Question 207312
Graph the 2 lines below; and from the graph, read off the point of intersection.
{{{system(y=2x+3,y=x+5)}}}
<pre><font size = 4 color = "indigo"><b>
Get 3 points on the first line:

Arbitrarily select, say, x = 0 and substitute 0 for x in the
first equation:

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

So the first point on the first line is (x,y) = (0,3)

---

Arbitrarily select, say, x = -2 and substitute -2 for x in the
first equation:

{{{y=2x+3}}}
{{{y=2(-2)+3}}}
{{{y=-4+3}}}
{{{y=-1}}}

So the second point on the first line is (x,y) = (-1,-1)

---

Arbitrarily select, say, x = 3 and substitute 3 for x in the
first equation:

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

So the third point of the first line is (x,y) = (3,9)

Plot those three points:

{{{drawing(400,400,-5,15,-5,15, graph(400,400,-5,15,-5,15),

line(0+.2,3,0-.2,3), line(0,3+.2,0,3-.2),
line(-1+.2,1,-1-.2,1), line(-1,1+.2,-1,1-.2),
line(3+.2,9,3-.2,9), line(3,9+.2,3,9-.2)

)}}}

Get a ruler and draw a straight line through them:

{{{drawing(400,400,-5,15,-5,15, graph(400,400,-5,15,-5,15,2x+3),

line(0+.2,3,0-.2,3), line(0,3+.2,0,3-.2),
line(-1+.2,1,-1-.2,1), line(-1,1+.2,-1,1-.2),
line(3+.2,9,3-.2,9), line(3,9+.2,3,9-.2)

)}}}

---------

Get 3 points on the second line:

Arbitrarily select, say, x = 1 and substitute 1 for x in the
second equation:

{{{y=x+5}}}
{{{y=(1)+5}}}
{{{y=1+5}}}
{{{y=6}}}

So the first point on the second line is (x,y) = (1,6)

---

Arbitrarily select, say, x = -1 and substitute -1 for x in the
second equation:

{{{y=x+5}}}
{{{y=(-1)+5}}}
{{{y=-1+5}}}
{{{y=4}}}

So the second point on the second line is (x,y) = (-1,4)

---

Arbitrarily select, say, x = 4 and substitute 4 for x in the
second equation:

{{{y=x+5}}}
{{{y=(4)+5}}}
{{{y=4+5}}}
{{{y=9}}}

So the third point of the second line is (x,y) = (4,9)

Plot those three points:

{{{drawing(400,400,-5,15,-5,15, graph(400,400,-5,15,-5,15,2x+3),

line(0+.2,3,0-.2,3), line(0,3+.2,0,3-.2),
line(-1+.2,1,-1-.2,1), line(-1,1+.2,-1,1-.2),
line(3+.2,9,3-.2,9), line(3,9+.2,3,9-.2),

line(4+.2,9,4-.2,9), line(4,9+.2,4,9-.2),
line(-1+.2,4,-1-.2,4), line(-1,4+.2,-1,4-.2),
line(1+.2,6,1-.2,6), line(1,6+.2,1,6-.2)


)}}}

Take your ruler and draw a straight line through
those three points:

{{{drawing(400,400,-5,15,-5,15, graph(400,400,-5,15,-5,15,2x+3,x+5),

line(0+.2,3,0-.2,3), line(0,3+.2,0,3-.2),
line(-1+.2,1,-1-.2,1), line(-1,1+.2,-1,1-.2),
line(3+.2,9,3-.2,9), line(3,9+.2,3,9-.2),

line(4+.2,9,4-.2,9), line(4,9+.2,4,9-.2),
line(-1+.2,4,-1-.2,4), line(-1,4+.2,-1,4-.2),
line(1+.2,6,1-.2,6), line(1,6+.2,1,6-.2)


)}}}

Now take your ruler and draw 2 lines from the point
where the two lines cross, one directly to the x-axis
and one directly to the y-axis:

{{{drawing(400,400,-5,15,-5,15, graph(400,400,-5,15,-5,15,2x+3,x+5),

line(0+.2,3,0-.2,3), line(0,3+.2,0,3-.2),
line(-1+.2,1,-1-.2,1), line(-1,1+.2,-1,1-.2),
line(3+.2,9,3-.2,9), line(3,9+.2,3,9-.2),

line(4+.2,9,4-.2,9), line(4,9+.2,4,9-.2),
line(-1+.2,4,-1-.2,4), line(-1,4+.2,-1,4-.2),
line(1+.2,6,1-.2,6), line(1,6+.2,1,6-.2),
line(2,7,0,7), line(2,7,2,0)

)}}}

Notice that these last two lines hit the x-axis
at 2 and the y-axis at 7.  So the solution is

{{{x=2}}}, {{{y=7}}}, sometimes written

(x,y) = (2,7)

To check we substitute {{{x=2}}} and {{{y=7}}}
in both equations:

{{{y=2x+3}}}
{{{7=2(2)+3}}}
{{{7=4+3}}}
{{{7=7}}}

It checks the first equation.  Now let's see if
it checks the second equation.

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

So it checks both equations, so it is correct.

Edwin</pre>