Math Problem Statement

Solve the equation: (1/3)x + 1/5 = -2((5/6)x + 5).

Solution

The equation provided in the image is:

13x+15=2(56x+5)\frac{1}{3}x + \frac{1}{5} = -2\left(\frac{5}{6}x + 5\right)

Let's go through the steps to solve it:

Step 1: Distribute the -2 on the right-hand side

2(56x+5)=256x25=106x10=53x10-2\left(\frac{5}{6}x + 5\right) = -2 \cdot \frac{5}{6}x - 2 \cdot 5 = -\frac{10}{6}x - 10 = -\frac{5}{3}x - 10

So, the equation becomes: 13x+15=53x10\frac{1}{3}x + \frac{1}{5} = -\frac{5}{3}x - 10

Step 2: Eliminate fractions by multiplying the entire equation by the least common denominator (LCD) of 3 and 5, which is 15.

Multiply every term by 15: 15(13x)+15(15)=15(53x10)15\left(\frac{1}{3}x\right) + 15\left(\frac{1}{5}\right) = 15\left(-\frac{5}{3}x - 10\right) This simplifies to: 5x+3=25x1505x + 3 = -25x - 150

Step 3: Move all terms involving xx to one side and constant terms to the other side.

5x+25x=15035x + 25x = -150 - 3 30x=15330x = -153

Step 4: Solve for xx

x=15330x = \frac{-153}{30}

Simplify the fraction: x=5110x = \frac{-51}{10}

Thus, the solution is: x=5110x = \frac{-51}{10}

This matches the solution given in the image.

Would you like further details or have any questions?

Here are five related questions:

  1. How do you simplify a fraction like 15330\frac{-153}{30}?
  2. What are the steps to clear fractions in an equation?
  3. How do you solve equations involving the distributive property?
  4. What is the significance of the least common denominator in equations with fractions?
  5. How can you check if your solution for xx is correct?

Tip: Always try to clear fractions early in solving equations for easier simplification!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Fractions
Distributive Property

Formulas

Linear equation formula: ax + b = cx + d
Distributive Property: a(b + c) = ab + ac

Theorems

Basic algebraic operations
Least Common Denominator (LCD)

Suitable Grade Level

Grades 8-10