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Cross-multiplication is a powerful technique used to solve proportions, which are equations where two ratios are set equal to each other. The key to cross-multiplication is the concept that if \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \). This essentially describes the method of 'multiplying across' the equals sign from the top of one fraction to the bottom of the other.
Why use cross-multiplication? Because it turns a proportion into a simple equation. For example, in the proportion \( \frac{1425}{1062} = \frac{z}{177} \), using cross-multiplication we get \( 1425 \times 177 = 1062 \times z \).

• This results in \( 252225 = 1062z \), a much simpler equation to handle than the original proportion.
• It eliminates the fractions, making subsequent calculations easier and clearer.

Understanding the logic behind cross-multiplication is crucial for solving many algebraic problems efficiently.

Once we have an equation like \( 252225 = 1062z \), the goal is now to solve for the unknown variable, \( z \). Solving equations involves finding the value of the variable that makes the equation true.
In this case, \( z \) is multiplied by 1062. To solve for \( z \), you'll need to isolate it on one side of the equation. Mathematically, this process means getting \( z \) by itself through algebraic manipulation.

• To isolate \( z \), you perform the inverse operation of multiplication, which is division.
• Divide both sides of the equation by 1062 to remove this coefficient from \( z \).

This results in \( z = \frac{252225}{1062} \), effectively determining the exact value of \( z \) necessary to maintain the balance of the original proportion.

Division plays a key role in solving mathematical equations, especially once cross-multiplication has been performed. After a proportion has been rearranged into an equation, division is often the next step used to isolate a variable.
When handling the equation \( z = \frac{252225}{1062} \), dividing 252225 by 1062 simplifies the expression and gives us the value of \( z \).

• Use either a calculator or computer software to perform the division efficiently.
• The result shows \( z \approx 237.53 \), revealing the approximate value, which is often rounded to a manageable number of decimal places in practical scenarios.

By understanding the underlying purpose of each division step, students find it easier to track and comprehend their solutions, ensuring accuracy in problem-solving across all mathematical applications.