Solve each linear equation. See Examples 1–3.
5/6x - 2x + 4/3 = 5/3

Verified step by step guidance

1. First, rewrite the equation clearly: \(\frac{5}{6}x - 2x + \frac{4}{3} = \frac{5}{3}\).
2. Identify the variable terms and constant terms. Combine the terms with \(x\): \(\frac{5}{6}x - 2x\).
3. To combine the \(x\) terms, express \$2x$ as a fraction with denominator 6: \(2x = \frac{12}{6}x\). Then, \(\frac{5}{6}x - \frac{12}{6}x = \left(\frac{5}{6} - \frac{12}{6}\right)x = -\frac{7}{6}x\).
4. Rewrite the equation as \(-\frac{7}{6}x + \frac{4}{3} = \frac{5}{3}\). Next, isolate the \(x\) term by subtracting \(\frac{4}{3}\) from both sides: \(-\frac{7}{6}x = \frac{5}{3} - \frac{4}{3}\).
5. Simplify the right side and then solve for \(x\) by multiplying both sides by the reciprocal of \(-\frac{7}{6}\), which is \(-\frac{6}{7}\).

Key Concepts

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. Techniques include combining like terms, using the distributive property, and performing inverse operations.

Working with Fractions in Equations

When equations contain fractions, it is important to find a common denominator to combine terms or clear fractions by multiplying both sides by the least common denominator (LCD). This simplifies the equation and makes it easier to isolate the variable without dealing with complex fractional expressions.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable raised to the same power. This step simplifies the equation by reducing the number of terms, making it easier to solve. For example, terms with x can be combined separately from constant terms.