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Linear equations form the backbone of many algebraic problems. They are equations that make a straight line when plotted on a graph. A linear equation typically looks like this: \( ax + b = c \), where \( x \) is the variable, and \( a \), \( b \), and \( c \) are constants. In our problem, we formed the equation \( x + 4x = 150 \) based on the relationship between the cable segments.
Linear equations feature prominently in systems of equations, where you can have multiple equations working together. When solving a problem with multiple conditions, such as knowing both the total length and the proportional relationship of the lengths of cable pieces, linear equations help to organize the information neatly.

An algebraic solution involves using algebra techniques to solve equations. In our example, the algebraic solution unfolds as follows: by setting up the equation \( x + 4x = 150 \), you can solve for the unknown variable \( x \).
This involves isolating \( x \) by combining like terms to get \( 5x = 150 \).

• First, combine the like terms: \( x + 4x = 5x \).
• Next, solve for \( x \) by dividing both sides of the equation by 5: \( x = \frac{150}{5} \).

This process highlights the simplicity and power of algebra in solving problems
involving numbers and unknowns, providing clear, logical steps to follow.

Defining variables is a critical first step in solving mathematical problems.
A variable represents an unknown that you're trying to find. In this context, when we say, 'Let's denote the length of the first piece as \( x \)', we are identifying \( x \) as our variable.
This helps us simplify the problem.

• The first piece of cable is \( x \) meters long, which simplifies the equation setup.
• We express the second piece as \( 4x \), making use of the given relationship.

Defining variables allows us to break down complex real-world problems into manageable mathematical expressions that can be systematically solved.

Setting up the equation is all about translating the problem's text into a mathematical language.
Equations bring structure and provide a pathway to solutions. To set up an equation, we analyze all given relationships from the problem and formulate them mathematically.

• From the problem, the total length is 150 meters—the equation \( x + 4x = 150 \) is formulated based on this total.
• This equation considers \( x \) for the length of the first piece and \( 4x \) for the length of the second piece.
• This structure allows the equation to reflect the actual relationship between the pieces and their lengths.

Understanding equation setup simplifies problem-solving, as you translate complex scenarios into solvable mathematical forms.