Question

here are some equations that are related to ( x - 0.1x + 2.70 = 56.70 ). each equation is a result of performing one or more moves on that original equation. each can also be interpreted in terms of noahs purchase.
for each equation, determine either what move was made or how the equation could be interpreted. (some examples are given here.) then, check if 60 is the solution of the equation.
equation a
( 100x - 10x + 270 = 5,670 )
● what was done?
● interpretation? the price is expressed in cents instead of dollars.
● same solution?
equation b
( x - 0.1x = 54 )
● what was done? subtract 2.70 from both sides of the equation.
● interpretation?
● same solution?
equation c
( 0.9x + 2.70 = 56.70 )
● what was done?
● interpretation? 10% off means paying 90% of the original price. 90% of the original price plus sales tax is $56.70.
● same solution?

Explanation:

Equation B

• Interpretation: If the total cost (original price with some reduction + tax) was \(\$56.70\), and the tax was \(\$2.70\), then \(x - 0.1x\) represents the price of the item after a \(10\%\) discount, and subtracting the tax (\(\$2.70\)) from the total (\(\$56.70\)) gives the discounted - price amount (\(\$54\)).
• Check for \(x = 60\):
• Substitute \(x = 60\) into the left - hand side of the equation \(x-0.1x\).
• Using the distributive property \(x-0.1x=(1 - 0.1)x=0.9x\).
• When \(x = 60\), \(0.9x=0.9\times60 = 54\). Since the right - hand side of the equation \(x - 0.1x=54\) is \(54\), \(x = 60\) is a solution.

Equation C

• What was done: Combine like terms (\(x-0.1x=(1 - 0.1)x = 0.9x\)) in the original equation \(x-0.1x + 2.70=56.70\).
• Check for \(x = 60\):
• Substitute \(x = 60\) into the left - hand side of the equation \(0.9x+2.70\).
• First, calculate \(0.9x\): \(0.9\times60=54\).
• Then add \(2.70\): \(54 + 2.70=56.70\). Since the left - hand side \(0.9x + 2.70\) equals the right - hand side \(56.70\) when \(x = 60\), \(x = 60\) is a solution.

Answer:

• Equation B:
• Interpretation: The price of the item after a \(10\%\) discount (before tax) is \(\$54\).
• Same solution: Yes, \(x = 60\) is a solution.
• Equation C:
• What was done: Combine like terms (\(x-0.1x\) to get \(0.9x\)).
• Same solution: Yes, \(x = 60\) is a solution.