Chapters

 

How To Solve Systems of Equations by the Substitution Method

1.

Work out the value of an unknown in one of the equations.

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2.

Substitute the expression of this unknown in the other equation, obtaining an equation with one unknown.

3.

Solve the equation.

4.

The value obtained is substituted into the other equation.

5.

The two values obtained are the solution of the system.

 

Q. Find the value of unknowns using the substitutional method.

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

 

1

Work out the value of x.

2x = 16 - 4y \qquad 3x = -6 + 4y

x = 8 - 2y \qquad 3x = -6 + 4y

 

2

Substitute into the other equation the value of x:

3(8 - 2y) = -6 + 4y

3

Solve the equation obtained:

24 - 6y = -6 + 4y

24 + 6 = 6y + 4y

30 = 10y

y = 3

 

4

Substitute the value obtained.

x = 8 - 2y

x = 8 - 2(3)

x = 8 - 6

x = 2

5

Solution:

x = 2, \qquad y = 3

 

\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

-8y + 9y = -3

y = -3

 

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

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

x = 4

 

 

 

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

3x = 7 - 2y

x = \frac { 7 - 2y }{ 3 }

 

4x - 3y = -2

4 . (\frac { 7 - 2y }{ 3 }) - 3y = -2

\frac { 28 - 8y }{ 3 } - 3y = -2

28 - 8y - 9y = -6

-17y = -34

y = 2

 

x = \frac { 7 - 2y }{ 3 }

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

x = 1

 

 

\left\{\begin{matrix} \frac { x + 3y }{ 2 } = 5 \\ 3x - y = 5y \end{matrix}\right

\left\{\begin{matrix} x + 3y = 10 \\ 3x = 5y + y \end{matrix}\right

x + 3y = 10

x = 10 - 3y

 

3x = 6y

3(10 - 3y) = 6y

30 - 9y = 6y

30 = 15y

y = 2

 

x = 10 - 3y

x = 10 - 3(2)

x = 4

 

 

\left\{\begin{matrix} \frac { x + 3y }{ 2 } = 5 \\ 4 - \frac { 2x - y }{ 2 } = 1 \end{matrix}\right

\left\{\begin{matrix} x + 3y = 10 \\ -2x + y = -6 \end{matrix}\right

x + 3y = 10

x = 10 - 3y

 

-2x + y = -6

-2(10 - 3y) + y = -6

-20 + 6y + y = -6

7y = 14

y = 2

 

x = 10 - 3y

x = 10 - 3(2)

x = 4

 

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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.