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Cross-multiplication is a handy method used to eliminate fractions in equations, especially when dealing with proportions. A proportion is essentially a statement that two ratios are equal, like \( \frac{2x+10}{6} = \frac{14}{3} \).
Cross-multiplication helps simplify these proportions by allowing you to work with whole numbers. The method involves multiplying the numerator (the top part of a fraction) of one ratio by the denominator (the bottom part) of the other ratio. This process is repeated with the opposite numerator and denominator.
In the given example, cross-multiplying involves two steps:

• Multiply \((2x + 10)\) by 3.
• Multiply 14 by 6.

This results in the equation \(3(2x + 10) = 6 \times 14\), providing an equation without fractions that's easier to solve. By performing cross-multiplication, you're setting the stage for neatly simplifying and solving the equation.

Once we have removed fractions through cross-multiplication, we need to solve the equation to find the value of the unknown variable, \(x\).
The equation from our example becomes \(6x + 30 = 84\). Solving an equation generally involves two main steps:

• Isolating the variable on one side of the equation.
• Simplifying to solve for the variable.

To isolate \(x\), the first step is to move any numbers not attached to \(x\) to the other side of the equation. This involves subtracting 30 from both sides, resulting in \(6x = 54\).
Next, to isolate \(x\) itself, divide both sides by the coefficient of \(x\), which is 6. This gives us \(x = \frac{54}{6}\), simplifying further to \(x = 9\). Solving equations allows us to determine the value of unknowns and complete the statement with a known number.

The distributive property is a fundamental property in algebra used to simplify expressions and equations. This property states that multiplying a sum by a number is the same as doing each multiplication separately. In mathematical terms, it looks like: \(a(b + c) = ab + ac\).
In our example, before isolating \(x\), we used the distributive property in the expression \(3(2x + 10)\). Here's how that works:

• Distribute the 3 to both \(2x\) and 10, resulting in \(6x + 30\).

By applying the distributive property, you're breaking down complex expressions into simpler parts. This makes it easier to perform further steps, like isolating variables and solving equations. It's crucial to handle the expressions separately to ensure that you accurately solve for the variables.