Question

directions: simplify each expression by combining

1. ( 8 y - 9 y )
2. ( 8 a + 6 + a + 1 )
3. ( - x + 2 + 15 x )
4. ( 6 d + 4 + d + 2 )
5. ( 9 a + 8 + 2 a + 3 + 5 a )
6. ( 7 x + 2 y + 5 + 5 x + 9 y + 1 )
7. ( 6 + 3 a + b + 4 a + 4 b + 5 )
8. ( 8 c ^ { 2 } - c + 3 + 2 c ^ { 2 } - c + 2 )
9. ( 6 a ^ { 2 } b ^ { 2 } + 3 a b ^ { 2 } + a ^ { 2 } b ^ { 2 } + 4 a b ^ { 2 } )
10. write the perimeter of the rectangle as a simplified expression.

Explanation:

Step1: Combine like terms for \(6 + 3a + b+4a + 4b + 5\)

Combine the constant terms \(6 + 5\) and the \(a\) - terms \(3a+4a\) and the \(b\) - terms \(b + 4b\).
$$(6 + 5)+(3a+4a)+(b + 4b)$$

Step2: Calculate each part

For the constant terms: \(6+5 = 11\). For the \(a\) - terms: \(3a+4a=(3 + 4)a=7a\). For the \(b\) - terms: \(b + 4b=(1 + 4)b = 5b\).
$$11+7a+5b$$

Step3: Combine like terms for the perimeter of the rectangle

The formula for the perimeter of a rectangle is \(P=2(l + w)\), where \(l=x + 1\) and \(w=x+3\).
$$P=2((x + 1)+(x + 3))$$

Step4: Simplify inside the parentheses first

\((x + 1)+(x + 3)=(x+x)+(1 + 3)=2x+4\)

Step5: Multiply by 2

\(P=2(2x + 4)=2\times2x+2\times4\)
$$P = 4x+8$$

Answer:

For \(6 + 3a + b+4a + 4b + 5\), the simplified expression is \(7a + 5b+11\). For the perimeter of the rectangle, the simplified expression is \(4x + 8\).