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The Distributive Property in algebra is a core concept that allows us to simplify expressions and is especially useful when dealing with expressions involving parentheses. Imagine you have an expression like \(2(6x - 1)\). Here, the number 2 is outside the parentheses, and it needs to multiply each term within the parentheses.

• First, we distribute the 2 to each term inside the parentheses: \(2 \cdot 6x\) and \(2 \cdot -1\).
• The result is \(12x - 2\).

The distributive property isn't only for positive numbers. In the expression \(-(x-7)\), you distribute a negative sign as \(-1\). Multiply \(-1\) by each term inside, giving \(-x + 7\). Combining these gives a clearer, simplified expression that's easier to work with in subsequent steps.

Once distribution has been applied, the next step in simplification is to combine like terms. Like terms are terms that have the same variable raised to the same power. They can be easily added or subtracted because they represent the same kind of thing.

In our example, after using the distributive property, we have: \(12x - 2 - x + 7\). Here:

• The terms \(12x\) and \(-x\) are like terms because both include the variable \(x\).
• The terms \(-2\) and \(+7\) are like terms because both are constants with no variables attached.

Combine the \(x\) terms by doing the arithmetic: \(12x - x = 11x\). Combine the constants: \(-2 + 7 = 5\). Hence, the expression simplifies to \(11x + 5\), which is much neater and easier to interpret.

A linear expression is an algebraic expression where the variables are raised to the power of one. These expressions are fundamental in algebra because they form the basis for more complex equations and functions.

The final simplified version of our exercise, \(11x + 5\), is a prime example of a linear expression. The variable \(x\) is to the first power, making it linear. Linear expressions like \(11x + 5\) are straightforward, with only one variable term and constant term, making them easy to plot on a coordinate graph as a straight line.

• Linear expressions appear in equations of the form \(ax + b = c\).
• They help in understanding constant change and relationships between quantities.

Grasping linear expressions is crucial, as they lay the groundwork for moving into systems of equations and algebraic functions, enhancing a student’s understanding of mathematics.