A 58 -ft-long cable must be cut into three pieces. One piece will be 6 ft longer than the shorte

step 1 All right. In this problem, we know we have a 58 foot long cable that's being cut into three pie

A 58 -ft-long cable must be cut into three pieces. One piece...

A 58 -ft-long cable must be cut into three pieces. One piece will be $6 \mathrm{ft}$ longer than the shortest piece, and the longest portion must be twice as long as the shortest. Find the length of the three pieces of cable.

Key Concepts

Translation of Word Problems

Translating word problems into mathematical statements involves reading the problem carefully, assigning variables, and then expressing the given relationships as equations. This step is crucial because it enables the solver to methodically set up a framework that captures the problem's essential conditions and constraints.

Solution Verification

Once a solution is found, it is important to check that it satisfies all the conditions stated in the problem. Verification confirms that the derived values for the unknowns are consistent with both the original relationships and the overall constraints, such as the total length in a division problem.

Algebraic Modeling

This concept involves representing real-world problems using variables and equations. In situations like dividing a cable into segments, algebraic modeling allows you to translate information and conditions into algebraic expressions, serving as a bridge between the problem's narrative and its mathematical formulation.

System of Linear Equations

Systems of linear equations consist of multiple equations that relate the same set of variables. When a problem describes relationships between unknown quantities—such as differences or multiplicative relationships—the resulting system of equations must be solved to find the values of these unknowns. This is a critical skill in algebra for addressing problems with multiple constraints.

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