Question 12569
First you need to get both equations into Slope Intercept Form ( y = mx + b )
<BR>
X + 2Y = 9 Subract X From Both sides
2Y = -X + 9 Divide Both Sides by 2
Y = -(1/2)X + 9 ( PURPLE LINE )
<BR>
-X + 6Y = -1 Add X to both sides
6Y = X - 1 Divide both sides by 6
y = (1/6)X - (1/6)(GREEN LINE )
<P>

Now we have to graph these lines, see the graph below
{{{graph(300,200, -5, 15, -5, 15, -(1/2)x + 9, (1/6)x - (1/6))}}}

<P> Now you have to estimate the intersection point... when you graph this on a calculator or a piece of graph paper you may get a more definitive answer, but for this exercise I am going to say those two lines intersect at (14,2)
<P>

Now do check them algebraically, you must take (14,2) and substitute 14 for X and 2 for Y, your goal is to get an equation that makes sense... 
for example 2 = 2 makes sense because that is true, while 3 =2 does not make sense.
<P>
If you get an equation that makes sense, then the point we chose is the real intersection point, if the equation does not make sense, then we chose the wrong point.
<P>
Equation 1: Y = -(1/2)X + 9 Substitute ( 14, 2 ) into the equation
2 = -(1/2)*14 + 9 Simplfy
2 = -7 + 9 Add
2 = 2 ( check, this makes sense, so (14, 2) lies on the purple line )
<P>
You always need to check both equations just to make sure!
<P>
Equation 2: y = (1/6)X - (1/6)Substitue ( 14, 2 ) into the equation
2 = (1/6)*14 - (1/6) Multiply
2 = (14/6) - (1/6) Subtract
2 = 13/6 or 2(1/6)

Since this equation does not make sense, that means that ( 14, 2 ) is not the intersection, although it is close, it is not right on, so you would have you use your graph paper to make a better estimate.