Chapter 3, Problem 1VR

Complete each statement with the correct term from the column on the right. Some of the choices may be used more than once and some may not be used at all.

line

half-plane

independent

dependent

consistent

inconsistent

pair

triple

A solution of a system of two equations in two variables is an ordered ________that makes both equations true. [3.1a]

To determine

To fill: The blanks provided in the statement “A solution of a system of two equations in two variables is an ordered _________ that makes both equations true”

Answer to Problem 1VR

Solution:

A solution of a system of two equations in two variables is an ordered pair that makes both equations true.

Explanation of Solution

Given information:

The provided options are:

Line

Half-plane.

Independent.

Dependent.

Consistent.

Inconsistent.

Pair.

Triple.

A system of two equations has two variables; a solution to that system is an ordered pair representing each variable that makes both equations true.

For Example:

The equations $2x-3y=5$ and $x=4y+5$ have a solution for the variables $x\text{ and }y$ as $1\text{ and }-1$ respectively which can be found as:

Consider the equations:

$2x-3y=5$ …… (1)

$x=4y+5$ …… (2)

Substitute $x=4y+5$ in equation (1).

$\begin{array}{c}2\left(4y+5\right)-3y=5\\ 8y+10-3y=5\\ 5y=-5\\ y=-1\end{array}$

Therefore, the value of variable $y$ is $-1$.

Substitute $y=-1$ in equation (1).

$\begin{array}{c}2x-3\left(-1\right)=5\\ 2x+3=5\\ 2x=2\\ x=1\end{array}$

Therefore the value of variable $x$ is $1$.

Hence, the solution to the equations is $x=1$ and $y=-1$.

As seen from the example above the solution is an ordered pair having a value of $\left(1,-1\right)$ representing both variables $\left(x,y\right)$.