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Cross multiplication is a mathematical technique used when we have a proportion, which is basically an equation stating that two ratios are equal, like \( \frac{a}{b} = \frac{c}{d} \). This tool helps eliminate the fractions and simplifies the process of solving for an unknown variable. Here's how it works. Start by cross-multiplying: multiply the numerator of each ratio by the denominator of the other ratio. Thus, the expression \( \frac{a}{b} = \frac{c}{d} \) becomes

• \( a \times d = b \times c \)

These cross-products are equal, forming a new equation, which is simpler to handle. For example, if we multiply across our original problem, \( \frac{x-2}{2} = \frac{4}{5} \), we get \( 5(x-2) = 2 \times 4 \). Now you’ve transformed the proportion problem into a simpler equation without fractions. Cross multiplication is handy because it makes equations easier to solve and is often one of the first steps when dealing with proportion problems.

Once a proportion problem is transformed using cross multiplication, the next step is solving the resulting linear equation. A linear equation is simply an equation where the highest power of the variable is 1, such as \( ax + b = c \). These are straightforward to solve. Let's go step by step.After cross multiplying our proportion \( \frac{x-2}{2} = \frac{4}{5} \), we ended up with the equation \( 5(x-2) = 8 \). The goal here is to isolate the variable, \( x \). Begin by using inverse operations.

• First, distribute any coefficient through terms within parentheses, if necessary.
• Then, in this case, add or subtract constants to bring terms involving the variable to one side of the equation.
• Finally, divide or multiply to solve for the variable itself.

For our problem, you’d distribute \( 5 \) over \( (x - 2) \), giving \( 5x - 10 = 8 \). From there, you add \( 10 \) to both sides to get \( 5x = 18 \), and divide by \( 5 \) to isolate \( x \). This gives you \( x = \frac{18}{5} \), or converted to decimal 3.6. Solving linear equations is about reversing operations to find the value of the unknown variable.

The distributive property is a fundamental algebraic concept used not only in solving proportions but also widely in mathematics to simplify expressions. It states that multiplying a sum by a number is the same as multiplying each addend separately by the number and then adding the products. Mathematically, it can be expressed as:

• \( a(b + c) = ab + ac \)

This property is especially helpful in simplifying expressions and solving equations. In our example problem, we see it when we took the equation \( 5(x - 2) = 8 \).By applying the distributive property, the expression \( 5(x - 2) \) became \( 5x - 10 \). You multiplied \( 5 \) by each term inside the parentheses: \( 5 \times x \) and \( 5 \times (-2) \). This step is essential before solving for \( x \), as it transitions from a complex-looking expression to a simpler form, making it easier to manage.Remember, the distributive property provides clarity and makes solving equations more efficient, especially when dealing with variables and constants together.