Chapter 6.1, Problem 13P

Reasoning If the ordered pair $(3,-2)$ satisfies one of the two linear equations in a system, how can you tell whether the point satisfies the other equation of the system? Explain.

To determine

To explain the given situation

Explanation of Solution

To understand the situation an example is assumed.

Let the system of equation are,

$\begin{array}{l}x+y=1-----(i)\\ x-y=5-----(ii)\end{array}$

Now, adding equation ( ii ) and equation ( i ),

$\begin{array}{l}\Rightarrow x+y+x-y=1+5\\ \Rightarrow 2x=6\\ \Rightarrow x=3\end{array}$

Substituting the value of $x$ in equation ( ii ),

  $\begin{array}{l}\Rightarrow 3-y=5\\ \Rightarrow y=-2\end{array}$

So, from the above solution the obtained ordered pair is (3,-2).

Since, the ordered pair is obtained by solving both the equation, hence (3,-2) should satisfy both the equations simultaneously because it is the unique solution to both the equation.

Let’s verify the above statement by putting (3,-2) in both the equations:

For Equation ( i ),

$\begin{array}{l}\Rightarrow x+y=1\\ \Rightarrow 3+(-2)=1\\ \Rightarrow 1=1\end{array}$

For Equation ( ii ),

$\begin{array}{l}\Rightarrow x-y=5\\ \Rightarrow 3-(-2)=5\\ \Rightarrow 5=5\end{array}$

Hence the above statement is verified.

Conclusion:

Hence, if an ordered pair satisfies one equation of the system of equations, then it will also satisfy the other one because the ordered pair is obtained by solving both the equations which makes it unique or rather common to both of them.