Question

write a system of equations that represents the situation. let y represent the total amount of money, in dollars, in each persons savings account in terms of the number of weeks, x, that he or she places money in the account. then solve for the indicated amounts. kai has $65 in her savings account. her friend emma has $119 in her savings account. kai plans to add $23 per week to her account. emma plans to add $14 per week. enter the answer in each space provided. use numbers instead of words. equation kais balance emmas balance

Explanation:

Step1: Define Kai's balance equation

Kai starts with $65 and adds $14 weekly. The total balance $y$ after $x$ weeks is initial amount plus weekly deposit times weeks:
$y = 14x + 65$

Step2: Define Emma's balance equation

Emma starts with $119 and adds $23 weekly. The total balance $y$ after $x$ weeks is initial amount plus weekly deposit times weeks:
$y = 23x + 119$

Step3: Solve for equal balances

Set the two equations equal to find when their balances are the same:
$14x + 65 = 23x + 119$

Step4: Rearrange to solve for $x$

Subtract $14x$ and $119$ from both sides:
$65 - 119 = 23x - 14x$
$-54 = 9x$
$x = \frac{-54}{9} = -6$

Step5: Analyze the result

A negative number of weeks means their balances were equal 6 weeks before the starting point. To find their current (week 0) balances:
Kai: $y = 14(0) + 65 = 65$
Emma: $y = 23(0) + 119 = 119$

Answer:

Kai's Balance Equation: $y = 14x + 65$
Emma's Balance Equation: $y = 23x + 119$
Their balances were equal 6 weeks before the given starting point. At the current starting week (x=0), Kai has $65 and Emma has $119.