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Difficulty: HardSolving One-Variable Linear Equations

A network engineer is configuring a load balancer to distribute traffic across two servers. The equation below models the traffic distribution, where tt represents the number of active connections:

5(2t−4)−3t=4(t+2)−(t−8)5(2t - 4) - 3t = 4(t + 2) - (t - 8)

Calculate the required number of active connections, tt, to successfully balance the server loads.

Answer: 9 connections

Answer

9
By successfully applying the distributive property on both sides, combining the variable and constant terms, and isolating the variable, the equation perfectly balances when the variable is exactly 9.

Step-by-Step Solution

1
Distribute the coefficients and negative signs across the parentheses on both sides of the equation.
10t−20−3t=4t+8−t+810t - 20 - 3t = 4t + 8 - t + 8
Distribution removes the parentheses, allowing individual terms to be combined.
2
Combine like terms on each side of the equation.
7t−20=3t+167t - 20 = 3t + 16
Simplifying each side individually makes it easier to balance the equation.
3
Subtract 3t3t from both sides of the equation.
4t−20=164t - 20 = 16
This groups all the variable terms on one side of the equation.
4
Add 20 to both sides and divide by 4.
4t=36  ⟹  t=94t = 36 \implies t = 9
Isolating the variable determines the final numerical solution.

Key Concept

Solving multi-step linear equations involving the distributive property and combining like terms.
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Topics

Solving One-Variable Linear Equations

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