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Cross-multiplication is a powerful technique used to solve equations involving proportions, particularly when you have two fractions set equal to each other. A proportion is simply an equation that states two ratios are equivalent.
For example, given a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa.

• You multiply \( a \) by \( d \)
• Then, you multiply \( b \) by \( c \)

This results in the equation \( ad = bc \), which is a linear equation that is usually much easier to work with.
Cross-multiplication helps to eliminate the fractions, simplifying the solving process and leading directly to dealing with a simple equation.

Linear equations are foundational to understanding cross-multiplication and solving proportions. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
In our worked example, after performing cross-multiplication, we obtain the equation \( 32a + 48 = 16 \). This is a linear equation because it has a constant term (48), a coefficient (32), and a variable term \( a \) in a simple form.

• Linear equations usually have one solution.
• They are expressed in the form \( ax + b = c \).
• They involve operations such as addition, subtraction, multiplication, and division.

Solving these equations involves isolating the variable (\( a \) in this case) on one side of the equation to find its value.

Isolating the variable is the key step in solving any equation, including those that arise from cross-multiplication. This process involves performing operations to get the variable alone on one side of the equation.
In our original problem, once we simplify via cross-multiplication to obtain \( 32a + 48 = 16 \), we need to isolate \( a \):

• First, subtract 48 from both sides of the equation to remove the constant term: \( 32a = -32 \).
• Then, divide by 32, which is the coefficient of \( a \), to fully isolate \( a \): \( a = -1 \).

These steps ensure that \( a \) is by itself, which allows us to easily see that its value is \( -1 \).
Isolating the variable helps us solve for unknowns in algebraic equations and better understand the relationships between quantities.