Solve each equation. 0.5x+ (4/3)x= x+10

Verified step by step guidance

1. Start by writing the given equation clearly: \(0.5x + \frac{4}{3}x = x + 10\).
2. Convert the decimal \(0.5\) to a fraction to make calculations easier: \(0.5x = \frac{1}{2}x\). So the equation becomes \(\frac{1}{2}x + \frac{4}{3}x = x + 10\).
3. Find a common denominator for the fractions on the left side to combine them. The denominators are 2 and 3, so the common denominator is 6. Rewrite each term with denominator 6: \(\frac{3}{6}x + \frac{8}{6}x = x + 10\).
4. Combine the fractions on the left side: \(\frac{3}{6}x + \frac{8}{6}x = \frac{11}{6}x\). Now the equation is \(\frac{11}{6}x = x + 10\).
5. To isolate \(x\), subtract \(x\) from both sides: \(\frac{11}{6}x - x = 10\). Rewrite \(x\) as \(\frac{6}{6}x\) to subtract easily: \(\frac{11}{6}x - \frac{6}{6}x = 10\). Then simplify the left side to get \(\frac{5}{6}x = 10\).

Key Concepts

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. In this equation, terms with the variable x on both sides can be combined to simplify the equation and isolate the variable.

Solving Linear Equations

Solving linear equations means finding the value of the variable that makes the equation true. This typically involves isolating the variable on one side by performing inverse operations such as addition, subtraction, multiplication, or division.

Working with Fractions and Decimals

This concept involves manipulating fractions and decimals within equations. Converting decimals to fractions or vice versa, and finding common denominators, helps simplify terms and solve the equation accurately.