow \times (-2) \rightarrow \end{matrix}\right \left\{\begin{matrix} 6x + 9y = -3 \\ -6x - 8y = 0 \end{matrix}\right

Adding both equations:

6x + 9y - 6x - 8y = -3 + 0

y = -3

 

Replacing the value of y in the first equation:

2x + 3y = -1

2x + 3(-3) = -1

2x - 9 = -1

2x = 8

x = 4

 

Solution:

x = 4, y = -3

 

\left\{\begin{matrix} 3x + 2y = 7 \\ 4x - 3y = -2 \end{matrix}\right

\left\{\begin{matrix} 3x + 2y = 7 \rightarrow \times 3 \rightarrow \\ 4x - 3y = -2 \rightarrow \times 2 \rightarrow \end{matrix}\right \left\{\begin{matrix} 9x + 6y = 21 \\ 8x - 6y = -4 \end{matrix}\right

Adding both equations

9x + 6y + 8x - 6y = 21 - 4

17x = 17

x = 1

 

Replacing the value of x in the first equation:

3x + 2y = 7

3 + 2y = 7

2y = 4

y = 2

 

\left\{\begin{matrix} \frac { x + y }{ 2 } = x - 1 \\ \frac { x - y }{ 2 } = y + 1 \end{matrix}\right

\left\{\begin{matrix} x + y = 2(x - 1) \\ x - y = 2(y + 1) \end{matrix}\right

\left\{\begin{matrix} x + y = 2x - 2 \\ x - y = 2y - 2 \end{matrix}\right

\left\{\begin{matrix} -x + y = -2 \\ x - 3y = 2 \end{matrix}\right

Adding both equations:

-x + y + x - 3y = -2 + 2

-2y = 0

y = 0

 

Plugging the value of y in the first equation:

\frac { x + y }{ 2 } = x - 1

\frac { x + 0 }{ 2 } = x - 1

x = 2x - 2

-x = -2

x = 2

 

\left\{\begin{matrix} \frac { x }{ 2 } + \frac { y }{ 3 } = 4 \\ \frac { x }{ 3 } + y = 1 \end{matrix}\right

\left\{\begin{matrix} 3x + 2y = 24 \\ x + 3y = 3 \rightarrow \times (-3) \rightarrow \end{matrix}\right \left\{\begin{matrix} 3x + 2y = 24 \\ -3x - 9y = -9 \end{matrix}\right

Adding both equations:

3x + 2y - 3x - 9y = 24 - 9

-7y = 15

y = - \frac { 15 }{ 7 }

 

Replacing the value of y in the third equation:

3x + 2y = 24

3x + 2(- \frac { 15 }{ 7 }) = 24

3x - \frac { 30 }{ 7 } = 24

21x - 30 = 168

21x = 198

x = \frac { 198 }{ 21 }

x = \frac { 66 }{ 7 }

 

\left\{\begin{matrix} \frac { x + 1 }{ 3 } + \frac { y - 1 }{ 2 } = 0 \\ \frac { x + 2y }{ 3 } - \frac { x + y + 2 }{ 4 } = 0  \end{matrix}\right

\left\{\begin{matrix} 2(x + 1) + 3(y - 1) = 0 \\ 4(x + 2y) - 3(x + y + 2) = 0 \end{matrix}\right

\left\{\begin{matrix} 2x + 2 + 3y - 3 =0 \\ 4x + 8y - 3x - 3y - 6 = 0 \end{matrix}\right

\left\{\begin{matrix} 2x + 3y = 1 \\ x + 5y = 6 \rightarrow \times (-2) \rightarrow \end{matrix}\right \left\{\begin{matrix} 2x + 3y = 1 \\ -2x - 10y = -12 \end{matrix}\right

Adding both equations:

2x + 3y - 2x - 10y = 1 - 12

-7y = -11

y = \frac { 11 }{ 7 }

 

Replacing the value of y in the third equation:

2x + 3y = 1

2x + 3 . (\frac { 11 }{ 7 }) = 1

2x + \frac { 33 }{ 7 } = 1

14x + 33 = 7

14x = -26

x = \frac { -26 }{ 14 }

x = \frac { -13 }{ 7 }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.