Describing Solutions to Systems of Linear Equations

a

The ordered pair

(- 2, 4)

satisfies the system if it makes a true statement in both equations. Let's substitute

x = - 2

and

y = 4

into each of the equations and simplify.

{y = - 7 x - 10 y = - 8 x + 12 (I) (II)

x = - 2

,

y = 4

{4 = ? - 7 (- 2) - 10 4 = ? 8 (- 2) + 12

(I):

- a (- b) = a \cdot b

{4 = ? 14 - 10 4 = ? 8 (- 2) + 12

(II):

a (- b) = - a \cdot b

{4 = ? 14 - 10 4 = ? - 16 + 12

AddSubTermsAdd and subtract terms

{4 = 4 4 \neq = - 4

One of our resulting statements,

4 \neq = - 4,

is not true. Thus, the point

(- 2, 4)

does not satisfy the system of equations.

b

We want to know if the ordered pair

(- 2, 4)

satisfies the following system of equations.

{y = - x + 2 y = 3 x + 10

We can investigate that by substituting

x = - 2

and

y = 4

into both equations and simplifying the resulting equation. The ordered pair satisfies the system only if both equations then are true.

{y = - x + 2 y = 3 x + 10 (I) (II)

x = - 2

,

y = 4

{4 = ? - (- 2) + 2 4 = ? 3 (- 2) + 10

(I):

- (- a) = a

{4 = ? 2 + 2 4 = ? 3 (- 2) + 10

(II):

a (- b) = - a \cdot b

{4 = ? 2 + 2 4 = ? - 6 + 10

AddTermsAdd terms

{4 = 4 4 = 4

Since both statements are true the point

(- 4, - 2)

is contained in the solution set of the system.

c

We have been asked to determine if the ordered pair

(- 2, 4)

is a solution to the following system of equations.

{- 6 x + 3 y = - 24 - 4 x - 4 y = - 24

Let's find out by substituting

x = - 2

and

y = 4

into both equations. If both equations are true the ordered pair is a solution.

{- 6 x + 3 y = - 24 - 4 x - 4 y = - 24 (I) (II)

x = - 2

,

y = 4

{- 6 (- 2) + 3 \cdot 4 = ? 24 4 (- 2) - 4 \cdot 4 = ? - 24

(I):

- a (- b) = a \cdot b

{12 + 3 \cdot 4 = ? 24 4 (- 2) - 4 \cdot 4 = ? - 24

(II):

a (- b) = - a \cdot b

{12 + 3 \cdot 4 = ? 24 - 8 - 4 \cdot 4 = ? - 24

(I), (II):

Multiply

{12 + 12 = ? 24 - 8 - 16 = ? - 24

AddSubTermsAdd and subtract terms

{24 = 24 - 24 = - 24

We have found that both statements are true. Therefore, the point

(- 2, 4)

is a solution to the system.

Describing Solutions to Systems of Linear Equations 1.1 - Solution