Problem:
Solve each of the following systems of equations. Find all solutions.
(a)\(\begin{align*} x+y&=-1,\\ 3x &= 4-3y. \end{align*} \)
(b)\(\begin{align*} 3x - 4y + 2&=0,\\ 10 - 10y& = 10y - 15x. \end{align*}\)
+150 the first one has no solutions, I will show you how, by step
1) we rewrite the equations : x+y=−1;3x=−3y+4
2) in the x+y=-1, we solve for x
x+y=−1
x+y+−y=−1+−y(Add -y to both sides)
x=−y−1
3)Substitute−y−1for x in 3x=−3y+4:
3x=−3y+4
3(−y−1)=−3y+4
−3y−3=−3y+4(Simplify both sides of the equation)
−3y−3+3y=−3y+4+3y(Add 3y to both sides)
−3=4
−3+3=4+3(Add 3 to both sides)
0=7
and so there are no solution
the second one:
1)Solve3x−4y+2=0 for x:
3x−4y+2=0
3x−4y+2+4y=0+4y(Add 4y to both sides)
3x+2=4y
3x+2+−2=4y+−2(Add -2 to both sides)
3x=4y−2
3x/3 = (4y-2)/3
(Divide both sides by 3)
x=4/3y + -2/3
Problem 2: Substitute 4/3y + -2/3 for x in -10y+10=15x-10
−10y+10=15x−10
−10y+10=15(4/3y+-2/3)-10
−10y+10=20y−20(Simplify both sides of the equation)
−10y+10+−20y=20y−20+−20y(Add -20y to both sides)
−30y+10=−20
−30y+10+−10=−20+−10(Add -10 to both sides)
−30y=−30
-30y/-30 = -30/-30, (Divide both sides by -30)
and Y=1
Now we solve for X
x=4/3y+-2/3
x=4/3(1) +-2/3
x=2/3
so x=2/3 and y=1
