Bolt & Surd Math Challenge: Step-by-Step Solutions

by Mei Lin 51 views

Hey guys! Let's dive into some math problems today. We're going to break down a tricky word problem about bolts and washers, and then tackle an equation involving surds (those square root things!). So grab your calculators and let's get started!

Question 7: Unraveling the Bolt and Washer Mystery

Our first challenge involves a trip to the hardware store! The hardware store problem states Alex bought 4 bolts and 6 washers for a total of $K 2. Now, we need to figure out the cost of each individual bolt. This sounds like a classic system of equations problem, but there's a slight catch. We only have one equation! That's right, with only one total cost given, we can't definitively find a unique price for both bolts and washers. There will be infinitely many possible combinations, but we can at least lay out the math and discuss the limitations.

Let's break it down step by step:

  1. Define our variables:

    • Let 'b' represent the cost of one bolt.
    • Let 'w' represent the cost of one washer.

  2. Formulate the equation:

    We know that 4 bolts plus 6 washers cost $K 2. We can write this as:

    4b + 6w = 2

    (Note: For clarity, I will assume K means the same units as the number 2, so it is just a multiplier, and the units would be dollars, which is implied by the context. If not, we need information on the units of K)

    This is our one and only equation. Because the prompt only provides one piece of information (the total cost), we have a single equation with two unknowns. This type of problem in mathematics is called an underdetermined system. Imagine trying to find two specific points on a map, but only being given one clue – like the distance between them. There are countless possible locations!

  3. The limitations of the equation:

    Unfortunately, we can't solve for unique values of 'b' and 'w' with just this single equation. We need another piece of information, like the cost of the washers or another purchase scenario (e.g., Alex bought 2 bolts and 3 washers for $1). Without that second equation, we can only express one variable in terms of the other.

  4. Expressing 'b' in terms of 'w' (or vice-versa):

    Let's solve for 'b' to see how the cost of a bolt depends on the cost of a washer:

    4b = 2 - 6w

    b = (2 - 6w) / 4

    b = 0.5 - 1.5w

    This equation tells us that the cost of a bolt ('b') is equal to 0.5 minus 1.5 times the cost of a washer ('w'). So, if we knew the price of a washer, we could plug it into this equation and find the corresponding price of a bolt. For example:

    • If a washer costs $0.10 (w = 0.10):

      b = 0.5 - 1.5(0.10)

      b = 0.5 - 0.15

      b = $0.35

    • If a washer costs $0.20 (w = 0.20):

      b = 0.5 - 1.5(0.20)

      b = 0.5 - 0.30

      b = $0.20

    Notice how the price of the bolt changes depending on the price of the washer. This highlights the infinite solutions possible with just one equation.

  5. What if we tried to solve it anyway?

    If we tried to use a calculator to