topic 3: multi-step equations & word problems
solve each equation.
17. ( 3(7 - 9k) + 23k = 4k - (24 - k) )
18. ( 7 - \frac{5}{2}(8n - 18) = 14 - 10(2n - 3) )
19. ( \frac{7x - 3}{3} = \frac{3x - 4}{8} )
20. if ( sa = \frac{1}{2}lp + b ), find ( p )
21. (crossed out) the width of a rectangle is four less than one half the length. if the perimeter of the rectangle is 94 meters, find the area of the rectangle.
22. (crossed out) find three consecutive odd numbers such that the sum of five times the smaller number and twice the larger number is 33 more than six times the median number.
topic 4: absolute value equations
solve each equation. be sure to check for extraneous solutions.
23. ( |-7 - 9x| = 2 )
24. ( \frac{|5n - 10|}{-2} = -15 )

Question

Accepted Answer

### Problem 17: $ 3(7 - 9k) + 23k = 4k - (24 - k) $
# Explanation:
## Step 1: Expand both sides
Expand the left - hand side: $ 3\times7-3\times9k + 23k=21-27k + 23k=21 - 4k $
Expand the right - hand side: $ 4k-24 + k=5k-24 $
So the equation becomes $ 21-4k = 5k-24 $

## Step 2: Move variable terms to one side
Add $ 4k $ to both sides: $ 21=5k + 4k-24 $, which simplifies to $ 21 = 9k-24 $

## Step 3: Move constant terms to the other side
Add 24 to both sides: $ 21 + 24=9k $, so $ 45 = 9k $

## Step 4: Solve for $ k $
Divide both sides by 9: $ k=\frac{45}{9}=5 $

# Answer: $ k = 5 $

### Problem 18: $ 7-\frac{5}{2}(8n - 18)=14-10(2n - 3) $
# Explanation:
## Step 1: Expand both sides
Expand the left - hand side: $ 7-\frac{5}{2}\times8n+\frac{5}{2}\times18=7 - 20n+45=52-20n $
Expand the right - hand side: $ 14-20n + 30=44-20n $
So the equation becomes $ 52-20n=44-20n $

## Step 2: Analyze the equation
Add $ 20n $ to both sides: $ 52=44 $, which is a contradiction.

# Answer: No solution

### Problem 19: $ \frac{7x - 3}{3}=\frac{3x - 4}{8} $
# Explanation:
## Step 1: Cross - multiply
Cross - multiply to get $ 8(7x - 3)=3(3x - 4) $

## Step 2: Expand both sides
Expand the left - hand side: $ 56x-24 $
Expand the right - hand side: $ 9x-12 $
So the equation becomes $ 56x-24 = 9x-12 $

## Step 3: Move variable terms to one side
Subtract $ 9x $ from both sides: $ 56x-9x-24=-12 $, which simplifies to $ 47x-24=-12 $

## Step 4: Move constant terms to the other side
Add 24 to both sides: $ 47x=-12 + 24=12 $

## Step 5: Solve for $ x $
$ x=\frac{12}{47} $

# Answer: $ x=\frac{12}{47} $

### Problem 20: If $ SA=\frac{1}{2}lp + B $, find $ p $
# Explanation:
## Step 1: Isolate the term with $ p $
Subtract $ B $ from both sides: $ SA - B=\frac{1}{2}lp $

## Step 2: Solve for $ p $
Multiply both sides by $ \frac{2}{l} $ (assuming $ l
eq0 $): $ p=\frac{2(SA - B)}{l} $

# Answer: $ p=\frac{2(SA - B)}{l} $

### Problem 21: The width of a rectangle is four less than one half the length. If the perimeter of the rectangle is 94 meters, find the area of the rectangle.
# Explanation:
## Step 1: Define variables
Let the length of the rectangle be $ l $ meters. Then the width $ w=\frac{1}{2}l-4 $

## Step 2: Use the perimeter formula
The perimeter of a rectangle $ P = 2(l + w) $. We know $ P = 94 $, so $ 2(l+\frac{1}{2}l - 4)=94 $

## Step 3: Simplify the equation
First, simplify inside the parentheses: $ l+\frac{1}{2}l-4=\frac{3}{2}l-4 $
Then the equation becomes $ 2(\frac{3}{2}l - 4)=94 $, which simplifies to $ 3l-8 = 94 $

## Step 4: Solve for $ l $
Add 8 to both sides: $ 3l=94 + 8=102 $
Divide by 3: $ l = 34 $

## Step 5: Find the width
$ w=\frac{1}{2}\times34-4=17 - 4 = 13 $

## Step 6: Calculate the area
The area of a rectangle $ A=l\times w $, so $ A = 34\times13 = 442 $ square meters.

# Answer: The area is 442 square meters.

### Problem 22: Find three consecutive odd numbers such that the sum of five times the smaller number and twice the larger number is 33 more than six times the median number.
# Explanation:
## Step 1: Define the numbers
Let the three consecutive odd numbers be $ n-2 $, $ n $, $ n + 2 $ (where $ n $ is the median, and it is odd)

## Step 2: Translate the problem into an equation
Five times the smaller number: $ 5(n - 2) $
Twice the larger number: $ 2(n + 2) $
Six times the median number: $ 6n $
The equation is $ 5(n - 2)+2(n + 2)=6n + 33 $

## Step 3: Expand and simplify the left - hand side
Expand: $ 5n-10 + 2n + 4=7n-6 $
So the equation becomes $ 7n-6=6n + 33 $

## Step 4: Solve for $ n $
Subtract $ 6n $ from both sides: $ n-6 = 33 $
Add 6 to both sides: $ n = 39 $

## Step 5: Find the three numbers
The smaller number: $ n-2=39 - 2 = 37 $
The median number: $ n = 39 $
The larger number: $ n + 2=39 + 2 = 41 $

# Answer: The three consecutive odd numbers are 37, 39, 41.

### Problem 23: $|-7 - 9x|=2$
# Explanation:
## Step 1: Use the definition of absolute value
If $|a|=b$ ($b\geq0$), then $a = b$ or $a=-b$. So we have two cases:
Case 1: $ - 7-9x=2 $
Case 2: $ - 7-9x=-2 $

## Step 2: Solve Case 1
$ - 9x=2 + 7=9 $, so $ x=-1 $

## Step 3: Solve Case 2
$ - 9x=-2 + 7 = 5 $, so $ x=-\frac{5}{9} $

# Answer: $ x=-1$ or $x =-\frac{5}{9}$

### Problem 24: $ \frac{|5n - 10|}{-2}=-15 $
# Explanation:
## Step 1: Isolate the absolute value
Multiply both sides by - 2: $|5n - 10|=(-15)\times(-2)=30$

## Step 2: Use the definition of absolute value
We have two cases:
Case 1: $ 5n-10 = 30 $
Case 2: $ 5n-10=-30 $

## Step 3: Solve Case 1
$ 5n=30 + 10 = 40 $, so $ n = 8 $

## Step 4: Solve Case 2
$ 5n=-30 + 10=-20 $, so $ n=-4 $

# Answer: $ n = 8$ or $n=-4$

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QUESTION IMAGE

Question

Explanation:

Problem 17: \( 3(7 - 9k) + 23k = 4k - (24 - k) \)

Step 1: Expand both sides

Step 2: Move variable terms to one side

Step 3: Move constant terms to the other side

Step 4: Solve for \( k \)

Step 1: Expand both sides

Step 2: Analyze the equation

Step 1: Cross - multiply

Step 2: Expand both sides

Step 3: Move variable terms to one side

Step 4: Move constant terms to the other side

Step 5: Solve for \( x \)

Answer:

Problem 18: \( 7-\frac{5}{2}(8n - 18)=14-10(2n - 3) \)