Question 295611
Find the ordered pair that is a solution to the graphs

{{{system(y=2x+3,y= x-2)}}}

a) (3,7)
<pre>
We substitute the first number of the ordered pair, which is 3,
in place of x and the second number, which is 7, in place of y 
in both equations and simplify.  

Substituting 3 for x, and 7 for y in the first one:

{{{y=2x+3}}}
{{{7=2(3)+3}}}
{{{7=6+3}}}
{{{7=9}}}

That's a false equation because 7 does not equal 9.
So the ordered paie (3.7) is not a solution to {{{y=2x+3}}}.

Substituting 3 for x, and 7 for y in the second one:

{{{y=x-2}}}
{{{7=3-2}}}
{{{7=1}}}
{{{7=1}}}

That's a false equation because 7 does not equal 1.
So the ordered paie (3.7) is not a solution to the
equation {{{y=x-2}}}.
 
So the ordered pair (3,7) is not a solution to
{{{system(y=2x+3,y= x-2)}}} because it is not a solution
to either one of the equations, so it cannot be a solution
to both.


If what we had ended up with in BOTH cases had been true equations,
then the ordered pair would have been a solution.  But in this 
case they were both false, so the ordered pair (3,7) is not a 
solution to {{{system(y=2x+3,y= x-2)}}}. 

</pre>
b) (0,7)
<pre>
We substitute the first number of the ordered pair, which is 0,
in place of x and the second number, which is 7, in place of y 
in both equations and simplify.  

Substituting 0 for x, and 7 for y in the first one:

{{{y=2x+3}}}
{{{7=2(0)+3}}}
{{{7=0+3}}}
{{{7=3}}}

That's a false equation because 7 does not equal 3.
So the ordered paie (0.7) is not a solution to {{{y=2x+3}}}.

Substituting 0 for x, and 7 for y in the second one:

{{{7=x-2}}}
{{{7=0-2}}}
{{{7=-2}}}
{{{7=1}}}
{{{7=1}}}

That's a false equation because 7 does not equal 1.
So the ordered paie (3.7) is not a solution to the
equation {{{y=x-2}}}.
 
So the ordered pair (3,7) is not a solution to
{{{system(y=2x+3,y= x-2)}}} because it is not a solution
to either one of the equations, so it cannot be a solution
to both.

If what we had ended up with in BOTH cases had been true equations,
then the ordered pair would have been a solution.  But in this 
case they were both false, so the ordered pair (0,7) is not a 
solution to {{{system(y=2x+3,y= x-2)}}}. 

</pre>
c) (7,2)
<pre>
We substitute the first number of the ordered pair, which is 7,
in place of x and the second number, which is 2, in place of y 
in both equations and simplify.  

Substituting 7 for x, and 2 for y in the first one:

{{{y=2x+3}}}
{{{2=2(7)+3}}}
{{{2=14+3}}}
{{{2=17}}}

That's a false equation because 2 does not equal 17.
So the ordered paie (7.2) is not a solution to {{{y=2x+3}}}.

Substituting 7 for x, and 2 for y in the second one:

{{{y=x-2}}}
{{{2=7-2}}}
{{{2=5}}}

That's a false equation because 2 does not equal 5.
So the ordered paie (7.2) is not a solution to the
equation {{{y=x-2}}}.
 
So the ordered pair (7,2) is not a solution to
{{{system(y=2x+3,y= x-2)}}} because it is not a solution
to either one of the equations, so it cannot be a solution
to both.

If what we had ended up with in BOTH cases had been true equations,
then the ordered pair would have been a solution.  But in this 
case they were both false, so the ordered pair (7,2) is not a 
solution to {{{system(y=2x+3,y= x-2)}}}.

</pre>
d) (-5, -7)
<pre>
We substitute the first number of the ordered pair, which is -5,
in place of x and the second number, which is -7, in place of y 
in both equations and simplify.  

Substituting -5 for x, and -7 for y in the first one:

{{{y=2x+3}}}
{{{-7=2(-5)+3}}}
{{{-7=-10+3}}}
{{{-7=-7}}}

That's a true equation because -7 does equal -7.
So the ordered paie (-5.-7) is a solution to {{{y=2x+3}}}.

Substituting -5 for x, and -7 for y in the second one:

{{{7=x-2}}}
{{{-7=-5-2}}}
{{{-7=-7}}}

That's a true equation because -7 does equal -7.
So the ordered paie (3.7) is a solution to the
equation {{{y=x-2}}}.
 
So the ordered pair (3,7) is a solution to
{{{system(y=2x+3,y= x-2)}}} because it is a solution
to BOTH of the equations.

So the correct choice is (d).

The significance of this is that if we graph both lines,
by plotting points, we get this.

{{{drawing(400,400,-10,10,-10,10, 
graph(400,400,-10,10,-10,10,2x+3),
graph(400,400,-10,10,-10,10,x-2)
)}}} 

We see that the point that is on both lines is the point
whose set of coordinates is the ordered pair (-5,-7), the
point where the two lines cross. That's because if you draw
a green vertical line and a green horizontal line through 
the point where the two red lines cross, like this:

{{{drawing(400,400,-10,10,-10,10, 
graph(400,400,-10,10,-10,10,2x+3),
graph(400,400,-10,10,-10,10,x-2),
green(line(-5,11,-5,-11)), green(line(11,-7,-11,-7)))
}}} 

that the vertical line crosses the x-axis at -5 and the
horizontal line crosses the y-axis at -7. 

Edwin</pre>