# Show how to solve each of the following systems of linear equations

Show how to solve each of the following systems of linear equations by elimination or substitution, and using matrix inverses

a) 3x-y=1

-2+y=1

b) 3x+2y=5

-2x+3y+14

• a).

Substitution method :

The system of equations are

3x - y = 1 → ( 1 )

- 2 + y= 1 → ( 2 )

Solve eq (2), since the y has a coefficient of 1.

- 2 + y = 1

y = 1 + 2

⇒ y = 3.

Substitute the value of y = 3 in eq (1) to find the valueof x.

3x - 3 = 1

3x = 1 + 3

3x = 4

⇒ x = 4/3.

The solution is x = 4/3 and y= 3.

Inverse matrix method :

The systemof equations are

3x - y = 1 → ( 1 )

- 2 + y= 1 → ( 2 )

Rewrite the equations as 3x - y = 1 and (0)x + y = 3.

Write the equations in matrices form AX  = B .

A  is coefficeint mattrix , is variable matrix and B  is constant matrix.

and .

The solution is x = 4/3 and y = 3.

• b).

Substitution method :

Consider the system as

3x + 2y = 5     → ( 1 )

- 2x + 3y = 14 → ( 2 )

Solve eq ( 2 ) for x.

2x = 3y - 14

⇒ x = (3y - 14)/2.

Substitute the value of x = (3y - 14)/2 in eq ( 1 ), and solve for y.

3[ (3y - 14)/2] + 2y = 5

9y - 42 + 4y = 10

13y = 10 + 42 = 52

y = 52/13 = 4.

Substitute the value of y = 4 in x = (3y - 14)/2, tofind the value of x.

x = (3 * 4 - 14)/2

= (12 - 14)/2

= - 2/2

x = - 1.

The solution is x = - 1 and y = 4.

Inverse matrix method :

The systemof equations are

3x + 2y = 5    → ( 1 )

- 2x + 3y= 14 → ( 2 )

Write the equations in matrices form AX  = B .

A  is coefficeint mattrix , is variable matrix and B  is constant matrix.

and .

.

The solution is x = - 1 and y = 4.