Math Problem Statement

Solve the following equation: (5/9)x - 1 - (1/3)x = 3

Solution

The equation from the image is:

59x113x=3\frac{5}{9}x - 1 - \frac{1}{3}x = 3

Let's solve it step by step:

  1. Combine like terms. The terms involving xx are 59x\frac{5}{9}x and 13x-\frac{1}{3}x. To combine them, we need a common denominator. The least common denominator between 9 and 3 is 9. So, rewrite 13x-\frac{1}{3}x as 39x-\frac{3}{9}x.

Now the equation looks like: 59x39x1=3\frac{5}{9}x - \frac{3}{9}x - 1 = 3

  1. Combine the xx-terms: (59x39x)=29x\left(\frac{5}{9}x - \frac{3}{9}x\right) = \frac{2}{9}x Now the equation becomes: 29x1=3\frac{2}{9}x - 1 = 3

  2. Add 1 to both sides to move the constant term to the right side: 29x=4\frac{2}{9}x = 4

  3. To solve for xx, multiply both sides by the reciprocal of 29\frac{2}{9}, which is 92\frac{9}{2}: x=4×92x = 4 \times \frac{9}{2}

  4. Simplify: x=362=18x = \frac{36}{2} = 18

So, the solution is: x=18x = 18

Would you like further details or have any questions? Here are some related questions:

  1. How do you find the least common denominator when combining fractions?
  2. Why is it important to combine like terms in an equation?
  3. How do you solve equations that involve fractions and constants?
  4. What happens if you multiply both sides of an equation by the reciprocal of a fraction?
  5. How can checking your solution help ensure accuracy in solving equations?

Tip: When solving equations with fractions, it's often helpful to clear the denominators by multiplying through by the least common denominator!

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Math Problem Analysis

Mathematical Concepts

Algebra
Equations with Fractions
Combining Like Terms
Solving Linear Equations

Formulas

ax + b = c
Common Denominator Method for Combining Fractions

Theorems

Properties of Equality
Solving Linear Equations with Fractions

Suitable Grade Level

Grades 8-10