Question

Find the value of \(x.\) $$(x-7):(x+7)=2: 9$$

Short answer

The value of \(x\) is 11.

Step-by-step solution

Set Up the Proportion

Write the given equation as a proportion: \(\frac{x-7}{x+7} = \frac{2}{9}\).

Cross-Multiply

Cross-multiply to eliminate the fractions: \(9(x-7) = 2(x+7)\).

Distribute the Multiplication

Distribute both sides of the equation: \(9x - 63 = 2x + 14\).

Collect Like Terms

Move all terms involving \(x\) to one side and constant terms to the other: \(9x - 2x = 14 + 63\).

Simplify the Equation

Simplify both sides of the equation: \(7x = 77\).

Solve for \(x\)

Divide both sides by 7 to find \(x\): \(x = \frac{77}{7}\).

Calculate the Value of \(x\)

Calculate \(x\): \(x = 11\).

Key concepts

Cross-Multiplication

Cross-multiplication is a handy technique used in solving proportions, which are statements that two ratios are equal. When you encounter 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 doing the same with the remaining numerator and denominator. This gives us two products, and setting them equal to each other forms an algebraic equation without fractions.

For example, in the exercise \(\frac{x-7}{x+7} = \frac{2}{9}\), by cross-multiplying we get \(9(x-7) = 2(x+7)\). This step is crucial as it transforms the proportion into a simpler algebraic equation, which is easier to solve. After cross-multiplication, the next steps involve distributing the multiplication, gathering like terms, and solving for the variable, ultimately providing the value of \(x\).

Proportions in Mathematics

Proportions are a central concept in mathematics, expressing the relationship that two ratios or fractions are equivalent. A proportion states that the two fractions are of equal value. When two ratios are set equal, they form a proportion, such as \(\frac{a}{b} = \frac{c}{d}\). Proportions are used to solve various problems, including those related to scaling, mixtures, and rates.

In the given exercise, the proportion is derived from an equation that compares two ratios. To find the value of \(x\), we need to ensure that the two sides of the proportion remain balanced as we perform algebraic operations to solve for \(x\). Recognizing that a proportion represents equality can assist in understanding the logic behind cross-multiplication and the steps that follow.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions with one or several variables. The goal when solving an algebraic equation is to find the value(s) of the variable(s) that make the equation true. In our exercise, the algebraic equation comes from a proportion and is solved through a sequence of steps:

• Identifying terms that represent the same kinds of value
• Using algebraic operations to isolate the variable on one side of the equation
• Simplifying the equation to reveal the solution for the variable

After cross-multiplication, the exercise involves distributing multiplication across terms, combining like terms to consolidate the equation, and then isolating the variable \(x\) to find its value. This series of operations is typical in solving algebraic equations, allowing students to uncover the unknown variable systematically.

Once \(x\) is isolated, we obtain \(x = 11\), which is the solution to our original proportion.