1) -31 = x + -13
3) -5 = k + 1
5) -12 = -4 - k
7) 2 + n = -10 - 2n
9) -5b - 1 + 8 = -8b - 11

Question

Accepted Answer

### Problem 1: $-31 = x + -13$
# Explanation:
## Step1: Add 13 to both sides
To isolate $ x $, we add 13 to both sides of the equation. This is based on the addition property of equality, which states that if we add the same number to both sides of an equation, the equality still holds.
$$
-31 + 13 = x + (-13) + 13
$$
## Step2: Simplify both sides
Simplify the left - hand side: $-31 + 13=-18$.
Simplify the right - hand side: $x+(-13)+13 = x$ (because $-13 + 13 = 0$).
So we have $x=-18$.
# Answer: $x = - 18$

### Problem 3: $-5=k + 1$
# Explanation:
## Step1: Subtract 1 from both sides
Using the subtraction property of equality (subtracting the same number from both sides of an equation keeps the equation balanced), we subtract 1 from both sides.
$$
-5-1=k + 1-1
$$
## Step2: Simplify both sides
Simplify the left - hand side: $-5-1=-6$.
Simplify the right - hand side: $k + 1-1=k$.
So $k=-6$.
# Answer: $k=-6$

### Problem 5: $-12=-4 - k$
# Explanation:
## Step1: Add 4 to both sides
We use the addition property of equality. Add 4 to both sides of the equation.
$$
-12 + 4=-4 - k+4
$$
## Step2: Simplify both sides
Simplify the left - hand side: $-12 + 4=-8$.
Simplify the right - hand side: $-4 - k+4=-k$ (because $-4 + 4 = 0$). So we have $-8=-k$.
## Step3: Multiply both sides by - 1
To solve for $k$, we multiply both sides of the equation $-8=-k$ by - 1.
$$
(-1)\times(-8)=(-1)\times(-k)
$$
## Step4: Simplify both sides
Simplify the left - hand side: $(-1)\times(-8) = 8$.
Simplify the right - hand side: $(-1)\times(-k)=k$.
So $k = 8$.
# Answer: $k = 8$

### Problem 7: $2 + n=-10-2n$
# Explanation:
## Step1: Add $2n$ to both sides
Using the addition property of equality, we add $2n$ to both sides to get all the $n$ terms on one side.
$$
2 + n+2n=-10-2n + 2n
$$
## Step2: Simplify both sides
Simplify the left - hand side: $2 + 3n$ (since $n+2n = 3n$).
Simplify the right - hand side: $-10$ (since $-2n+2n = 0$). So the equation becomes $2 + 3n=-10$.
## Step3: Subtract 2 from both sides
Using the subtraction property of equality, subtract 2 from both sides.
$$
2+3n - 2=-10 - 2
$$
## Step4: Simplify both sides
Simplify the left - hand side: $3n$ (since $2-2 = 0$).
Simplify the right - hand side: $-12$ (since $-10-2=-12$). So we have $3n=-12$.
## Step5: Divide both sides by 3
Using the division property of equality (dividing both sides of an equation by the same non - zero number keeps the equation balanced), divide both sides by 3.
$$
n=\frac{-12}{3}
$$
## Step6: Simplify the fraction
$\frac{-12}{3}=-4$. So $n=-4$.
# Answer: $n=-4$

### Problem 9: $-5b-1 + 8=-8b-11$
# Explanation:
## Step1: Simplify the left - hand side
Combine the constant terms on the left - hand side. $-1 + 8 = 7$, so the equation becomes $-5b+7=-8b-11$.
## Step2: Add $8b$ to both sides
Using the addition property of equality, add $8b$ to both sides to get all the $b$ terms on one side.
$$
-5b+7 + 8b=-8b-11+8b
$$
## Step3: Simplify both sides
Simplify the left - hand side: $3b + 7$ (since $-5b+8b = 3b$).
Simplify the right - hand side: $-11$ (since $-8b + 8b=0$). So the equation is $3b + 7=-11$.
## Step4: Subtract 7 from both sides
Using the subtraction property of equality, subtract 7 from both sides.
$$
3b+7 - 7=-11 - 7
$$
## Step5: Simplify both sides
Simplify the left - hand side: $3b$ (since $7 - 7 = 0$).
Simplify the right - hand side: $-18$ (since $-11-7=-18$). So we have $3b=-18$.
## Step6: Divide both sides by 3
Using the division property of equality, divide both sides by 3.
$$
b=\frac{-18}{3}
$$
## Step7: Simplify the fraction
$\frac{-18}{3}=-6$. So $b=-6$.
# Answer: $b=-6$

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QUESTION IMAGE

Question

Explanation:

Problem 1: $-31 = x + -13$

Step1: Add 13 to both sides

Step2: Simplify both sides

Step1: Subtract 1 from both sides

Step2: Simplify both sides

Step1: Add 4 to both sides

Step2: Simplify both sides

Step3: Multiply both sides by - 1

Step4: Simplify both sides

Answer:

Problem 3: \(-5=k + 1\)