Question 188887
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Solving systems of linear equations.Solve by substitution 
2x+y=10, EQN 1
3x-4y=26, EQN 2


In EQN 1 we get, {{{y=10-2x}}}--> EQN 3, substitute in EQN 2


{{{3x-4(highlight(10-2x))=26}}}
{{{3x-40+8x=26}}}
{{{11x=26+40=66}}} ---->{{{cross(11)x/cross(11)=cross(66)6/cross(11)}}}
{{{highlight(x=6)}}}, substitute in EQN 3:


{{{y=10-2(6)=10-12=highlight(-2)}}}


Intersect at point <font color=blue>(6,-2)</font>
{{{drawing(300,300,-4,10,-8,5,graph(300,300,-4,10,-8,5,10-2x,(3x-26)/4),blue(circle(6,-2,.15)))}}} ---><font color=red>RED, EQN1</font>; <font color=green>GREEN, EQN 2</font>


y=2x+3, EQN 1
3x+2y=6, EQN 2


Substitute EQN 1 in EQN 2:
{{{3x+2highlight((2x+3))=6}}} 
{{{3x+4x+6=6}}}
{{{7x=6-6=0}}} ----> {{{highlight(x=0)}}}, subst. in EQN 1


{{{y=2(0)+3}}} ----> {{{highlight(y=3)}}}


Intersect at point <font color=blue>(0,3)</font>
{{{drawing(300,300,-4,10,-8,5,graph(300,300,-4,10,-8,5,2x+3,(6-3x)/2),blue(circle(0,3,.15)))}}} ---> <font color=red>RED, EQN 1</font>;<font color=green> GREEN, EQN 2</font>



Solving systems of linear equations. Solve by addition elimination 
3x-6y=5, EQN 1
2x-4y=8, EQN 2 


Multiply EQN 1 by -4, EQN 2 by 6, and then "add":
{{{3x-6y=5}}}x{{{-4}}} ----> {{{cross(-12x)+cross(24y)=-20}}}
{{{2x-4y=8}}}x{{{6}}} -----> {{{cross(12x)-cross(24y)=48}}}
*[Tex \ \ \ \ \ \ \ \ \ \ \ \ \ -------------------]
*[Tex \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0=28] no solution. These lines don't intersect:


We'll look for the intercepts:
In EQN 1, let fy=0
{{{3x-6(0)=5}}} ---> {{{cross(3)x/cross(3)=5/3}}}
{{{red(x=5/3)}}}
let fx=0
{{{3(0)-6y=5}}} ---> {{{cross(-6)y/cross(-6)=5/-6}}}
{{{red(y=-5/6)}}}


In EQN 2, let Fy=0
{{{2x-4(0)=8}}} ---> {{{cross(2)x/cross(2)=cross(8)4/cross(2)}}}
{{{red(x=4)}}}
Let fx=0
{{{2(0)-4y=8}}} ---> {{{cross(-4)y/cross(-4)=cross(8)2/cross(-4)}}}
{{{red(y=-2)}}}

{{{drawing(300,300,-4,10,-8,5,graph(300,300,-4,10,-8,5,(3x-5)/6,(2x-8)/4),blue(circle(5/3,0,.15)),blue(circle(0,-5/6,.15)),blue(circle(4,0,.15)),blue(circle(0,-2,.15)))}}} ---> <font color=red>RED, </font>EQN 1; <font color=green>GREEN, EQN 2</font>
Since they line on the same plane they are parallel



Thank you,
Jojo</font>