# Solve for z (2z-6)/5+16/10=(4z+4)/10

2z-65+1610=4z+410
Move 4z+410 to the left side of the equation by subtracting it from both sides.
2z-65+1610-4z+410=0
Simplify 2z-65+1610-4z+410.
Find the common denominator.
Multiply 2z-65 by 22.
2z-65⋅22+1610-4z+410=0
Combine.
(2z-6)⋅25⋅2+1610-4z+410=0
Reorder the factors of 5⋅2.
(2z-6)⋅22⋅5+1610-4z+410=0
Multiply 2 by 5.
(2z-6)⋅210+1610-4z+410=0
(2z-6)⋅210+1610-4z+410=0
Combine the numerators over the common denominator.
(2z-6)⋅2+16-(4z+4)10=0
Cancel the common factor of (2z-6)⋅2+16-(4z+4) and 10.
Factor -1 out of (2z-6)⋅2.
-((-2z+6)⋅2)+16-(4z+4)10=0
Rewrite 16 as -1(-16).
-((-2z+6)⋅2)-1(-16)-(4z+4)10=0
Factor -1 out of -((-2z+6)⋅2)-1(-16).
-((-2z+6)⋅2-16)-(4z+4)10=0
Factor -1 out of -((-2z+6)⋅2-16)-(4z+4).
-((-2z+6)⋅2-16+4z+4)10=0
Rewrite -((-2z+6)⋅2-16+4z+4) as -1((-2z+6)⋅2-16+4z+4).
-1((-2z+6)⋅2-16+4z+4)10=0
Factor 2 out of -1((-2z+6)⋅2-16+4z+4).
2(-1((-z+3)⋅2-8+2z+2))10=0
Cancel the common factors.
Factor 2 out of 10.
2(-1((-z+3)⋅2-8+2z+2))2(5)=0
Cancel the common factor.
2(-1((-z+3)⋅2-8+2z+2))2⋅5=0
Rewrite the expression.
-1((-z+3)⋅2-8+2z+2)5=0
-1((-z+3)⋅2-8+2z+2)5=0
-1((-z+3)⋅2-8+2z+2)5=0
Simplify the numerator.
-1((-z+3)⋅2+2z-6)5=0
Factor 2 out of (-z+3)⋅2+2z-6.
Factor 2 out of (-z+3)⋅2.
-1(2⋅(-z+3)+2z-6)5=0
Factor 2 out of 2z.
-1(2⋅(-z+3)+2(z)-6)5=0
Factor 2 out of -6.
-1(2⋅(-z+3)+2z+2⋅-3)5=0
Factor 2 out of 2⋅(-z+3)+2z.
-1(2⋅(-z+3+z)+2⋅-3)5=0
Factor 2 out of 2⋅(-z+3+z)+2⋅-3.
-1(2(-z+3+z-3))5=0
-1(2(-z+3+z-3))5=0
-1(2(0+3-3))5=0
-1(2(3-3))5=0
Subtract 3 from 3.
-1⋅2⋅05=0
Combine exponents.
Multiply -1 by 2.
-2⋅05=0
Multiply -2 by 0.
05=0
05=0
05=0
Divide 0 by 5.
0=0
0=0
Since 0=0, the equation will always be true for any value of z.
All real numbers
The result can be shown in multiple forms.
All real numbers
Interval Notation:
(-∞,∞)
Solve for z (2z-6)/5+16/10=(4z+4)/10

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