1Cademy - Solving a Catch-Up Motion Problem Using a System of Equations by Substitution

Learn Before

Problem-Solving Strategy for Systems of Linear Equations

Example

Solving a Catch-Up Motion Problem Using a System of Equations by Substitution

Apply the seven-step problem-solving strategy for systems of linear equations to a uniform motion catch-up scenario, using two variables and substitution to solve the resulting system.

Problem: Joni left St. Louis on the interstate, driving west towards Denver at $65$ miles per hour. Half an hour later, Kelly left St. Louis on the same route, driving $78$ miles per hour. How long will it take Kelly to catch up to Joni?

Read the problem and draw a diagram showing both drivers traveling the same route from St. Louis toward Denver, with Joni at $65$ mph and Kelly (leaving $\frac{1}{2}$ hour later) at $78$ mph. Create a rate–time–distance table.
Identify: The travel time for each driver.
Name: Let $j$ = Joni's driving time (in hours) and $k$ = Kelly's driving time (in hours). Both rates are known, so multiply rate by time to fill in the distance column:

	Rate (mph)	Time (hrs)	Distance (miles)
Joni	$65$	$j$	$65j$
Kelly	$78$	$k$	$78k$

Translate into a system of equations. Kelly catches Joni when they have traveled the same distance: $65j = 78k$ . Since Kelly left $\frac{1}{2}$ hour later, her time is $\frac{1}{2}$ hour less than Joni's: $k = j - \frac{1}{2}$ . The system is:

$\left\{\begin{array}{l} k = j - \frac{1}{2} \\ 65j = 78k \end{array}\right.$

Solve by substitution. Substitute $k = j - \frac{1}{2}$ into the second equation:

$65j = 78\left(j - \frac{1}{2}\right)$

Distribute: $65j = 78j - 39$ . Subtract $78j$ from both sides: $-13j = -39$ . Divide by $-13$ : $j = 3$ .

Substitute $j = 3$ into the first equation: $k = 3 - \frac{1}{2} = \frac{5}{2} = 2\frac{1}{2}$ .

Check: Joni: $65 \times 3 = 195$ miles. Kelly: $78 \times 2\frac{1}{2} = 195$ miles. Both distances are equal. $\checkmark$
Answer: Kelly will catch up to Joni in $2\frac{1}{2}$ hours. By then, Joni will have been driving for $3$ hours.

This example demonstrates the catch-up scenario solved with a system of two equations in two variables. Two key observations produce the system: (1) when the faster traveler catches the slower one, both have covered the same distance, giving the equal-distance equation $65j = 78k$ ; and (2) the later departure means the faster traveler's time is shorter by the delay, giving the time-relationship equation $k = j - \frac{1}{2}$ . Because the first equation is already solved for $k$ , substitution is the natural method. Earlier uniform motion examples expressed both unknowns through a single variable, but here each traveler receives its own variable, making the setup more direct.

Updated 2026-04-24

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After