Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Vicky and Diana each opened a savings account on the same day. Vicky started by putting $\$404$ in her account, and she will deposit an additional $\$188$ each week. Diana made an initial deposit of $\$442$ , and she will add $\$186$ more each week. Eventually, Vicky and Diana will each have the same amount saved. What is that amount? How many weeks will that take? $\newline$ Vicky and Diana will each have saved a total of $\$$ _ after _ weeks.

Full solution

Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Vicky and Diana each opened a savings account on the same day. Vicky started by putting

\$404

in her account, and she will deposit an additional

\$188

each week. Diana made an initial deposit of

\$442

, and she will add

\$186

more each week. Eventually, Vicky and Diana will each have the same amount saved. What is that amount? How many weeks will that take?

\newline

Vicky and Diana will each have saved a total of

\$

_____ after _____ weeks.

Define Equations: Step $1$ : Define the equations based on the problem statement. $\newline$ Vicky's savings after $t$ weeks: $V(t) = 404 + 188t$ $\newline$ Diana's savings after $t$ weeks: $D(t) = 442 + 186t$ $\newline$ We need to find when $V(t) = D(t)$ .
Set Up Equation: Step $2$ : Set up the equation to find when their savings are equal. $\newline$ $404 + 188t = 442 + 186t$ $\newline$ Subtract $186t$ from both sides: $188t - 186t = 442 - 404$ $\newline$ Simplify: $2t = 38$
Solve for t: Step $3$ : Solve for t. $\newline$ Divide both sides by $2$ : $t = \frac{38}{2}$ $\newline$ $t = 19$ weeks.
Substitute and Calculate: Step $4$ : Substitute $t$ back into either $V(t)$ or $D(t)$ to find the amount saved. $\newline$ Using $V(t)$ : $V(19) = 404 + 188 \times 19$ $\newline$ Calculate: $V(19) = 404 + 3572$ $\newline$ $V(19) = 3976$ dollars.