Math Problem Statement

A set of five math problems involving subtraction, simplifying expressions, and factoring.

Solution

Let me analyze and solve each question in detail:


Question 1:

Subtract:
(4y7)(y7)(4y - 7) - (y - 7)

Solution:

  1. Distribute the negative sign to the second parenthesis:
    (4y7)y+7(4y - 7) - y + 7
  2. Combine like terms:
    • For yy: 4yy=3y4y - y = 3y
    • For constants: 7+7=0-7 + 7 = 0

Final Answer:
3y3y


Question 2:

Rewrite the expression 13d(9d4)13d - (-9d - 4) without parentheses.

Solution:

  1. Distribute the negative sign to (9d4)(-9d - 4):
    13d+9d+413d + 9d + 4
  2. Combine like terms:
    • For dd: 13d+9d=22d13d + 9d = 22d
    • The constant remains +4+4.

Final Answer:
22d+422d + 4


Question 3:

A student received a coupon for 14% off the total purchase price at a clothing store. Let cc be the original price of the purchase. The expression c0.14cc - 0.14c represents the new price. Write an equivalent expression to show another way to represent the new price.

Solution:
Factor cc out from c0.14cc - 0.14c:
c(10.14)c(1 - 0.14) Simplify:
c(0.86)c(0.86)

Final Answer:
0.86c0.86c


Question 4:

Simplify the expression:
(0.25n0.3)(0.8n0.25)(0.25n - 0.3) - (0.8n - 0.25)

Solution:

  1. Distribute the negative sign to the second parenthesis:
    0.25n0.30.8n+0.250.25n - 0.3 - 0.8n + 0.25
  2. Combine like terms:
    • For nn: 0.25n0.8n=0.55n0.25n - 0.8n = -0.55n
    • For constants: 0.3+0.25=0.05-0.3 + 0.25 = -0.05

Final Answer:
0.55n0.05-0.55n - 0.05


Question 5:

Each month, a shopkeeper spends 5x+115x + 11 dollars on rent and electricity. If he spends 2x32x - 3 dollars on rent, how much does he spend on electricity?

Solution:

  1. Total expense = Rent + Electricity:
    5x+115x + 11 Rent = 2x32x - 3
    Electricity = Total expense - Rent:
    (5x+11)(2x3)(5x + 11) - (2x - 3)
  2. Simplify:
    Distribute the negative sign:
    5x+112x+35x + 11 - 2x + 3 Combine like terms:
    • For xx: 5x2x=3x5x - 2x = 3x
    • For constants: 11+3=1411 + 3 = 14

Final Answer:
3x+143x + 14


Summary of Answers:

  1. 3y3y
  2. 22d+422d + 4
  3. 0.86c0.86c
  4. 0.55n0.05-0.55n - 0.05
  5. 3x+143x + 14

Would you like detailed explanations for any specific question?

Here are 5 related questions to practice:

  1. Simplify (6a5)(3a+7)(6a - 5) - (3a + 7).
  2. Rewrite 15x(10x2)15x - (-10x - 2) without parentheses.
  3. Write an equivalent expression for b0.25bb - 0.25b.
  4. Simplify (0.4y0.6)(0.9y0.3)(0.4y - 0.6) - (0.9y - 0.3).
  5. A shopkeeper spends 6y+86y + 8 dollars, with 3y43y - 4 on rent. How much does he spend on electricity?

Tip: Always distribute negative signs carefully when simplifying expressions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplification
Factoring
Combining Like Terms

Formulas

Distributive Property: a(b + c) = ab + ac
Factoring: ab - ac = a(b - c)

Theorems

Basic Properties of Arithmetic and Algebra

Suitable Grade Level

Grades 6-8