Hello!
I'll solve the first two systems.
1)
[tex] \left \{ {{5x+2y=21} \atop {-x-y=-9}} \right. [/tex]
Multiply the second equation in the system by 2 to be able to cancel out y-terms:
[tex]-x-y=-9[/tex]
[tex]2(-x-y=-9)[/tex]
[tex]-2x-2y=-18[/tex]
Now, add the equations together:
[tex] \left \ {{5x+2y=21} \atop {-2x-2y=-18}} \right. [/tex]
_____________
[tex]3x-0=3[/tex]
[tex]3x=3[/tex]
[tex]x=1[/tex]
Plug 1 for x into one of the original equations to find y:
[tex]5x+2y=21[/tex]
[tex]5(1)+2y=21[/tex]
[tex]5+2y = 21[/tex]
[tex]2y=16[/tex]
[tex]y =8[/tex]
Check work by plugging your values for x and y back into both original equations:
[tex] \left \ {{5x+2y=21} \atop {-x-y=-9}} \right. [/tex]
[tex] \left \ {{5(1)+2(8)=21} \atop {-(1)-(8)=-9}} \right. [/tex]
[tex] \left \ {{5+16=21} \atop {-1-8=-9}} \right. [/tex]
Both are true; x = 1 and y = 8.
2)
[tex] \left \ {{3x+2y=13} \atop {3x+4y=1}} \right. [/tex]
For this system, to be able to cancel out x-terms, you can multiply either equation by -1; I'll use the bottom equation:
[tex]-1(3x+4y=1)[/tex]
[tex]-3x-4y=-1[/tex]
Now add the equations together:
[tex] \left \ {{3x+2y=-13} \atop {-3x-4y=-1}} \right. [/tex]
_____________
[tex]0-2y=-14[/tex]
[tex]-2y = -14[/tex]
[tex]y=7[/tex]
Plug 7 for y into either of the original equation to solve for x:
[tex]3x+2(7)=-13[/tex]
[tex]3x+14=-13[/tex]
[tex]3x = -27[/tex]
[tex]x=-9[/tex]
Plug -9 for x and 7 for y into each original equation to check work:
[tex] \left \ {{3(-9)+2(7)=-13} \atop {3(-9)+4(7)=1}} \right. [/tex]
[tex] \left \ {{-27+14=-13} \atop {-27+28=1}} \right. [/tex]
Both are correct; x = -9 and y = 7.
Apply these steps to the last two systems and you're good to go. (:
