EXAMPLE 1 Finding a Combined Area Architecture A replica of the Parthenon, a temple in ancient Greece, was built in Nashville, Tennessee, in 1897. The diagram below shows the approximate dimensions of two adjacent rooms inside the replica. You can find the total area in two ways as shown in Example 1.

EXAMPLE 1 Finding a Combined Area Two methods can be used to find the total area of the two rooms. METHOD 1 Find the area of each room, and then find the total area. Area = 63(44) + 63(98) = 2772 + 6174 = 8946 square feet

EXAMPLE 1 Finding a Combined Area METHOD 2 Find the total length, and then multiply by the common width. Area = 63(44 + 98) = 63 (142) = 8946 square feet The total area of the two rooms is 8946 square feet. ANSWER

Using the Distributive Property EXAMPLE 2 Using the Distributive Property a. –5(x + 10) = –5x + (–5)(10) Distributive property = –5x + (–50) Multiply. = –5x – 50 Simplify. b. 3[1 – 20 + (–5)] = 3(1) – 3(20) + 3(–5) Distributive property = 3 – 60 + (–15) Multiply. = 3 + (–60) + (–15) Add the opposite of 60. = –72 Add.

GUIDED PRACTICE for Examples 1 and 2 Use the distributive property to find the area of the figure. 1. Find the total length, and then multiply with common width. Area = 10 ( 12 + 22 ) = 10 ( 34 ) = 340 ft2

GUIDED PRACTICE for Examples 1 and 2 Use the distributive property to find the area of the figure. 2. Find the total length, and then multiply with common width. Area = 14 ( 3 + 9 ) = 14 ( 12 ) = 168 m2

GUIDED PRACTICE for Examples 1 and 2 Use the distributive property to evaluate or write an equivalent expression. 3. –2(5 + 12) –2(5 + 12) = –2(5) + (–2)(12) Distributive property = –10 + (–24) Multiply. = –34 Add.

GUIDED PRACTICE for Examples 1 and 2 Use the distributive property to evaluate or write an equivalent expression. 4. –4(–7 – 10) –4(–7 – 10) = –4(–7) – (–4)(–10) Distributive property = 28 + 40 Multiply. = 68 Add.

GUIDED PRACTICE for Examples 1 and 2 Use the distributive property to evaluate or write an equivalent expression. 5. 2(w – 8) 2(w – 8) = 2w – (2)(8) Distributive property = 2w – (16) Multiply. = 2w – 16 Simplify.

GUIDED PRACTICE for Examples 1 and 2 Use the distributive property to evaluate or write an equivalent expression. 6. –8(z + 25) –8(z + 25) = –8z + (–8)(25) Distributive property = –8z + (–200) Multiply. = –8z – 200 Simplify.

EXAMPLE 3 Combining Like Terms a. 3x + 4x = (3 + 4)x = 7x b. Distributive property = 7x Add inside grouping symbols. b. –9y + 7y + 5z = (–9 + 7)y + 5z Distributive property = –2y + 5z Add inside grouping symbols.

Simplifying an Expression EXAMPLE 4 Simplifying an Expression a. 2(4 + x) + x = 8 + 2x + x Distributive property = 8 + 3x Combine like terms. b. –5(3x – 6) + 7x = –15x + 30 + 7x Distributive property = –8x + 30 Combine like terms.

Simplify the expression by combining like terms. GUIDED PRACTICE for Examples 3 and 4 Simplify the expression by combining like terms. 7. 2(x + 4) + 3x – 5 2(x + 4) + 3x – 5 = 2x + 8 + 3x – 5 Distributive property = 5x + 8 + (–5) Add. = 5x + 3 Combine like terms.

Simplify the expression by combining like terms. GUIDED PRACTICE for Examples 3 and 4 Simplify the expression by combining like terms. 8. 5y + 9z – 7 – 3y 5y + 9z – 7 – 3y = (5 – 3)y + 9z – 7 Distributive property = 2y + 9z – 7 Combine like terms.

Simplify the expression by combining like terms. GUIDED PRACTICE for Examples 3 and 4 Simplify the expression by combining like terms. 9. –3(6x + 2y) + 22x –3(6x + 2y) + 22x = –18x – 6y + 22x Distributive property = 4x – 6y Combine like terms.

Sec. 2.8

EXAMPLE 1 Naming Points in a Coordinate Plane Give the coordinates of the point. A B C SOLUTION Point A is 3 units to the right of the origin and 1.5 units up. So, the x-coordinate is 3 and the y-coordinate is 1.5. The coordinates of A are (3, 1.5). Point B is 3 units to the left of the origin and 2 units down. So, the x-coordinate is –3 and the y-coordinate is –2. The coordinates of B are (–3, –2).

EXAMPLE 1 Naming Points in a Coordinate Plane Point C is 2 units up from the origin. So, the x-coordinate is 0 and the y-coordinate is 2. The coordinates of C are (0, 2).

GUIDED PRACTICE for Example 1 Use the graph. Give the coordinates of the point. D SOLUTION Point D is 3 units to the right of the origin and 4 units down. So, the x-coordinate is 3 and the y-coordinate is –4. The coordinates of D are (3, –4).

GUIDED PRACTICE for Example 1 Use the graph. Give the coordinates of the point. E SOLUTION Point E is 2.5 units to the left of the origin and 2 units up. So, the x-coordinate is –2.5 and the y-coordinate is 2. The coordinates of D are (–2.5, 2).

GUIDED PRACTICE for Example 1 Use the graph. Give the coordinates of the point. F SOLUTION Point F is 3 units to the left of the origin. So, the x-coordinate is 0 and the y-coordinate is –3. The coordinates of F are (0, –3).

EXAMPLE 2 Graphing Points in a Coordinate Plane Plot the point and describe its location. A (4, –2) B (–1, 2.5) C (0 , –3) SOLUTION Begin at the origin, move 4 units to the right, then 2 units down. Point A lies in Quadrant IV. Begin at the origin, move 1 unit to the left, then 2.5 units up. Point B lies in Quadrant II.

EXAMPLE 2 Graphing Points in a Coordinate Plane Begin at the origin, move 3 units down. Point C lies on the y-axis.

EXAMPLE 3 Solve a Multi-Step Problem Archaeology On a field trip, students are exploring an archaeological site. They rope off a region to explore as shown. Identify the shape of the region and find its perimeter. The units on the scale are feet.

EXAMPLE 3 Solve a Multi-Step Problem SOLUTION STEP 1 Notice that points A, B, C, and D form a rectangle. Find the coordinates of the vertices. A(–30, 20), B(30, 20), C(30, –20), D (–30, –20) STEP 2 Find the horizontal distance from A to B to find the length l. x-coordinate of B x-coordinate of A = l – = –30 – 30 –60 = = 60

EXAMPLE 3 Solve a Multi-Step Problem STEP 3 Find the vertical distance from A to D to find the width w. y-coordinate of D y-coordinate of A = w – = 20 – (–20) 40 = = 40 STEP 4 Find the perimeter: 2l + 2w = 2(60) + 2(40) = 200. ANSWER The region’s perimeter is 200 units 10 feet per unit = 2000 feet.

GUIDED PRACTICE for Examples 2 and 3 Plot the point and describe its location. R (–3, 4) Begin at the origin and move 3 units to the left, then 4 units up. Point R lies in Quadrant II. S (1, –2.5) Begin at the origin and move 1 unit right, then 2.5 units down. Point S lies in Quadrant IV.

GUIDED PRACTICE for Examples 2 and 3 Plot the point and describe its location. T (0.5 , 3) Begin at the origin and move 0.5 unit to the right, then 3 units up. Point T lies in Quadrant I. U (–3, 4) Begin at the origin and move 4 units to the left. Point U lies on the x-axis.

GUIDED PRACTICE for Examples 2 and 3 Plot the point and describe its location. Move points A and B in Example 3 to form a new rectangle. Find the perimeter. Answers may vary. Sample answer: Moving A and B to A(–30, 0) and B(30, 0), the perimeter is 160 units. ANSWER