The Distributive Property & Combining Like Terms

y + xy 2(x + 5) = 2(x) + 2(5) 2x + 10 a(b + c) = ab + ac USING THE DISTRIBUTIVE PROPERTY THE DISTRIBUTIVE PROPERTY The product of a and (b + c): 2(x + 5) = 2(x) + 2(5) = 2x + 10 a(b + c) = ab + ac (b + c)a = ba + ca (x + 5)2 = (x)2 + (5)2 = 2x + 10 y(1 + x) = y(1) +y(x) = y + xy (1 + 5x)2 = (1)2 + (5x)2 = 2 + 10x BACK

= –7 + 3x (–3)(1 + x) = (–3)(1) + (–3)(x) = –3 – 3x –(y – 5) USING THE DISTRIBUTIVE PROPERTY Remember that a factor must multiply each term of an expression. Distribute the –3. (–3)(1 + x) = (–3)(1) + (–3)(x) Simplify. = –3 – 3x –(y – 5) = (y)(–1) + (–5)(–1) Distribute the –1. = –y + 5 Simplify. = (–1)(7) + (–1)(–3x) –a = –1 • a –(7 – 3x) Simplify. = –7 + 3x Forgetting to distribute the negative sign when multiplying by a negative factor is a common error. BACK

= 5x2 - 4 8x + 3x + y = 11x +y 4x2 + 2 + x2 - 6 = 4x2 +1 x2 + 2 - 6 SIMPLIFYING BY COMBINING LIKE TERMS Like Terms are expressions that contain the same variable raised to the same power like 4mn and 7mn. 8x + 3x + y = 11x +y The 8x and 3x are like terms and can be combined. 4x2 + 2 + x2 - 6 = 4x2 +1 x2 + 2 - 6 Group like terms. = 5x2 - 4 Combine like terms. BACK

Combine like terms with parenthesis… BACK

Objectives Use the Distributive Property to solve equations. Solve real-world problems by using multistep equations.

Key Skills Solve 3(x - 4) = 48 Solve 3(x - 4) = 48 Use the distributive property. 3 * x + 3 * -4 = 48 Multiply Add both sides of the = sign. 3x - 12 = 48 +12 +12 Divide both sides of the = sign by 3. 3x = 60 3 3 x = 20

Key Skills TRY THIS Solve 5(x +2) = 35 Solve 5(x +2) = 35 Use the distributive property. 5 * x + 5 * 2 = 35 Multiply Subtract 10 from both sides of the = sign. 5x + 10 = 35 -10 -10 Divide both sides of the = sign by 3. 5x = 25 5 5 x = 5

Key Skills Combine like terms that are on the same side of the = sign. Solve 3y – 8 –y = 6 Solve 3y – 8 –y = 6 2y – 8 = 6 Add 8 to both sides of the = sign. + 8 + 8 2y = 14 Divide both sides of the equal sign by 2 2 2 y = 7

Key Skills TRY THIS Combine like terms that are on the same side of the = sign. Solve 6y +2 –3y = 8 Solve 6y +2 –3y = 8 3y +2 = 8 Subtract 2 from both sides of the = sign. -2 -2 3y = 6 Divide both sides of the equal sign by 3 3 3 y = 2

Key Skills Solve 4x -8(x + 1) = 8 4x -8(x + 1) = 8 Distribute -8 Multiply 4x -8x -8 = 8 4x -8x -8 = 8 Combine like terms. -4x -8 = 8 Add 8 to both sides of the = sign. +8 +8 -4x = 16 Divide both sides of the equal sign by -4 -4 -4 x = -4

Key Skills Use the Distributive Property to solve multistep equations. Solve 8x – 2(3x – 4) = 5x – 7. Solve 8x – 2(3x – 4) = 5x – 7. Distributive -2 8x + (–2)(3x) + (–2)(–4) = 5x – 7 Multiply. Combine like terms that are on the same side of the = sign. 8x – 6x + 8 = 5x – 7 8x – 6x + 8 = 5x – 7 2x + 8 = 5x – 7

Subtract 2x from both sides of the = sign. 2x + 8 = 5x – 7 -2x -2x 8 = 3x -7 Add 7 to both sides of the = sign. +7 +7 Divide both sides of the = sign by 3 15 = 3x 3 3 5 = x

Key Skills TRY THIS Use the Distributive Property to solve multistep equations. Solve 4y – 7(y +6) = 5y – 2. Solve 4y – 7(y +6) = 5y – 2. Distributive -7 4y + (–7)(y) + (–7)(+6) = 5y – 2 Multiply. Combine like terms that are on the same side of the = sign. 4y – 7y -42 = 5y – 2 4y – 7y -42 = 5y – 2 -3y -42 = 5y – 2

Add 3y to both sides of the = sign. -3y -42 = 5y – 2 +3y +3y -42 = 8y -2 Add 2 to both sides of the = sign. +2 +2 Divide both sides of the = sign by 8 -40 = 8y 8 8 -5 = y