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Create an equation that contains:

at least one variable
at least one coefficient
at least one constant
at least two terms
at least two expressions
Your equation also must utilize at least one of the following properties:

Commutative Property
Associative Property
Distributive Property
Then, write-out a situation (CREATE A WORD PROBLEM) that would be described by the equation.

7 answers

Equation

Let's create the equation:

\[ 3x + 5 = 2(x + 1) \]

Components:

  • Variable: \( x \)
  • Coefficient: \( 3 \) and \( 2 \)
  • Constant: \( 5 \) and \( 1 \)
  • Terms: \( 3x \), \( 5 \), \( 2(x + 1) \) (which simplifies to \( 2x + 2 \))
  • Expressions: \( 3x + 5 \) and \( 2(x + 1) \)

Property Used

The equation utilizes the Distributive Property, where \( 2 \) is distributed to both \( x \) and \( 1 \) in \( 2(x + 1) \).

Word Problem

Maria is trying to save money to buy a new bike. She currently has \( 5 \) dollars saved but wants to add more over the next few weeks. She plans to save \( 3 \) dollars per week.

Meanwhile, her friend John also wants to save money for the same bike. He plans to save \( 2 \) dollars each week and has \( 1 \) extra dollar as a bonus for his birthday.

After \( x \) weeks, both Maria’s total savings and John’s total savings will be equal.

Write an equation to figure out how many weeks it will take for their total savings to be the same.

all in one and simple

Equation

\[ 3x + 5 = 2(x + 1) \]

Word Problem

Maria has saved $5 and plans to save $3 each week. Her friend John has $1 and saves $2 each week. Write an equation to find out how many weeks (\(x\)) it will take for their savings to be equal.

5(a+7)−2a+10(a−3)=2(4a+5)
Identify the following parts of the equation: the expression(s), variable(s), coefficient(s), constant(s), factor(s) and terms.
SOLVE the equation.
While solving, identify if and when you use the following properties: Commutative Properties, Associative Properties, Distributive Property

Let's analyze the equation step by step:

Given Equation

\[ 5(a + 7) - 2a + 10(a - 3) = 2(4a + 5) \]

Identifying Different Parts of the Equation

  • Expressions:

    • \(5(a + 7)\)
    • \(-2a\)
    • \(10(a - 3)\)
    • \(2(4a + 5)\)

  • Variable(s):

    • \(a\)

  • Coefficient(s):

    • \(5\) (from \(5(a + 7)\))
    • \(-2\) (from \(-2a\))
    • \(10\) (from \(10(a - 3)\))
    • \(2\) (from \(2(4a + 5)\))

  • Constant(s):

    • \(7\) (from \(5(a + 7)\))
    • \(-3\) (from \(10(a - 3)\))
    • \(5\) (from \(2(4a + 5)\))

  • Factor(s):

    • \(a + 7\) (from \(5(a + 7)\))
    • \(a - 3\) (from \(10(a - 3)\))
    • \(4a + 5\) (from \(2(4a + 5)\))

  • Terms:

    • \(5(a + 7)\)
    • \(-2a\)
    • \(10(a - 3)\)
    • \(2(4a + 5)\)

Solving the Equation

  1. Distributive Property: Apply the distributive property to expand both sides.

    \[ 5(a + 7) = 5a + 35 \] \[ 10(a - 3) = 10a - 30 \] \[ 2(4a + 5) = 8a + 10 \]

  2. Substitute these expansions back into the equation:

    \[ 5a + 35 - 2a + 10a - 30 = 8a + 10 \]

  3. Combine Like Terms: On the left side, we combine \(5a\), \(-2a\), and \(10a\):

    \[ (5a - 2a + 10a) + (35 - 30) = 8a + 10 \]

    Simplifying gives:

    \[ 13a + 5 = 8a + 10 \]

  4. Transposing Terms: Use the properties here:

    • Commutative Property: We can reorder the terms when moving \(8a\) to the left side.

    \[ 13a - 8a + 5 = 10 \]

    This simplifies to:

    \[ 5a + 5 = 10 \]

  5. Subtract 5 from both sides:

    \[ 5a = 5 \]

  6. Divide by 5:

    \[ a = 1 \]

Summary of Properties Used

  • Distributive Property: Used to expand the terms on both sides of the equation.
  • Commutative Property: Assumed when rearranging terms during the simplification.
  • Associative Property: Not explicitly needed in this example as we didn't change groupings of operations.

Solution

The value of \(a\) that satisfies the equation is:

\[ a = 1 \]

all in one simple thing

Equation

\[ 5(a + 7) - 2a + 10(a - 3) = 2(4a + 5) \]

Identify Parts

  • Expressions: \(5(a + 7)\), \(-2a\), \(10(a - 3)\), \(2(4a + 5)\)
  • Variable: \(a\)
  • Coefficients: \(5\), \(-2\), \(10\), \(2\)
  • Constants: \(7\), \(-3\), \(5\)
  • Factors: \(a + 7\), \(a - 3\), \(4a + 5\)
  • Terms: \(5(a + 7)\), \(-2a\), \(10(a - 3)\), \(2(4a + 5)\)

Solve the Equation

  1. Distributive Property:

    \[ 5(a + 7) \rightarrow 5a + 35 \] \[ 10(a - 3) \rightarrow 10a - 30 \] \[ 2(4a + 5) \rightarrow 8a + 10 \]

  2. Substitute:

    \[ 5a + 35 - 2a + 10a - 30 = 8a + 10 \]

  3. Combine like terms:

    \[ (5a - 2a + 10a) + (35 - 30) = 8a + 10 \rightarrow 13a + 5 = 8a + 10 \]

  4. Rearrange terms:

    \[ 13a - 8a + 5 = 10 \rightarrow 5a + 5 = 10 \]

  5. Solve for \(a\):

    \[ 5a = 5 \rightarrow a = 1 \]

Final Answer

The solution is \( a = 1 \).

Properties Used

  • Distributive Property: To expand terms.
  • Commutative Property: To rearrange terms.
  • Associative Property: Not explicitly used here.