An executive of Trident Com- munications recently traveled...

An executive of Trident Com- munications recently traveled to London, Paris, and Rome. He paid $$\$ 180$$, $$\$ 230$$, and $$\$ 160$$ per night for lodging in London, Paris, and Rome, respectively, and his hotel bills totaled $$\$ 2660 .$$ He spent $$\$ 110$$, $$\$ 120$$, and $$\$ 90$$ per day for his meals in London, Paris, and Rome, respectively, and his expenses for meals totaled $$\$ 1520 .$$ If he spent as many days in London as he did in Paris and Rome combined, how many days did he stay in each city?

Key Concepts

Constraint Relationships

In many word problems, additional constraints tie the variables together, reducing the number of independent variables. These constraints, such as relating the number of days in one location to the combined days in other locations, are essential for accurately setting up the system of equations and finding a unique solution.

Algebraic Solution Methods

Techniques such as substitution and elimination are used to solve systems of linear equations. These methods allow one to reduce the system step-by-step by expressing one variable in terms of others and substituting back into the equations, ultimately determining the numerical values of all the unknowns.

System of Linear Equations

A system of linear equations involves multiple equations that capture the relationships among unknown quantities. In problems like these, each condition (such as total expenses or specific cost rates) can be modeled as a linear equation, and the solution to the problem is found by determining the values of the unknowns that satisfy all these equations simultaneously.

Variable Assignment and Translation

Assigning variables to unknown quantities is a critical first step in translating a word problem into a mathematical form. This process involves identifying the key quantities (such as the number of days stayed or individual expenditures) and expressing the problem’s conditions as algebraic equations in terms of these variables.

Transcript

00:01 Here we're finding the cost of business travel for new york city and washington, d .c.

00:07 So for this problem, x is going to represent new york city, y is going to represent washington, d .c.

00:14 We're told that two nights in new york city, so 2x, plus three nights in washington, d .c.

00:24 Equals a total of $2 ,772.

00:27 We're also told that four nights in new york city, $1 ,000.

00:31 New york city and two nights in washington dc would equal $3 ,488.

00:39 So we want to find what the cost for each city would be.

00:43 In order to do that, we're going to start with one of these equations and solve for either x or y.

00:48 For this one, we're going to use this second equation and we're going to solve for y.

00:52 So we're going to start by subtracting 4x on both sides.

01:00 And so we're going to have 2y equals negative 4x.

01:05 Plus 3 ,488 and we're going to divide by 2 and when i do i'm left with y equals negative 2x plus 1 ,744.

01:24 Now we know that y equals this we're going to put that in place of y in the first equation so we're going to come back to the first equation we're going to have 2x plus 3 and in place of y we're going to use this equation right here so negative 2x plus 1 ,744 equals 2 ,772.

01:49 We're going to solve to find what x is, and we're going to have 2x.

01:53 3 times negative 2x is negative 6x, and 3 times 1 ,744 is 5 ,232...

Please give Ace some feedback