Question 174554
{{{system( 7X + 3Y = 14 ,  
5X + 9Y = 10)}}}
<pre><font size = 4 color = "indigo"><b>
Notice the 3Y in the first equation and the 9Y
in the second equation.  If you multiply the
first equation by -3, the 3Y will become -9Y and
then if you add the two equations, the Y's will
cancel.  So let's multiply the first equation by
-3:

{{{system( (-3)7X + (-3)3Y = (-3)14 ,  
5X + 9Y = 10)}}}

Simplifying:

{{{system(-21X - 9Y = -42 ,  
5X + 9Y = 10)}}}

Now we add corresponding terms.
-21X added to 5X gives -16X,
-9Y added to 9Y gives 0,
-42 added to 10 gives -32, so
by adding vertically term by 
term we have the equation:

{{{-16X + 0 = -32}}}
{{{-16X = -32}}}
Divide both sides by -16:

{{{X = 2}}}

Now pick either one of the original
equations and substitute 2 for X:

{{{matrix(8,1,
7X + 3Y = 14,
7(2) + 3Y = 14,
14 + 3Y = 14,
"Subtract_14_from_both_sides:",
3Y = 0,
"Divide_both_sides_by_3",
Y = 0/3,
Y = 0 )}}}

-----

{{{system(4X + Y = 16 ,    2X + 3Y = 24)}}}

Notice the 4X in the first equation and the 2X
in the second equation.  If you multiply the
second equation by -2, the 2X will become -4X and
then if you add the two equations, the X's will
cancel.  So let's multiply the second equation by
-2:

{{{system( 4X + Y = 16 ,  
(-2)2X + (-2)3Y = (-2)24)}}}

Simplifying:

{{{system(4X + Y = 16 ,  
-4X - 6Y = -48)}}}

Now we add corresponding terms.
4X added to -4X gives 0,
Y added to -6Y gives -5Y,
16 added to -48 gives -32, so
by adding vertically term by 
term we have the equation:

{{{0 - 5Y = -32}}}
{{{-5Y = -32}}}
Divide both sides by -5:

{{{Y = (-32)/(-5)}}}
{{{Y = 32/5}}}

We could substitute this fraction in
for Y as we did in the previous problem.
However, unlike the previous problem,
{{{32/5}}} is difficult to subtitute.
So instead we will start over:

{{{system(4X + Y = 16 ,    2X + 3Y = 24)}}}

and this time we'll eliminate Y:

Notice the Y in the first equation and the 3Y
in the second equation.  If you multiply the
first equation by -3, the Y will become -3Y and
then if you add the two equations, the Y's will
cancel.  So let's multiply the first equation by
-3:

{{{system( (-3)4X + (-3)Y = (-3)16 ,  
2X + 3Y = 24)}}}

Simplifying:

{{{system(-12X - 3Y = -48 ,  
2X + 3Y = 24)}}}

Now we add corresponding terms.
-12X added to 2X gives -10X,
-3Y added to 3Y gives 0,
-48 added to 24 gives -24, so
by adding vertically term by 
term we have the equation:

{{{-10X + 0 = -24}}}
{{{-10X = -24}}}
Divide both sides by -16:

{{{X = (-24)/(-10)}}} 

{{{X = 12/5}}}

---

{{{system(12X + Y = 25 ,  8X - 2Y = 14)}}}

You do this one by yourself.  It's just like the first one.

Answer: {{{X=2}}}, {{{Y=1}}}

Edwin</pre>