We want to complete each definition with one of the given vocabulary terms, then provide an example. Let's take a look at each definition, one at a time.

Definition $1$

Consider the given definition.

Any ordered pair that makes all equations in the system true is a $\underline{solution of a system of linear equations} .$

We will begin by making sense of the given information. An ordered pair satisfying an equation is a solution to the equation. Let's look at the following example.

Ordered Pair : Equation : (3, 1) 2 x + 3 y = 9

Here, the ordered pair makes the equation true.

2 x + 3 y = 9

SubstituteII

$x = 3$ , $y = 1$

2 (3) + 3 (1) = 9

Multiply

6 + 3 = 9

AddTerms

Add terms

9 = 9 ✓

The ordered pair

(3, 1)

is a solution of the equation. In a similar vein, an ordered pair that makes all equations in a system of linear equations is called a solution to the system of equations.

Any ordered pair that makes all equations in the system true is a $\underline{solution of a system of linear equations} .$

Let's now give an example of a system of linear equations and an ordered pair that makes all equations in the system true.

Example

Consider the following system of linear equations and the ordered pair.

Ordered Pair : System of Equations : (3, - 2) {3 x + 7 y = - 5 x + 2 y = - 1 (I) (II)

Here, the ordered pair

(3, - 2)

satisfies both equations in the system simultaneously.

{3 x + 7 y = - 5 x + 2 y = - 1 (I) (II)

$(I), (II):$ $x = 3$ , $y = 1$

{3 (3) + 7 (- 2) = - 5 3 + 2 (- 2) = - 1 (I) (II)

Multiply

{9 + (- 14) = - 5 3 + (- 4) = - 1 (I) (II)

AddNeg

$a + (- b) = a - b$

{9 - 14 = - 5 3 - 4 = - 1 (I) (II)

SubTerms

Subtract terms

{- 5 = - 5 ✓ - 1 = - 1 ✓ (I) (II)

The ordered pair

(3, - 2)

makes both equations true. Therefore, it is a solution of the system.

Definition $2$

Let's take a look at the next definition.

A $\underline{system of linear equations}$ is formed by two or more linear equations that use the same variables.

If the variables of two or more linear equations are common, then we can say that these equations work together. When two or more linear equations work together, they are called a system of linear equations.

A $\underline{system of linear equations}$ is formed by two or more linear equations that use the same variables.

Example

There are infinitely many examples of a system of linear equation. Let's form a system of

3

linear equations with the variables

x,

y,

and

z .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x + y - 3 z = 5 x - 2 y + z = - 3 - x + y + 2 z = 6 (I) (II) (III)

Summary

Finally, let's summarize our findings by making a table with the definitions and examples.

Definition	Example
Any ordered pair that makes all equations in the system true is a $\underline{solution of a system of linear equations} .$	$System : Solution : {3 x + 7 y = - 5 x + 2 y = - 1 (I) (II) (3, - 2)$
A $\underline{system of linear equations}$ is formed by two or more linear equations that use the same variables.	$⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x + y - 3 z = 5 x - 2 y + z = - 3 - x + y + 2 z = 6 (I) (II) (III)$

Definition 1

Example

Definition 2

Example

Summary

Definition $1$

Definition $2$

Exercise