Question 169709
 Explain how to apply elimination in solving a system of equations.
a. Explain how to apply substitution in solving a system of equations.
b. Demonstrate each technique in solving the system

{{{system(3x + 9y = 12,5x - 4y = 3)}}}

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By elimination:

First eliminate {{{x}}} and find {{{y}}}.
Multiply the first equation through by {{{-5}}} 
and the second equation through by {{{3}}}:

{{{system(-15x -45y = -60,15x - 12y = 9)}}}

Now add vertically term by term:

{{{-57y=-51}}}

Divide both sides by {{{-57}}}

{{{y=(-51)/(-57)}}}

{{{y=17/19}}}

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{{{system(3x + 9y = 12,5x - 4y = 3)}}}

Next eliminate {{{y}}} and find {{{x}}}.
Multiply the first equation through by 
{{{4}}} and the second equation through 
by {{{9}}}:

{{{system(12x +36y = 48,45x - 36y = 27)}}}

Now add vertically term by term:

{{{57x=75}}}

Divide both sides by {{{57}}}

{{{x=75/57}}}

{{{x=25/19}}}

Solution: {{{matrix(1,11, "(", x, ",", y, ")", "=", "(", 25/19, ",", 17/19,")" )}}}  

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Now do the same problem from scratch using
substitution:

{{{system(3x + 9y = 12,5x - 4y = 3)}}}

Solve the first equation for {{{x}}}:

{{{3x + 9y = 12}}}

Add {{{-9x}}} to both sides:

{{{3x=12-9y}}}

Divide through every term by 3:

{{{(3x)/3=12/3-(9y)/3}}}

Simplify:

{{{x=4-3y}}}

Substitute {{{(4-3y)}}} for {{{x}}}
in the second equation:

{{{5x - 4y = 3)}}}
{{{5(4-3y) - 4y = 3)}}}
Distribute:
{{{20-15y-4y=3}}}
{{{20-19y=3}}}
Add {{{-20}}} to both sides:
{{{-19y=3-20}}}
{{{-19y=-17}}}
Divide both sides by {{{-19}}}
{{{(-19y)/(-19)=(-17)/(-19)}}}
{{{y=17/19}}}

Now substitute {{{(17/19)}}} for {{{y}}}
in

{{{x=4-3y}}}
{{{x=4-3(17/19)}}}
Write the {{{4}}} as {{{4/1}}} and 
write the {{{3}}} as {{{3/1}}}

{{{x=4/1-(3/1)(17/19)}}}
{{{x=4/1-51/19}}}

The LCD is {{{19}}}
Multiply the {{{4/1}}} by {{{19/19}}}

{{{x=(4/1)(19/19)-51/19}}}
{{{x=76/19-51/19}}}
{{{x=25/19}}}

Solution: {{{matrix(1,11, "(", x, ",", y, ")", "=", "(", 25/19, ",", 17/19,")" )}}} 

Edwin</pre>