to solve for the value of one of the variables. We will once again apply the Elimination Method, but this time it will be similar to when using it in a system with only two variables.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ y + 5 z = - 7 y + z = 1 x = 2 y + 3 z

▼

(I):

Solve by elimination

SysEqnSub

$(I):$ Subtract $(II)$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ y + 5 z - (y + z) = - 7 - (1) y + z = 1 x = 2 y + 3 z

Distr

$(I):$ Distribute $- 1$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ y + 5 z - y - z = - 7 - 1 y + z = 1 x = 2 y + 3 z

SubTerms

$(I):$ Subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 4 z = - 8 y + z = 1 x = 2 y + 3 z

DivEqn

$(I):$ $LHS / 4 = RHS / 4$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y + z = 1 x = 2 y + 3 z

Now that we know that

z = - 2,

we can substitute it into the second equation to find the value of

y .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y + z = 1 x = 2 y + 3 z

Substitute

$(II):$ $z = - 2$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y + (- 2) = 1 x = 2 y + 3 z

RemovePar

$(II):$ Remove parentheses

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y - 2 = 1 x = 2 y + 3 z

AddEqn

$(II):$ $LHS + 2 = RHS + 2$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y = 3 x = 2 y + 3 z

The value of

y

is

3 .

Let's substitute both values into the third equation to find

x .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y = 3 x = 2 y + 3 z

SubstituteII

$(III):$ $y = 3$ , $z = - 2$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y = 3 x = 2 (3) + 3 (- 2)

Multiply

$(III):$ Multiply

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y = 3 x = 6 - 6

SubTerm

$(III):$ Subtract term

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ z = - 2 y = 3 x = 0

The solution to the system is

(0, 3, - 2) .

This is the singular point at which all three planes intersect. Now, we can check our solution by substituting the values into the system.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ x - y + 2 z = - 7 y + z = 1 x = 2 y + 3 z (I) (II) (III)

$(I), (II), (III):$ Substitute values

⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ 0 - 3 + 2 (- 2) = ? - 7 3 + (- 2) = ? 1 0 = ? 2 (3) + 3 (- 2)

$(I), (II), (III):$ Multiply

⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ 0 - 3 - 4 = ? - 7 3 + (- 2) = ? 1 0 = ? 6 - 6

$(I), (II), (III):$ Add and subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 7 = - 7 ✓ 1 = 1 ✓ 0 = 0 ✓

Since the substitution of our answers into the given equations resulted in three identities, we know that our solution is correct.

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