Studdy Solution
STEP 1
1. Each equation is a linear equation in one variable.
2. The goal is to solve for the variable in each equation.
3. The equations involve basic algebraic operations, including distribution, combining like terms, and isolating the variable.
STEP 2
1. Distribute any constants through parentheses.
2. Combine like terms on each side of the equation.
3. Isolate the variable by performing algebraic operations.
4. Check the solution by substituting it back into the original equation.
Let's solve each equation one at a time.
**Equation 5: −8(3+3n)+2(n+1)=−22**
STEP 3
Distribute the constants:
−8×3+(−8)×3n+2×n+2×1=−22
−24−24n+2n+2=−22
STEP 4
Combine like terms:
−24n+2n=−22n
−24+2=−22
So, the equation becomes:
−22n−22=−22
STEP 5
Add 22 to both sides to isolate the term with n:
−22n=0
Divide both sides by −22:
n=0
STEP 6
Check the solution by substituting n=0 back into the original equation:
−8(3+3(0))+2(0+1)=−22
−24+2=−22
The solution is correct.
**Equation 6: −76=−4(3+k)+2(−k−8)**
STEP_1:
Distribute the constants:
−4×3−4×k+2×(−k)+2×(−8)=−76
−12−4k−2k−16=−76
STEP_2:
Combine like terms:
−4k−2k=−6k
−12−16=−28
So, the equation becomes:
−6k−28=−76
STEP_3:
Add 28 to both sides:
−6k=−48
Divide both sides by −6:
k=8
STEP_4:
Check the solution by substituting k=8 back into the original equation:
−76=−4(3+8)+2(−8−8)
−76=−44−32
−76=−76
The solution is correct.
**Equation 7: 7(4+5n)+2(1−2n)=−32**
STEP_1:
Distribute the constants:
7×4+7×5n+2×1−2×2n=−32
28+35n+2−4n=−32
STEP_2:
Combine like terms:
35n−4n=31n
28+2=30
So, the equation becomes:
31n+30=−32
STEP_3:
Subtract 30 from both sides:
31n=−62
Divide both sides by 31:
n=−2
STEP_4:
Check the solution by substituting n=−2 back into the original equation:
7(4+5(−2))+2(1−2(−2))=−32
7(4−10)+2(1+4)=−32
−42+10=−32
The solution is correct.
**Equation 8: 8(6x+6)−8(x+2)=−48**
STEP_1:
Distribute the constants:
8×6x+8×6−8×x−8×2=−48
48x+48−8x−16=−48
STEP_2:
Combine like terms:
48x−8x=40x
48−16=32
So, the equation becomes:
40x+32=−48
STEP_3:
Subtract 32 from both sides:
40x=−80
Divide both sides by 40:
x=−2
STEP_4:
Check the solution by substituting x=−2 back into the original equation:
8(6(−2)+6)−8(−2+2)=−48
8(−12+6)=−48
−48=−48
The solution is correct.
**Equation 9: −49=3(b−3)+4(1+2b)**
STEP_1:
Distribute the constants:
3×b−3×3+4×1+4×2b=−49
3b−9+4+8b=−49
STEP_2:
Combine like terms:
3b+8b=11b
−9+4=−5
So, the equation becomes:
11b−5=−49
STEP_3:
Add 5 to both sides:
11b=−44
Divide both sides by 11:
b=−4
STEP_4:
Check the solution by substituting b=−4 back into the original equation:
−49=3(−4−3)+4(1+2(−4))
−49=−21−28
−49=−49
The solution is correct.
**Equation 10: −65=−8(1−8r)+7(8+7r)**
STEP_1:
Distribute the constants:
−8×1+8×8r+7×8+7×7r=−65
−8+64r+56+49r=−65
STEP_2:
Combine like terms:
64r+49r=113r
−8+56=48
So, the equation becomes:
113r+48=−65
STEP_3:
Subtract 48 from both sides:
113r=−113
Divide both sides by 113:
r=−1
STEP_4:
Check the solution by substituting r=−1 back into the original equation:
−65=−8(1−8(−1))+7(8+7(−1))
−65=−8(1+8)+7(8−7)
−65=−72+7
−65=−65
The solution is correct.
