Solve this system using matrices:

x1 − x2 + x3 = 6

2x1 − x2 + x3 = 5

3x1 + x2 + 3x3 = −20

x1 = x2 = x3 =

Step-by-step explanation:

A system of linear equations is one which may be written in the form

a11x1 + a12x2 + · · · + a1nxn = b1 (1)

a21x1 + a22x2 + · · · + a2nxn = b2 (2)

.

am1x1 + am2x2 + · · · + amnxn = bm (m)

Here, all of the coefficients aij and all of the right hand sides bi are assumed to be known constants. All of the

xi

’s are assumed to be unknowns, that we are to solve for. Note that every left hand side is a sum of terms of

the form constant × x

Solving Linear Systems of Equations

We now introduce, by way of several examples, the systematic procedure for solving systems of linear

equations.

Here is a system of three equations in three unknowns.

x1+ x2 + x3 = 4 (1)

x1+ 2x2 + 3x3 = 9 (2)

2x1+ 3x2 + x3 = 7 (3)

We can reduce the system down to two equations in two unknowns by using the first equation to solve for x1

in terms of x2 and x3

x1 = 4 − x2 − x3 (1’)

1

and substituting this solution into the remaining two equations

(2) (4 − x2 − x3) + 2x2+3x3 = 9 =⇒ x2+2x3 = 5

(3) 2(4 − x2 − x3) + 3x2+ x3 = 7 =⇒ x2− x3 = −1