a) For the first system of equations use elimination because you have a +y+y in the first equation and a -y−y in the second equation. Add the two equations directly:
" "x + y = 4 x+y=4
"+ "2x - y = -2+ 2x−y=−2
" " 3x = 2; " "x = 2/3 3x=2; x=23
Substitute xx back into one of the equations to find yy:
2/3 + y = 4/1*3/323+y=41⋅33
y = 12/3 - 2/3 = 10/3y=123−23=103
Solution a) (2/3, 10/3)(23,103)
To check to see if this is correct, put this point into the second equation:
2/1*2/3 - 10/3 = -6/3 = -221⋅23−103=−63=−2
b) Rearrange the first equation to get b = 2a + 7b=2a+7
Substitute this equation into the second equation:
-5a -3(2a + 7) = 15−5a−3(2a+7)=15
Distribute: -5a -6a -21 = 15−5a−6a−21=15
Add like-terms: -11a -21 +21 = 15 + 21−11a−21+21=15+21
Simplify: -11a = 36−11a=36
Divide by -11: a = -36/11 = -3 3/11−11:a=−3611=−3311
Substitute this value into b = 2a + 7b=2a+7 to find bb:
b = 2*-36/11 + 7/1 * 11/11b=2⋅−3611+71⋅1111
b = -72/11 + 77/11 = 5/11b=−7211+7711=511
Check the answer by inputting it into the second equation:
-5/1*-36/11 - 3/1*5/11 = 180/11 - 15/11 = 165/11 = 15−51⋅−3611−31⋅511=18011−1511=16511=15
