Question

1. $6r - 5 = 1 + 8r - 24$
2. $-5 + 7m - 20 = 20 - 2m$
3. $-6(n - 3) = 5(4 - n)$
4. $-\frac{1}{2}(12v - 20) + 4 = -12 - 8v$

Explanation:

Problem 10: Solve for $r$

Step1: Simplify right-hand side

$6r - 5 = 8r - 23$

Step2: Move $r$ terms to left

$6r - 8r - 5 = -23$

Step3: Simplify left $r$ terms

$-2r - 5 = -23$

Step4: Isolate $-2r$

$-2r = -23 + 5$

Step5: Calculate right-hand side

$-2r = -18$

Step6: Solve for $r$

$r = \frac{-18}{-2} = 9$

Problem 12: Solve for $m$

Step1: Simplify left-hand side

$7m - 25 = 20 - 2m$

Step2: Move $m$ terms to left

$7m + 2m - 25 = 20$

Step3: Simplify left $m$ terms

$9m - 25 = 20$

Step4: Isolate $9m$

$9m = 20 + 25$

Step5: Calculate right-hand side

$9m = 45$

Step6: Solve for $m$

$m = \frac{45}{9} = 5$

Problem 14: Solve for $n$

Step1: Expand both sides

$-6n + 18 = 20 - 5n$

Step2: Move $n$ terms to left

$-6n + 5n + 18 = 20$

Step3: Simplify left $n$ terms

$-n + 18 = 20$

Step4: Isolate $-n$

$-n = 20 - 18$

Step5: Calculate right-hand side

$-n = 2$

Step6: Solve for $n$

$n = -2$

Problem 16: Solve for $v$

Step1: Expand left-hand side

$-6v + 10 + 4 = -12 - 8v$

Step2: Simplify left-hand side

$-6v + 14 = -12 - 8v$

Step3: Move $v$ terms to left

$-6v + 8v + 14 = -12$

Step4: Simplify left $v$ terms

$2v + 14 = -12$

Step5: Isolate $2v$

$2v = -12 - 14$

Step6: Calculate right-hand side

$2v = -26$

Step7: Solve for $v$

$v = \frac{-26}{2} = -13$

Answer:

1. $r = 9$
2. $m = 5$
3. $n = -2$
4. $v = -13$