To simplify the given expression \( 6 - 4[x + 1 - (y + 3)] + 5y \), follow these steps:
1. Simplify inside the innermost parentheses:
First, focus on the expression inside the parentheses:
[tex]\[ x + 1 - (y + 3) \][/tex]
Distribute the negative sign inside the parentheses:
[tex]\[ x + 1 - y - 3 \][/tex]
Combine like terms:
[tex]\[ x - y - 2 \][/tex]
2. Substitute the simplified expression back into the original problem:
Replace \( x + 1 - (y + 3) \) with \( x - y - 2 \) in the original expression:
[tex]\[ 6 - 4[x - y - 2] + 5y \][/tex]
3. Distribute the \(-4\) across the terms inside the brackets:
Distribute \(-4\) to each term inside the brackets \( x - y - 2 \):
[tex]\[ -4 \times x = -4x \][/tex]
[tex]\[ -4 \times -y = 4y \][/tex]
[tex]\[ -4 \times -2 = 8 \][/tex]
So, the expression becomes:
[tex]\[ 6 - 4x + 4y + 8 + 5y \][/tex]
4. Combine like terms:
First, combine the constants \( 6 \) and \( 8 \):
[tex]\[ 6 + 8 = 14 \][/tex]
Next, combine the terms involving \( y \):
[tex]\[ 4y + 5y = 9y \][/tex]
So, the simplified expression is:
[tex]\[ 14 - 4x + 9y \][/tex]
Thus, the simplified form of the given expression \( 6 - 4[x + 1 - (y + 3)] + 5y \) is:
[tex]\[ \boxed{14 - 4x + 9y} \][/tex]
