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Difficulty: MediumSolving Systems of Linear Equations

A motor pool contains a total of 24 vehicles, consisting exclusively of transport trucks and utility vehicles. The number of transport trucks is 6 less than twice the number of utility vehicles. How many utility vehicles are in the motor pool?

Answer: 10 / ten / 10 utility vehicles

Answer

10
By defining tt as transport trucks and uu as utility vehicles, we can create the system t+u=24t + u = 24 and t=2u−6t = 2u - 6. Substituting the value of tt into the first equation yields (2u−6)+u=24(2u - 6) + u = 24. Combining the terms gives 3u−6=243u - 6 = 24. Adding 6 to both sides results in 3u=303u = 30, and dividing by 3 shows there are 1010 utility vehicles.

Step-by-Step Solution

1
Define variables for the unknown quantities.
Let tt be the number of transport trucks and uu be the number of utility vehicles.
This allows us to translate the word problem into a system of algebraic equations.
2
Set up the system of equations based on the problem statement.
Equation 1: t+u=24t + u = 24 (total vehicles).
Equation 2: t=2u−6t = 2u - 6 (transport trucks are 6 less than twice the utility vehicles).
These equations accurately represent the given conditions in mathematical terms.
3
Substitute the expression for tt from Equation 2 into Equation 1.
(2u−6)+u=24(2u - 6) + u = 24
Substitution eliminates the variable tt, leaving a single-variable equation that can be solved directly for uu.
4
Combine like terms and solve for uu.
3u−6=24⇒3u=30⇒u=103u - 6 = 24 \Rightarrow 3u = 30 \Rightarrow u = 10
Isolating uu gives the exact number of utility vehicles requested by the problem.

Key Concept

Solving Systems of Linear Equations Using Substitution

Alternative Method

This can also be solved by elimination if we rearrange the second equation to −t+2u=6-t + 2u = 6. Adding this to t+u=24t + u = 24 yields 3u=303u = 30, which gives u=10u = 10.
Estimated Time:1m 0s
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Topics

Solving Systems of Linear Equations

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