Solving equations with grouping symbols?

Question

Solve each equation. Check your solution
1) 3(a-5)=18
2) 32=4(x+9)
3) 2(d+6)=3d-1
4) 6(n-3)=4(n+2.1)
5) 12-h=-h+3
6) 3(2g+4)=6(g+2)

Answer 1

simple. distribute the variable. for example:
1)3(a-5)=18
3*a - 3*5 = 18
combine like terms and solve.
3a-15=18
3a=33
a=11

Do the same for the rest.

Answer 2

1) Simplifying
3(a + -5) = 18

Reorder the terms:
3(-5 + a) = 18
(-5 * 3 + a * 3) = 18
(-15 + 3a) = 18

Solving
-15 + 3a = 18

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '15' to each side of the equation.
-15 + 15 + 3a = 18 + 15

Combine like terms: -15 + 15 = 0
0 + 3a = 18 + 15
3a = 18 + 15

Combine like terms: 18 + 15 = 33
3a = 33

Divide each side by '3'.
a = 11

Simplifying
a = 11

2) Simplifying
32 = 4(x + 9)

Reorder the terms:
32 = 4(9 + x)
32 = (9 * 4 + x * 4)
32 = (36 + 4x)

Solving
32 = 36 + 4x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-4x' to each side of the equation.
32 + -4x = 36 + 4x + -4x

Combine like terms: 4x + -4x = 0
32 + -4x = 36 + 0
32 + -4x = 36

Add '-32' to each side of the equation.
32 + -32 + -4x = 36 + -32

Combine like terms: 32 + -32 = 0
0 + -4x = 36 + -32
-4x = 36 + -32

Combine like terms: 36 + -32 = 4
-4x = 4

Divide each side by '-4'.
x = -1

Simplifying
x = -1

3) Simplifying
2(d + 6) = 3d + -1

Reorder the terms:
2(6 + d) = 3d + -1
(6 * 2 + d * 2) = 3d + -1
(12 + 2d) = 3d + -1

Reorder the terms:
12 + 2d = -1 + 3d

Solving
12 + 2d = -1 + 3d

Solving for variable 'd'.

Move all terms containing d to the left, all other terms to the right.

Add '-3d' to each side of the equation.
12 + 2d + -3d = -1 + 3d + -3d

Combine like terms: 2d + -3d = -1d
12 + -1d = -1 + 3d + -3d

Combine like terms: 3d + -3d = 0
12 + -1d = -1 + 0
12 + -1d = -1

Add '-12' to each side of the equation.
12 + -12 + -1d = -1 + -12

Combine like terms: 12 + -12 = 0
0 + -1d = -1 + -12
-1d = -1 + -12

Combine like terms: -1 + -12 = -13
-1d = -13

Divide each side by '-1'.
d = 13

Simplifying
d = 13

4) Simplifying
6(n + -3) = 4(n + 2.1)

Reorder the terms:
6(-3 + n) = 4(n + 2.1)
(-3 * 6 + n * 6) = 4(n + 2.1)
(-18 + 6n) = 4(n + 2.1)

Reorder the terms:
-18 + 6n = 4(2.1 + n)
-18 + 6n = (2.1 * 4 + n * 4)
-18 + 6n = (8.4 + 4n)

Solving
-18 + 6n = 8.4 + 4n

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-4n' to each side of the equation.
-18 + 6n + -4n = 8.4 + 4n + -4n

Combine like terms: 6n + -4n = 2n
-18 + 2n = 8.4 + 4n + -4n

Combine like terms: 4n + -4n = 0
-18 + 2n = 8.4 + 0
-18 + 2n = 8.4

Add '18' to each side of the equation.
-18 + 18 + 2n = 8.4 + 18

Combine like terms: -18 + 18 = 0
0 + 2n = 8.4 + 18
2n = 8.4 + 18

Combine like terms: 8.4 + 18 = 26.4
2n = 26.4

Divide each side by '2'.
n = 13.2

Simplifying
n = 13.2


Simplifying
12 + -1h = -1h + 3

Reorder the terms:
12 + -1h = 3 + -1h

Add 'h' to each side of the equation.
12 + -1h + h = 3 + -1h + h

Combine like terms: -1h + h = 0
12 + 0 = 3 + -1h + h
12 = 3 + -1h + h

Combine like terms: -1h + h = 0
12 = 3 + 0
12 = 3

Solving
12 = 3

This equation is invalid, the left and right sides are not equal, therefore there is no solution.


6) Simplifying
3(2g + 4) = 6(g + 2)

Reorder the terms:
3(4 + 2g) = 6(g + 2)
(4 * 3 + 2g * 3) = 6(g + 2)
(12 + 6g) = 6(g + 2)

Reorder the terms:
12 + 6g = 6(2 + g)
12 + 6g = (2 * 6 + g * 6)
12 + 6g = (12 + 6g)

Add '-12' to each side of the equation.
12 + -12 + 6g = 12 + -12 + 6g

Combine like terms: 12 + -12 = 0
0 + 6g = 12 + -12 + 6g
6g = 12 + -12 + 6g

Combine like terms: 12 + -12 = 0
6g = 0 + 6g
6g = 6g

Add '-6g' to each side of the equation.
6g + -6g = 6g + -6g

Combine like terms: 6g + -6g = 0
0 = 6g + -6g

Combine like terms: 6g + -6g = 0
0 = 0

Solving
0 = 0

This equation is an identity, all real numbers are solutions.

Answer 3

You must know this stuff. If you have a term in ( ) preceeded by a constant, you have to "multiply out" the term FIRST. Then you group "like terms" and solve. Three steps.

For example, problem 6.
STEP 1. 6g+12 = 6g +12
STEP 2. 6g-6g = 12 -12
STEP 3. 0 = 0
The problem preparer tossed in a "curve ball", but it illustrates what you need to do.

Answer 4

Just multiply it all out and solve for the variable, not hard. example

3*a+(3*-5)=18
3a-15=18
3a=18+15
a=33/3