Problem 123 - Elementary Algebra

Question

Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system \(\left\\{\begin{array}{l}56 s=70 t \\\ s=t+\frac{1}{2}\end{array}\right.\) (a) for \(t\) to find out how long it will take Tina to catch up to Stephanie. (b) what is the value of \(s,\) the number of hours Stephanie will have driven before Tina catches up to her?

Step-by-Step Solution

Answer

Tina will catch up to Stephanie in 2 hours. Stephanie will have driven for 2.5 hours.

Step 1: Express t from the second equation

Start with the second equation given:the equation is \[ s = t + \frac{1}{2} \]Express this as: \(t = s - \frac{1}{2}\)

Step 2: Substitute t into the first equation

Then, substitute this value of \(t\) into the first equation:\[56s = 70(s - \frac{1}{2})\]

Step 3: Simplify the equation

Next, simplify the equation to solve for \(s\):\[56s = 70s - 35\]

Step 4: Solve for s

Rearrange the equation:\[56s - 70s = -35\]\[-14s = -35\]\[s = \frac{35}{14} = 2.5\]

Step 5: Solve for t

Finally, substitute \(s\) back into the equation for \(t\):\[t = 2.5 - \frac{1}{2} = 2\]

Key Concepts

Distance, Rate, and Time

Distance, rate, and time are key concepts in motion problems. The relationship between them can be described by the formula: distance = rate × time. In this problem, Stephanie and Tina travel the same route, but at different speeds and starting times. To understand who catches up when, let's break it down. Stephanie travels at 56 miles per hour, while Tina travels at 70 miles per hour, but starts 30 minutes later (half an hour). Knowing these rates and the elapsed time, we can set up equations to solve their exact meeting point. Using the above formula helps us understand how far each traveled in a certain time, and then use this to understand how long it will take for their distances to meet.

Substitution Method

The substitution method is a common algebraic technique for solving systems of linear equations. It involves solving one equation for one variable and substituting that expression into another equation. Here, we have the system: \[ \begin{array}{l} 56 s = 70 t \ s = t + \frac{1}{2} \end{array} \] We start by expressing one variable in terms of the other. First, solve the second equation for t: \[t = s - \frac{1}{2}\]. This equation tells us the time difference between Stephanie and Tina. Then, substitute t into the first equation and solve for s: \[ 56s = 70(s - \frac{1}{2}) \ 56s = 70s - 35 \]. After simplifying, we get: \[ -14s = -35 \ s = \frac{35}{14} = 2.5 \]. This means it takes Stephanie 2.5 hours before Tina catches up.

Solving Linear Equations

Solving linear equations involves finding the values of variables that satisfy given linear relationships. For our question, this process follows through several steps:
First, arranging the equation correctly.
Rewriting the initial system, we have two equations: \[s = t + \frac{1}{2} \]. We substitute this into the second equation: \[56s = 70(s - \frac{1}{2}) \], and simplify it as follows:
Simplify and combine like terms:
\[ 56s = 70s - 35 \ 56s - 70s = -35 \ -14s = -35 \]. Solving for s, we get \[ s = \frac{35}{14} = 2.5 \].
Plugging back to find t, the time for Tina is:
\(t = 2.5 - \frac{1}{2} = 2\).
Thus, Tina takes 2 hours to catch up with Stephanie, who drives for 2.5 hours.