Certainly! Let's simplify each of the given expressions step-by-step.
### Expression 1: [tex]\( 4 \cdot (x + 8) - 7x \)[/tex]
1. Distribute the 4 in the first term:
[tex]\[ 4 \cdot (x + 8) = 4x + 32 \][/tex]
2. Now substitute this back into the expression:
[tex]\[ 4x + 32 - 7x \][/tex]
3. Combine like terms:
[tex]\[ 4x - 7x + 32 = -3x + 32 \][/tex]
So, the simplified form of [tex]\( 4 \cdot (x + 8) - 7x \)[/tex] is:
[tex]\[ -3x + 32 \][/tex]
### Expression 2: [tex]\( -(9 - t) - 7t \)[/tex]
1. Distribute the negative sign in the first term:
[tex]\[ -(9 - t) = -9 + t \][/tex]
2. Now substitute this back into the expression:
[tex]\[ -9 + t - 7t \][/tex]
3. Combine like terms:
[tex]\[ -9 + t - 7t = -9 - 6t \][/tex]
So, the simplified form of [tex]\( -(9 - t) - 7t \)[/tex] is:
[tex]\[ -9 - 6t \][/tex]
### Expression 3: [tex]\( 9x + 6 \cdot (7x - 8) + 13 \)[/tex]
1. Distribute the 6 in the second term:
[tex]\[ 6 \cdot (7x - 8) = 42x - 48 \][/tex]
2. Now substitute this back into the expression:
[tex]\[ 9x + 42x - 48 + 13 \][/tex]
3. Combine like terms:
[tex]\[ 9x + 42x - 48 + 13 = 51x - 35 \][/tex]
So, the simplified form of [tex]\( 9x + 6 \cdot (7x - 8) + 13 \)[/tex] is:
[tex]\[ 51x - 35 \][/tex]
### Expression 4: [tex]\( -3 \cdot (-y + 2) - 2 \cdot (y + 7) \)[/tex]
1. Distribute the -3 in the first term:
[tex]\[ -3 \cdot (-y + 2) = 3y - 6 \][/tex]
2. Distribute the -2 in the second term:
[tex]\[ -2 \cdot (y + 7) = -2y - 14 \][/tex]
3. Now substitute both results back into the expression:
[tex]\[ 3y - 6 - 2y - 14 \][/tex]
4. Combine like terms:
[tex]\[ 3y - 2y - 6 - 14 = y - 20 \][/tex]
So, the simplified form of [tex]\( -3 \cdot (-y + 2) - 2 \cdot (y + 7) \)[/tex] is:
[tex]\[ y - 20 \][/tex]
Therefore, the final simplified results are:
1. [tex]\( 4 \cdot (x + 8) - 7x = -3x + 32 \)[/tex]
2. [tex]\( -(9 - t) - 7t = -9 - 6t \)[/tex]
3. [tex]\( 9x + 6 \cdot (7x - 8) + 13 = 51x - 35 \)[/tex]
4. [tex]\( -3 \cdot (-y + 2) - 2 \cdot (y + 7) = y - 20 \)[/tex]
