Hugo decided to pay $2 a week to his brother to buy a bicycle. The equation $y-10=-2(x-10)$ models the amount of money he owes for the bike, where $x$ is the number of weeks and $y$ is the amount of money he still needs to pay.

How much did the bicycle cost?
After how many weeks will Hugo finish paying for the bike?
Graph the equation: $y-10=-2(x-10)$

Question

06-11-2026
Mathematics

Answered

Hugo decided to pay $2 a week to his brother to buy a bicycle. The equation $y-10=-2(x-10)$ models the amount of money he owes for the bike, where $x$ is the number of weeks and $y$ is the amount of money he still needs to pay.

How much did the bicycle cost?
After how many weeks will Hugo finish paying for the bike?
Graph the equation: $y-10=-2(x-10)$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T08:20:47-05:00

The linear model is expressed as $y = -2x + 30$.
Setting $x = 0$, the initial cost of the bicycle is found to be $\boxed{\$30}$.
Setting $y = 0$, the time to pay off the debt is calculated as $\boxed{15}$ weeks.

Explanation

Analysis of the Linear Model

The problem provides a linear model for the amount of money owed for a bicycle. The equation is given as $y - 10 = -2(x - 10)$, where $y$ represents the amount of money owed in dollars and $x$ represents the number of weeks elapsed. We are tasked with determining two values: the total cost of the bicycle (which corresponds to the amount owed at $x=0$) and the number of weeks required for Hugo to finish paying off the debt (which corresponds to the time when the debt $y=0$).

Simplifying the Equation

To make the equation easier to interpret, we convert it into the slope-intercept form, $y = mx + b$, where $m$ is the rate of payment and $b$ is the initial cost. Starting with $y - 10 = -2(x - 10)$, we distribute the $-2$:
\[y - 10 = -2x + 20\]
Adding 10 to both sides yields:
\[y = -2x + 30\]
In this form, we can clearly see the initial value is $30$ and the weekly reduction is $2$.

Calculating the Initial Cost

To find the initial cost of the bicycle, we evaluate the amount owed $y$ at the start, which occurs at $x = 0$ weeks. Substituting $x = 0$ into our simplified equation $y = -2x + 30$:
\[y = -2(0) + 30 = 30\]
Therefore, the bicycle cost $\$30$.

Determining the Payoff Duration

To find out when the debt is fully paid, we need to find the number of weeks $x$ such that the amount remaining to be paid $y$ is $0$. Setting $y = 0$ in our equation $y = -2x + 30$:
\[0 = -2x + 30\]
Rearranging to solve for $x$:
\[2x = 30\]
\[x = 15\]
Hugo will finish paying for the bike in 15 weeks.

Examples

This linear model is a fundamental tool for financial planning, such as budgeting for a purchase or managing a loan. By modeling debt as a linear function $y = mx + b$, individuals can easily predict when their remaining balance will reach zero or identify how much they started with. Such models are essential for tracking savings progress, calculating time-to-completion for projects, or managing installments in everyday life.