Math Problem Statement

Solve the given set of algebraic equations, such as 5(3x + 1) + 11 = 16, 2(x + 3) - 12 = -26, etc.

Solution

Let's solve each of the equations from the given exercise step by step:


a) 5(3x+1)+11=165(3x + 1) + 11 = 16

  1. Expand:
    15x+5+11=1615x + 5 + 11 = 16
    15x+16=1615x + 16 = 16

  2. Simplify:
    15x=015x = 0

  3. Solve:
    x=0x = 0


b) 2(x+3)12=262(x + 3) - 12 = -26

  1. Expand:
    2x+612=262x + 6 - 12 = -26
    2x6=262x - 6 = -26

  2. Simplify:
    2x=202x = -20

  3. Solve:
    x=10x = -10


c) 3(4x+8)=3x63(4x + 8) = -3x - 6

  1. Expand:
    12x+24=3x612x + 24 = -3x - 6

  2. Combine terms:
    12x+3x=62412x + 3x = -6 - 24
    15x=3015x = -30

  3. Solve:
    x=2x = -2


d) 4(x+7)=x174(-x + 7) = x - 17

  1. Expand:
    4x+28=x17-4x + 28 = x - 17

  2. Combine terms:
    4xx=1728-4x - x = -17 - 28
    5x=45-5x = -45

  3. Solve:
    x=9x = 9


e) y32=7(y8)y - 32 = 7(-y - 8)

  1. Expand:
    y32=7y56y - 32 = -7y - 56

  2. Combine terms:
    y+7y=56+32y + 7y = -56 + 32
    8y=248y = -24

  3. Solve:
    y=3y = -3


f) 2y+5=5(y+1)+122y + 5 = 5(y + 1) + 12

  1. Expand:
    2y+5=5y+5+122y + 5 = 5y + 5 + 12
    2y+5=5y+172y + 5 = 5y + 17

  2. Combine terms:
    2y5y=1752y - 5y = 17 - 5
    3y=12-3y = 12

  3. Solve:
    y=4y = -4


g) 4z+3=(3z+1)2z4z + 3 = (3z + 1) \cdot 2 - z

  1. Expand the right-hand side:
    4z+3=6z+2z4z + 3 = 6z + 2 - z
    4z+3=5z+24z + 3 = 5z + 2

  2. Combine terms:
    4z5z=234z - 5z = 2 - 3
    z=1-z = -1

  3. Solve:
    z=1z = 1


h) (3z+12)5+z=68(3z + 12) \cdot 5 + z = 68

  1. Expand the left-hand side:
    15z+60+z=6815z + 60 + z = 68
    16z+60=6816z + 60 = 68

  2. Simplify:
    16z=816z = 8

  3. Solve:
    z=12z = \frac{1}{2}


Would you like a detailed explanation of any of these steps?

Here are 5 related questions to further your learning:

  1. Can you solve a quadratic equation like ax2+bx+c=0ax^2 + bx + c = 0?
  2. How do you handle equations with fractions, e.g., x+12=3\frac{x+1}{2} = 3?
  3. Can you solve systems of equations, such as two equations with two variables?
  4. How do you check if your solution satisfies the original equation?
  5. Can you solve equations involving absolute values, e.g., x+3=7|x+3| = 7?

Tip: Always simplify the equation step-by-step and combine like terms to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Simplification
Algebraic Operations

Formulas

Distributive Property: a(b + c) = ab + ac
Combining Like Terms

Theorems

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Suitable Grade Level

Grades 8-10