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Cross-multiplication is a technique used to solve proportions or equations that involve fractions. When you have an equation in the form \(\frac{a}{b} = \frac{c}{d}\), cross-multiplication involves multiplying the numerator of one side by the denominator of the other side. This helps eliminate the fractions, simplifying the equation.

Here's how it works: multiply \(a\) from the left side by \(d\) from the right side to get \(a \cdot d\). Then, multiply \(b\) by \(c\) to get \(b \cdot c\). Set these two products equal to each other:

• \(a \cdot d = b \cdot c\)

This transformation turns the equation into one that is easier to solve without fractions. It simplifies the calculations significantly, especially in cases like the exercise example where \(4 \times 7 = x \times 2\). Solving this is much simpler than dealing with fractions directly.

In summary, cross-multiplication is a reliable and straightforward method for solving proportions that helps break down complex problems into simpler arithmetic.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. They form the basis for constructing equations, which we often need to solve. In the given problem, the expression \(\frac{4}{x}\) represents a fraction where \(4\) is constant and \(x\) is the variable.

Algebraic expressions are versatile.
They allow us to express relationships between different quantities. In the example, the equality of \(\frac{4}{x}\) and \(\frac{2}{7}\) sets a relationship between these quantities.

• The variable \(x\) stands for an unknown value that we need to find.
• Constants like \(4\) and \(7\) are numbers that have fixed values.

By manipulating algebraic expressions, we can simplify and solve equations, which helps further in determining unknowns. Understanding how to work with algebraic expressions is crucial as it paves the way for effective problem-solving in algebra.

Equation solving is the process of finding the value of the variables that make an equation true. It usually involves a series of logical steps.

For example: in the equation \(4 \times 7 = x \times 2\), our goal is to isolate \(x\) on one side of the equation. Here's a simplified approach to solving this equation:

• Start with the equation: \(28 = 2x\), derived from \(4 \times 7\).
• Next, divide both sides by \(2\) to isolate \(x\): \(x = \frac{28}{2}\).
• Finally, simplify the equation to get \(x = 14\).

Once \(x\) is determined, you can substitute it back into any related expression, such as \(2x\), to find derived values.

Equation solving is a fundamental skill in mathematics, enabling us to find unknown values and solve real-world problems. By systematically using operations like addition, subtraction, multiplication, and division, we can tackle a wide range of mathematical challenges.