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You have been given the following equation to solve as a challenge by your friend. Create a list of steps that you could follow to find the solution. Make sure to include all steps in the correct order.

5(x+3)–7(x–4)=38
Write a real-life situation to fit the following equation. Then, solve the equation in two different ways. Show all the steps. Finally, choose which solution method you liked best and explain your reason.

8(x+9)=144

1 answer

Steps to Solve the Equation 5(x+3)–7(x–4)=38

  1. Distribute the Terms: Apply the distributive property to both groups in the equation.

    • \(5(x) + 5(3) - 7(x) + 7(4) = 38\)

  2. Expand: Write out the expanded form.

    • \(5x + 15 - 7x + 28 = 38\)

  3. Combine Like Terms: Combine the \(x\) terms and the constant terms on the left side.

    • \((5x - 7x) + (15 + 28) = 38\)
    • \(-2x + 43 = 38\)

  4. Isolate the Variable: Subtract 43 from both sides to isolate terms with \(x\).

    • \(-2x + 43 - 43 = 38 - 43\)
    • \(-2x = -5\)

  5. Solve for \(x\): Divide both sides by -2.

    • \(x = \frac{-5}{-2}\)
    • \(x = \frac{5}{2}\) or \(x = 2.5\)

Real-Life Situation for the Equation 8(x + 9) = 144

Situation: Sarah is organizing a party and needs to purchase party bags. Each party bag costs \(x\) dollars, and she wants to buy as many as 9 bags for her guests. If she spends a total of $144 on party bags, how much does each bag cost?

Solving the Equation 8(x + 9) = 144

Method 1: Distributive Property

  1. Distribute on the Left Side:

    • \(8x + 72 = 144\)

  2. Isolate the Variable: Subtract 72 from both sides.

    • \(8x + 72 - 72 = 144 - 72\)
    • \(8x = 72\)

  3. Solve for \(x\): Divide both sides by 8.

    • \(x = \frac{72}{8}\)
    • \(x = 9\)

Method 2: Solving Directly

  1. Divide Both Sides by 8: Before distributing, simplify the equation directly by dividing both sides.

    • \(\frac{8(x + 9)}{8} = \frac{144}{8}\)
    • \(x + 9 = 18\)

  2. Isolate \(x\): Subtract 9 from both sides.

    • \(x + 9 - 9 = 18 - 9\)
    • \(x = 9\)

Conclusion: Choosing the Preferred Solution Method

I prefer Method 1 (using the distributive property) because it clearly shows each algebraic step leading to the solution, making it easier to follow for someone who may be less familiar with algebra. Additionally, it reinforces the understanding of the distributive property, which is a fundamental concept in algebra. Conversely, while Method 2 is quicker and leads to the solution, it may skip over important intermediate steps that contribute to a deeper understanding of the problem-solving process.