Solve the Systems of Equations by the Elimination Method

Exercise 1

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

Exercise 2

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

Exercise 3

\left\{\begin{matrix} 3x + 2y = 24 \\ x + 3y = 3 \end{matrix}\right

Exercise 4

\left\{\begin{matrix} x + y = 3500 \\ x - \frac { 10x }{ 100 } + y - \frac { 8y }{ 100 } = 3170 \end{matrix}\right

Exercise 5

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

Exercise 6

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

Exercise 7

\left\{\begin{matrix} x + y = 2000 \\ x + \frac { 10 x }{ 100 } + y + \frac { 15y }{ 100 } = 2260 \end{matrix}\right

Exercise 8

\left\{\begin{matrix} x + y = 58 \\ 2x + 4y = 168 \end{matrix}\right

 

Solution of exercise 1

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

\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

Adding both equations:

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

-20y = -60

y = 3

 

Plugging the value of y in any of the equation above:

2x + 4y = 16

2x + 4(3) = 16

2x + 12 = 16

2x = 4

x = 2

 

Solution of exercise 2

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

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

 

Plugging the value of x:

3x + 2y = 7

3 + 2y = 7

2y = 4

y = 2

 

Solution of exercise 3

\left\{\begin{matrix} 3x + 2y = 24 \\ x + 3y = 3 \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

Subtracting equation 3 from equation 1:

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

3x + 9y - 3x - 2y = -15

7y = -15

y = - \frac { 15 }{ 7 }

 

Plugging the value of y in the first equation:

3x + 2y = 24

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

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

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

3x = \frac { 30 + 168 }{ 7 }

3x = \frac { 198 }{ 7 }

x = \frac { 198 }{ 7 } \times \frac { 1 }{ 3 }

x = \frac { 66 }{ 7 }

 

Solution of exercise 4

\left\{\begin{matrix} x + y = 3500 \\ x - \frac { 10x }{ 100 } + y - \frac { 8y }{ 100 } = 3170 \end{matrix}\right

\left\{\begin{matrix} x + y = 3500 \rightarrow \times (-90) \rightarrow \\ x - \frac { 10x }{ 100 } + y - \frac { 8y }{ 100 } = 3170 \end{matrix}\right \left\{\begin{matrix} -90x - 90y = -315000 \\ 90x + 92y = 317000 \end{matrix}\right

Adding both equations:

-90x - 90y +90x + 92y = -315000 + 317000

2y = 2000

y = 1000

 

x + y = 3500

x + 1000 = 3500

x = 3500 - 1000 = 2500

Solution of exercise 5

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

3x + 4y = 0

3x = -4y

x = -\frac { 4y }{ 3 }

 

2x + 3y = -1

2 (-\frac { 4y }{ 3 }) + 3y = -1

-\frac { 8y }{ 3 } + 3y = -1

\frac { -8y + 9y }{ 3 } = -1

y = -3

 

x = -\frac { 4y }{ 3 }

x = -\frac { 4(-3) }{ 3 }

x = \frac { 12 }{ 3 }

x = 4

 

Solution of exercise 6

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

-x + y = -2

y = -2 + x

 

Plugging the value of y in the second equation:

x - 3y  = 2

x - 3(-2 + x)  = 2

x + 6 - 3x = 2

-2x = -4

x = 2

 

Plugging the value of y in the third equation:

y = -2 + x

y = -2 + 2

y = 0

 

Solution of exercise 7

\left\{\begin{matrix} x + y = 2000 \\ x + \frac { 10 x }{ 100 } + y + \frac { 15y }{ 100 } = 2260 \end{matrix}\right

\left\{\begin{matrix} x + y = 2000 \times (-110)\\ \frac { 10 x +100x + 100y +15y }{ 100 }  = 2260 \end{matrix}\right \left\{\begin{matrix} -110x -110y = -220000 \\ 110x + 115y = 226000 \end{matrix}\right

Adding both equations:

-110x -110y + 110x + 115y = -220000 + 226000

5y = 6000

y = 1200

 

Plugging the value of y in the first equation:

x + 1200 = 2000

x = 800

 

Solution of exercise 8

\left\{\begin{matrix} x + y = 58 \\ 2x + 4y = 168 \end{matrix}\right

\left\{\begin{matrix} x + y = 58 \rightarrow \times (-2) \rightarrow \\ 2x + 4y = 168 \end{matrix}\right \left\{\begin{matrix} -2x - 2y = -116 \\ 2x + 4y = 168 \end{matrix}\right

Adding both equations:

-2x - 2y + 2x + 4y = -116 + 168

2y = 52

y = 26

 

x + y = 58

x + 26 = 58

x = 58 - 26

x = 32

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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.