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Cross-multiplication is a helpful technique used to solve equations involving two fractions set equal to each other. It simplifies the process of finding the unknown variable by eliminating the fractions early in the solution. In cross-multiplication, you multiply the numerator of one fraction by the denominator of the other fraction and set them equal.

Here's how it works, step by step:

• Identify the two fractions on either side of the equality sign.
• Take the numerator of the first fraction and multiply it with the denominator of the second fraction.
• Do the same with the numerator of the second fraction and the denominator of the first fraction.

These products are then equal to each other, forming a simplified equation that can be solved using algebraic methods.

In our exercise, starting with \(\frac{x-2}{x} = \frac{x}{x+3}\), cross-multiplication transforms it to \(x(x+3) = x(x-2)\).

Algebraic expressions are mathematical phrases that contain numbers, variables, and operation signs. They are crucial in forming equations and allow us to generalize mathematical rules. In solving problems, you often need to simplify and manipulate these expressions to isolate variables.

Working with algebraic expressions involves:

• Understanding the order of operations when simplifying.
• Combining like terms which have the same variables raised to the same power.
• Expanding or factoring expressions to simplify operations.

In our example, once the equation is simplified by cross-multiplication, we expand the algebraic expressions: \(x(x+3) = x(x-2)\) becomes \(x^2 + 3x = x^2 - 2x\). Here, \(x^2\) is a common term on both sides, which helps in further simplification.

Solving linear equations is about finding the value of the unknown variable. This often involves simplifying expressions, using techniques like adding, subtracting, multiplying, or dividing both sides of the equation. The goal is to isolate the variable on one side with a coefficient of 1.

Steps to solve a linear equation can include:

• Eliminating any fractions via methods like cross-multiplication.
• Combining like terms and simplifying each side of the equation.
• Isolating the variable by moving terms from one side to the other.

In our example, once we've expanded the expressions, we can subtract \(x^2\) from both sides to simplify: \(3x + 2x = 0\), which when combined gives \(5x = 0\). To find \(x\), you simply divide both sides by 5, leading to \(x = 0\). This final step showcases how linear equations can often be solved by straightforward arithmetic operations when simplified enough.