Equivalent Expressions

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Answer Key & Explanations

Equivalent Expressions · Grade 7 · Worksheet 3

1. Rewrite 6(2x + 1) + 4x in simplest form. Answer: 16x + 6 Solution: Distribute the 6 to the terms inside the parentheses: 6(2x + 1) = 6 × 2x + 6 × 1 = 12x + 6. Add the 4x: 12x + 6 + 4x. Combine like terms (12x and 4x): 12x + 4x = 16x.
   Full step-by-step solution

   Step 1: Distribute the 6 to the terms inside the parentheses: 6(2x + 1) = 6 × 2x + 6 × 1 = 12x + 6. Step 2: Add the 4x: 12x + 6 + 4x. Step 3: Combine like terms (12x and 4x): 12x + 4x = 16x. Step 4: The simplified expression is 16x + 6. Final answer: 16x + 6.

2. Sophia draws a rectangle on a coordinate grid with vertices at (1, 6), (11, 6), (11, 16), and (1, 16). She then shades the region inside the rectangle but outside a smaller rectangle with vertices at (1, 6), (6, 6), (6, 16), and (1, 16). Write an expression for the area of the shaded region in two different equivalent forms: one expanded and one factored. Then determine the area of the shaded region. Answer: 50 Solution: Identify the dimensions of the large rectangle. The vertices are (1,6) to (11,6) so width = 11 - 1 = 10 units. Height = 16 - 6 = 10 units.
   Full step-by-step solution

   Step 1: Identify the dimensions of the large rectangle. The vertices are (1,6) to (11,6) so width = 11 - 1 = 10 units. Height = 16 - 6 = 10 units. Area of large rectangle = 10 x 10 = 100 square units. Step 2: Identify the dimensions of the small rectangle. The vertices are (1,6) to (6,6) so width = 6 - 1 = 5 units. Height = 16 - 6 = 10 units. Area of small rectangle = 5 x 10 = 50 square units. Step 3: The shaded region is the large rectangle minus the small rectangle. Expression: 10 x 10 - 5 x 10 = 100 - 50 = 50 square units. Step 4: Write the expression in expanded form: (10 x 10) - (5 x 10) = 100 - 50. Step 5: Write the expression in factored form using the distributive property: 10(10 - 5) = 10 x 5 = 50. Step 6: Both forms give the same area: 50 square units. The answer is 50.

3. Rewrite 7(3x + 9) - 2(4x - 5) in simplest form. Answer: 13x + 73 Solution: Combine the two expressions: (21x + 63) + (-8x + 10) = 21x - 8x + 63 + 10. Combine like terms: 21x - 8x = 13x, and 63 + 10 = 73. The simplified expression is 13x + 73.
   Full step-by-step solution

   Step 1: Apply the distributive property to 7(3x + 9): 7 × 3x = 21x and 7 × 9 = 63, so 7(3x + 9) = 21x + 63. Step 2: Apply the distributive property to -2(4x - 5): -2 × 4x = -8x and -2 × (-5) = +10, so -2(4x - 5) = -8x + 10. Step 3: Combine the two expressions: (21x + 63) + (-8x + 10) = 21x - 8x + 63 + 10. Step 4: Combine like terms: 21x - 8x = 13x, and 63 + 10 = 73. Step 5: The simplified expression is 13x + 73. Final answer: 13x + 73.

4. A rectangular garden has a length that is 3 meters more than twice its width. If the perimeter of the garden is 42 meters, what is the width of the garden in meters? Answer: 6 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). \( P = 2 \times (\text{length} + \text{width}) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). Given \( P = 42 \), we have: \( 2 \times (l + w) = 42 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 3 \) into the perimeter equation: \( 2 \times ( (2w + 3) + w ) = 42 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 42 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 3 = 21 \). Subtract 3 from both sides: \( 3w = 18 \). Divide by 3: \( w = 6 \). --- **Step 6: Conclusion** The width of the garden is 6 meters.

5. A school is planning a field trip and needs to transport 245 students. Each school bus can carry 48 students. If the school also needs to reserve 3 buses for chaperones and teachers, how many total buses are needed for the entire trip? Answer: 9 Solution: Calculate how many buses are needed for the students 245 students ÷ 48 students per bus = 5.104 buses Since we can't have a fraction of a bus, we round up to 6 buses for students The problem states 3 buses are needed for chaperones and teachers 6 buses for students + 3 buses for…
   Full step-by-step solution

   Step 1: Calculate how many buses are needed for the students 245 students ÷ 48 students per bus = 5.104 buses Since we can't have a fraction of a bus, we round up to 6 buses for students Step 2: Add the buses needed for chaperones and teachers The problem states 3 buses are needed for chaperones and teachers Step 3: Calculate total buses needed 6 buses for students + 3 buses for chaperones/teachers = 9 total buses The answer is 9 buses.

6. Rewrite 4(6x + 8) - 12x in simplest form. Answer: 12x + 32 Solution: Distribute the 4 to both terms inside the parentheses: 4(6x + 8) = 4 × 6x + 4 × 8 = 24x + 32. Now the expression is 24x + 32 - 12x. Combine like terms: 24x - 12x = 12x.
   Full step-by-step solution

   Step 1: Distribute the 4 to both terms inside the parentheses: 4(6x + 8) = 4 × 6x + 4 × 8 = 24x + 32. Step 2: Now the expression is 24x + 32 - 12x. Step 3: Combine like terms: 24x - 12x = 12x. Step 4: The constant term 32 remains. Step 5: The simplified expression is 12x + 32. Final answer: 12x + 32

7. A rectangular prism is drawn with dimensions: length = 12 cm, width = 8 cm, and height = 15 cm. If you were to draw all the visible edges of this prism, how many edges would be visible if it's sitting on its largest face and you're viewing it from the front corner? Answer: 9 Solution: A rectangular prism has 12 total edges (4 length edges, 4 width edges, 4 height edges). When sitting on its largest face, the bottom face rests on the surface. The largest face is 12 cm × 15 cm = 180 cm².
   Full step-by-step solution

   Step 1: A rectangular prism has 12 total edges (4 length edges, 4 width edges, 4 height edges). Step 2: When sitting on its largest face, the bottom face rests on the surface. The largest face is 12 cm × 15 cm = 180 cm². Step 3: From a front corner view, the bottom face edges touching the surface are hidden from view. This hides 3 edges (2 length edges and 1 width edge of the bottom face). Step 4: The back vertical edges are also hidden since we're viewing from the front. This hides 2 more edges. Step 5: Calculate visible edges: 12 total edges - 3 hidden bottom edges - 2 hidden back edges = 7 edges. Step 6: However, we must also consider that some edges overlap in the 2D projection. Actually, from a front corner view, you can see 3 edges of the front face, 3 edges of the side face, and 3 edges of the top face that don't overlap. Step 7: The visible edges are: front bottom length (1), front vertical left (1), front vertical right (1), side bottom width (1), side top width (1), side top-back vertical (1), top front length (1), top side width (1), and top-back length (1). Step 8: Counting these: 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 9 visible edges. The answer is 9.

8. A rectangular prism is drawn with vertices at (0,0,0), (8,0,0), (8,5,0), (0,5,0), (0,0,3), (8,0,3), (8,5,3), and (0,5,3). If you draw all the space diagonals of this prism (lines connecting vertices that are not on the same face), how many distinct space diagonals does the rectangular prism have? Answer: 4 Solution: A rectangular prism has 8 vertices total. Space diagonals connect vertices that are not on the same face and are not already connected by an edge.
   Full step-by-step solution

   Step 1: A rectangular prism has 8 vertices total. Step 2: Space diagonals connect vertices that are not on the same face and are not already connected by an edge. Step 3: From each vertex, we can draw space diagonals to the 3 vertices that are not on any of the same faces. Step 4: From vertex (0,0,0), the space diagonals go to (8,5,3), (8,5,0), and (0,5,3). Step 5: Since each space diagonal is counted twice (once from each end), we divide the total by 2. Step 6: Calculation: (8 vertices × 3 space diagonals per vertex) ÷ 2 = 24 ÷ 2 = 4. Step 7: Therefore, the rectangular prism has 4 distinct space diagonals.