Q9. - Algebra 2

Question

9. State for which of the following ordered pairs $\left(\text{x},\text{y}\right)$ is $\text{x}+\text{y}>3$ and $\text{x}-\text{y}<-2$.

 

A. $\left(0,3\right)$                                  B. $\left(3,4\right)$

 

C. $\left(5,3\right)$                                 D. $\left(2,5\right)$

Step-by-Step Solution

Answer

The answer is the ordered pair $\left(2,5\right)$ and the correct option is D.

Step 1 - State the definition of a variable.

A variable is a term that is used to represent an unknown value.

Step 2 - Solve the equations.

The equations are:

$x+y>3$                                                                                                            …… (1)

$x-y<-2$                                                                                                        …… (2)

The given options are:

$\left(0,3\right)$, $\left(3,4\right)$, $\left(5,3\right)$ and $\left(2,5\right)$.

The values x and y has to satify both the equation.

So, substitute each pair of x and y in the equations to check which pair satifies both the equations.

For $\left(0,3\right)$, $x=0$ and $y=3$.

Substitute x and y values in equation 1:

$\begin{array}{c}x+y>3\\ 0+3>3\\ 3>3\end{array}$

This not possible as it is incorrect.

Similarly, Substitute x and y values in equation 2:

$\begin{array}{c}x-y<-2\\ 0-3<-2\\ -3<-2\end{array}$

This is correct.

Here $\left(0,3\right)$ is satisfying equation 2 and not equation 1. As both the equations has to be satisfied, so $\left(0,3\right)$ is not the correct choice for $\left(x,y\right)$.

Step 3 - Check for the point 3 , 4 .

For $\left(3,4\right)$, $x=3$ and $y=4$.

Substitute x and y values in equation 1:

$\begin{array}{c}x+y>3\\ 3+4>3\\ 7>3\end{array}$

This is correct.

Similarly, Substitute x and y values in equation 2:

$\begin{array}{c}x-y<-2\\ 3-4<-2\\ -1<-2\end{array}$

This is not correct as $-1>-2$.

Here $\left(3,4\right)$ is satisfying equation 1 and not equation 2. As both the equations has to be satisfied, so $\left(3,4\right)$ is not the correct choice for $\left(x,y\right)$.

Step 4 - Check for the point 5 , 3 .

For $\left(5,3\right)$, $x=5$ and $y=3$.

Substitute x and y values in equation 1:

$\begin{array}{c}x+y>3\\ 5+3>3\\ 8>3\end{array}$

This is correct.

Similarly, Substitute x and y values in equation 2:

$\begin{array}{c}x-y<-2\\ 5-3<-2\\ 2<-2\end{array}$

This is not correct as $2>-2$.

Here $\left(5,3\right)$ is satisfying equation 1 and not equation 2. As both the equations has to be satisfied, so $\left(5,3\right)$ is not the correct choice for $\left(x,y\right)$.

Step 5 - Check for the point 2 , 5 .

For $\left(2,5\right)$, $x=2$ and $y=5$.

Substitute x and y values in equation 1:

$\begin{array}{c}x+y>3\\ 2+5>3\\ 7>3\end{array}$

This is correct.

Similarly, Substitute x and y values in equation 2:

$\begin{array}{c}x-y<-2\\ 2-5<-2\\ -3<-2\end{array}$

This is also correct.

Here $\left(2,5\right)$ is satisfying equation 1 and equation 2. As both the equations are satisfied, so $\left(2,5\right)$ is the correct choice for $\left(x,y\right)$.

Step 6 - State the conclusion.

The ordered pair $\left(2,5\right)$ is $x+y>3$ and $x-y<-2$. Therefore, option D is correct.