3) $5[a-3(4 a-5 b)]+4[3(2 a-b)-2(b-2 a)]+15 a=$
4) $2 a-(-3 x+2\{-a+3 x-2[-a+b-(2+a)]\})+3 x+8-4 b=$
5) $-4\{m-2 n-[3 n+2(2 n-m)+5(4 n-3 m)]\}+5(m-n)+67 m=$
6) $a-(a+b)-3\{-2 a+[-2 a+b+2(b-1)-a-b+1]\}-3+7 b=$

Simplify expression 3: 5 [ a − 3 ( 4 a − 5 b )] + 4 [ 3 ( 2 a − b ) − 2 ( b − 2 a )] + 15 a = 55 b
Simplify expression 4: 2 a − ( − 3 x + 2 { − a + 3 x − 2 [ − a + b − ( 2 + a )]}) + 3 x + 8 − 4 b = − 4 a
Simplify expression 5: − 4 { m − 2 n − [ 3 n + 2 ( 2 n − m ) + 5 ( 4 n − 3 m )]} + 5 ( m − n ) + 67 m = 111 n
Simplify expression 6: a − ( a + b ) − 3 { − 2 a + [ − 2 a + b + 2 ( b − 1 ) − a − b + 1 ]} − 3 + 7 b = 15 a

55 b , − 4 a , 111 n , 15 a ​
Explanation

Problem Analysis We will simplify the four given expressions step by step, using the distributive property and combining like terms.

Simplifying Expression 3 Let's simplify expression 3: 5 [ a − 3 ( 4 a − 5 b )] + 4 [ 3 ( 2 a − b ) − 2 ( b − 2 a )] + 15 a = First, we simplify the innermost parentheses: 5 [ a − 12 a + 15 b ] + 4 [ 6 a − 3 b − 2 b + 4 a ] + 15 a = 5 [ − 11 a + 15 b ] + 4 [ 10 a − 5 b ] + 15 a = − 55 a + 75 b + 40 a − 20 b + 15 a = ( − 55 + 40 + 15 ) a + ( 75 − 20 ) b = 0 a + 55 b = 55 b

Simplifying Expression 4 Now, let's simplify expression 4: 2 a − ( − 3 x + 2 { − a + 3 x − 2 [ − a + b − ( 2 + a )]}) + 3 x + 8 − 4 b = 2 a − ( − 3 x + 2 { − a + 3 x − 2 [ − a + b − 2 − a ]}) + 3 x + 8 − 4 b = 2 a − ( − 3 x + 2 { − a + 3 x − 2 [ − 2 a + b − 2 ]}) + 3 x + 8 − 4 b = 2 a − ( − 3 x + 2 { − a + 3 x + 4 a − 2 b + 4 }) + 3 x + 8 − 4 b = 2 a − ( − 3 x + 2 { 3 a + 3 x − 2 b + 4 }) + 3 x + 8 − 4 b = 2 a − ( − 3 x + 6 a + 6 x − 4 b + 8 ) + 3 x + 8 − 4 b = 2 a + 3 x − 6 a − 6 x + 4 b − 8 + 3 x + 8 − 4 b = ( 2 − 6 ) a + ( 3 − 6 + 3 ) x + ( 4 − 4 ) b + ( − 8 + 8 ) = − 4 a + 0 x + 0 b + 0 = − 4 a

Simplifying Expression 5 Next, let's simplify expression 5: − 4 { m − 2 n − [ 3 n + 2 ( 2 n − m ) + 5 ( 4 n − 3 m )]} + 5 ( m − n ) + 67 m = − 4 { m − 2 n − [ 3 n + 4 n − 2 m + 20 n − 15 m ]} + 5 m − 5 n + 67 m = − 4 { m − 2 n − [ 27 n − 17 m ]} + 72 m − 5 n = − 4 { m − 2 n − 27 n + 17 m } + 72 m − 5 n = − 4 { 18 m − 29 n } + 72 m − 5 n = − 72 m + 116 n + 72 m − 5 n = ( − 72 + 72 ) m + ( 116 − 5 ) n = 0 m + 111 n = 111 n

Simplifying Expression 6 Finally, let's simplify expression 6: a − ( a + b ) − 3 { − 2 a + [ − 2 a + b + 2 ( b − 1 ) − a − b + 1 ]} − 3 + 7 b = a − a − b − 3 { − 2 a + [ − 2 a + b + 2 b − 2 − a − b + 1 ]} − 3 + 7 b = − b − 3 { − 2 a + [ − 3 a + 2 b − 1 ]} − 3 + 7 b = − b − 3 { − 2 a − 3 a + 2 b − 1 } − 3 + 7 b = − b − 3 { − 5 a + 2 b − 1 } − 3 + 7 b = − b + 15 a − 6 b + 3 − 3 + 7 b = 15 a + ( − 1 − 6 + 7 ) b + ( 3 − 3 ) = 15 a + 0 b + 0 = 15 a


Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields. For example, in physics, you might need to simplify an expression representing the force acting on an object. In computer science, simplifying expressions can optimize code. In economics, you might simplify cost or revenue functions to analyze profitability. These simplifications make complex problems easier to understand and solve.

Answered by GinnyAnswer | 2025-07-03

The simplified results of the given expressions are: 55b, -4a, 111n, and 15a. Each expression was simplified step by step using the distributive property and combining like terms. This process is fundamental in algebra for solving equations and simplifying complex expressions.
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Answered by Anonymous | 2025-07-04