Question

example 6: ian is borrowing $1000 from his parents to buy a notebook computer. he plans to pay them back at the rate of $60 per month. ken is borrowing $600 from his parents to purchase a snowboard. he plans to pay his parents back at the rate of $20 per month.
a. write an equation a to present ians situation.
y = 60x+1000
b. write an equation b to represent kens situation.
y = 20x+600
c. determine algebraically and state in how many months the two boys will owe the same amount.
d. ian claims that he will have his loan paid off 6 months after he and ken owe the same amount. determine and state if ian is correct. explain your reasoning.

Explanation:

Step1: Set equations equal for part c

Set the two - equations equal to find when they owe the same amount. Let $y$ be the amount owed and $x$ be the number of months. We have $60x + 1000=20x + 600$.

Step2: Rearrange terms

Subtract $20x$ from both sides: $60x-20x + 1000=20x-20x + 600$, which simplifies to $40x+1000 = 600$. Then subtract 1000 from both sides: $40x+1000 - 1000=600 - 1000$, getting $40x=-400$.

Step3: Solve for $x$

Divide both sides by 40: $x=\frac{-400}{40}=-10$. But since the number of months can't be negative, there is an error. The correct equations should be $y = 1000-60x$ (amount Ian owes after $x$ months) and $y = 600 - 20x$ (amount Ken owes after $x$ months). Setting them equal: $1000-60x=600 - 20x$. Add $60x$ to both sides: $1000=600 - 20x+60x$, so $1000=600 + 40x$. Subtract 600 from both sides: $400 = 40x$. Then $x = 10$ months.

Step4: Analyze Ian's claim in part d

Ian's loan equation is $y = 1000-60x$. When the loan is paid off, $y = 0$. So $0=1000-60x$, then $60x=1000$, and $x=\frac{1000}{60}=\frac{50}{3}\approx16.67$ months. Ken's loan equation is $y = 600 - 20x$. When the loan is paid off, $y = 0$, so $0=600 - 20x$, then $20x=600$ and $x = 30$ months. The difference is $30-\frac{50}{3}=\frac{90 - 50}{3}=\frac{40}{3}\approx13.33
eq6$. So Ian is incorrect.

Answer:

a. $y = 1000-60x$
b. $y = 600 - 20x$
c. 10 months
d. Ian is incorrect.