# Solve for z 1/2*(z+1)+1/4*(z+3)=3-1/3*(z+2)

12⋅(z+1)+14⋅(z+3)=3-13⋅(z+2)
Simplify 12⋅(z+1)+14⋅(z+3).
Simplify each term.
Apply the distributive property.
12z+12⋅1+14⋅(z+3)=3-13⋅(z+2)
Combine 12 and z.
z2+12⋅1+14⋅(z+3)=3-13⋅(z+2)
Multiply 12 by 1.
z2+12+14⋅(z+3)=3-13⋅(z+2)
Apply the distributive property.
z2+12+14z+14⋅3=3-13⋅(z+2)
Combine 14 and z.
z2+12+z4+14⋅3=3-13⋅(z+2)
Combine 14 and 3.
z2+12+z4+34=3-13⋅(z+2)
z2+12+z4+34=3-13⋅(z+2)
To write z2 as a fraction with a common denominator, multiply by 22.
z2⋅22+z4+12+34=3-13⋅(z+2)
Write each expression with a common denominator of 4, by multiplying each by an appropriate factor of 1.
Multiply z2 and 22.
z⋅22⋅2+z4+12+34=3-13⋅(z+2)
Multiply 2 by 2.
z⋅24+z4+12+34=3-13⋅(z+2)
z⋅24+z4+12+34=3-13⋅(z+2)
Combine the numerators over the common denominator.
z⋅2+z4+12+34=3-13⋅(z+2)
Reorder z and 2.
2⋅z+z4+12+34=3-13⋅(z+2)
3⋅z4+12+34=3-13⋅(z+2)
3⋅z4+12+34=3-13⋅(z+2)
To write 12 as a fraction with a common denominator, multiply by 22.
3z4+12⋅22+34=3-13⋅(z+2)
Write each expression with a common denominator of 4, by multiplying each by an appropriate factor of 1.
Multiply 12 and 22.
3z4+22⋅2+34=3-13⋅(z+2)
Multiply 2 by 2.
3z4+24+34=3-13⋅(z+2)
3z4+24+34=3-13⋅(z+2)
Combine the numerators over the common denominator.
3z4+2+34=3-13⋅(z+2)
3z4+54=3-13⋅(z+2)
3z4+54=3-13⋅(z+2)
Simplify 3-13⋅(z+2).
Simplify each term.
Apply the distributive property.
3z4+54=3-13z-13⋅2
Combine z and 13.
3z4+54=3-z3-13⋅2
Multiply -13⋅2.
Multiply 2 by -1.
3z4+54=3-z3-2(13)
Combine -2 and 13.
3z4+54=3-z3+-23
3z4+54=3-z3+-23
Move the negative in front of the fraction.
3z4+54=3-z3-23
3z4+54=3-z3-23
To write 3 as a fraction with a common denominator, multiply by 33.
3z4+54=-z3+3⋅33-23
Combine 3 and 33.
3z4+54=-z3+3⋅33-23
Combine the numerators over the common denominator.
3z4+54=-z3+3⋅3-23
Simplify the numerator.
Multiply 3 by 3.
3z4+54=-z3+9-23
Subtract 2 from 9.
3z4+54=-z3+73
3z4+54=-z3+73
3z4+54=-z3+73
Move all terms containing z to the left side of the equation.
Add z3 to both sides of the equation.
3z4+54+z3=73
To write 3z4 as a fraction with a common denominator, multiply by 33.
3z4⋅33+z3+54=73
To write z3 as a fraction with a common denominator, multiply by 44.
3z4⋅33+z3⋅44+54=73
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
Multiply 3z4 and 33.
3z⋅34⋅3+z3⋅44+54=73
Multiply 4 by 3.
3z⋅312+z3⋅44+54=73
Multiply z3 and 44.
3z⋅312+z⋅43⋅4+54=73
Multiply 3 by 4.
3z⋅312+z⋅412+54=73
3z⋅312+z⋅412+54=73
Combine the numerators over the common denominator.
3z⋅3+z⋅412+54=73
Simplify each term.
Simplify the numerator.
Factor z out of 3z⋅3+z⋅4.
Factor z out of 3z⋅3.
z(3⋅3)+z⋅412+54=73
Factor z out of z⋅4.
z(3⋅3)+z(4)12+54=73
Factor z out of z(3⋅3)+z(4).
z(3⋅3+4)12+54=73
z(3⋅3+4)12+54=73
Multiply 3 by 3.
z(9+4)12+54=73
z⋅1312+54=73
z⋅1312+54=73
Move 13 to the left of z.
13z12+54=73
13z12+54=73
13z12+54=73
Move all terms not containing z to the right side of the equation.
Subtract 54 from both sides of the equation.
13z12=73-54
To write 73 as a fraction with a common denominator, multiply by 44.
13z12=73⋅44-54
To write -54 as a fraction with a common denominator, multiply by 33.
13z12=73⋅44-54⋅33
Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.
Multiply 73 and 44.
13z12=7⋅43⋅4-54⋅33
Multiply 3 by 4.
13z12=7⋅412-54⋅33
Multiply 54 and 33.
13z12=7⋅412-5⋅34⋅3
Multiply 4 by 3.
13z12=7⋅412-5⋅312
13z12=7⋅412-5⋅312
Combine the numerators over the common denominator.
13z12=7⋅4-5⋅312
Simplify the numerator.
Multiply 7 by 4.
13z12=28-5⋅312
Multiply -5 by 3.
13z12=28-1512
Subtract 15 from 28.
13z12=1312
13z12=1312
13z12=1312
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
13z=13
Divide each term by 13 and simplify.
Divide each term in 13z=13 by 13.
13z13=1313
Cancel the common factor of 13.
Cancel the common factor.
13z13=1313
Divide z by 1.
z=1313
z=1313
Divide 13 by 13.
z=1
z=1
Solve for z 1/2*(z+1)+1/4*(z+3)=3-1/3*(z+2)

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