Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 = 2x/3 + 1

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Identify the given equation: \(\frac{3x}{5} = \frac{2x}{3} + 1\).

Find the least common denominator (LCD) of the fractions involved. Here, the denominators are 5 and 3, so the LCD is 15.

Multiply every term on both sides of the equation by the LCD (15) to eliminate the denominators: \(15 \times \frac{3x}{5} = 15 \times \frac{2x}{3} + 15 \times 1\).

Simplify each term after multiplication: \(15 \times \frac{3x}{5} = 3 \times 3x = 9x\), \(15 \times \frac{2x}{3} = 5 \times 2x = 10x\), and \(15 \times 1 = 15\).

Rewrite the equation without fractions: \(9x = 10x + 15\). Then, solve for \(x\) by isolating the variable on one side.

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주요 개념

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Solving Linear Equations

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. Solving such equations involves isolating the variable on one side to find its value. This often requires performing inverse operations like addition, subtraction, multiplication, or division.

Clearing Fractions by Finding a Common Denominator

When an equation contains fractions, multiplying both sides by the least common denominator (LCD) eliminates the denominators, simplifying the equation. This step helps avoid dealing with fractions directly and makes solving the equation more straightforward.

Properties of Equality

The properties of equality allow you to perform the same operation on both sides of an equation without changing its solution. These include adding, subtracting, multiplying, or dividing both sides by the same nonzero number, which is essential for maintaining balance while solving equations.