Math Problem Statement

Solve multi-step equations with fractions as shown in the image.

Solution

Let's solve the multi-step equations with fractions from the image.


a) 6x(12+13)=2x+36x \left( \frac{1}{2} + \frac{1}{3} \right) = 2x + 3

  1. First, simplify the terms inside the parentheses: 12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} So the equation becomes: 6x56=2x+36x \cdot \frac{5}{6} = 2x + 3
  2. Simplify 6x566x \cdot \frac{5}{6}: x5=5xx \cdot 5 = 5x Now the equation is: 5x=2x+35x = 2x + 3
  3. Subtract 2x2x from both sides: 5x2x=35x - 2x = 3 3x=33x = 3
  4. Solve for xx: x=1x = 1

b) 23(x+5)=49\frac{2}{3}(x + 5) = \frac{4}{9}

  1. Distribute 23\frac{2}{3} over (x+5)(x + 5): 23x+235=23x+103\frac{2}{3}x + \frac{2}{3} \cdot 5 = \frac{2}{3}x + \frac{10}{3} So the equation becomes: 23x+103=49\frac{2}{3}x + \frac{10}{3} = \frac{4}{9}
  2. To eliminate the fractions, multiply the entire equation by 9: 9(23x+103)=9499 \cdot \left( \frac{2}{3}x + \frac{10}{3} \right) = 9 \cdot \frac{4}{9} 6x+30=46x + 30 = 4
  3. Subtract 30 from both sides: 6x=4306x = 4 - 30 6x=266x = -26
  4. Solve for xx: x=266=133x = \frac{-26}{6} = \frac{-13}{3}

c) 15(x2+34)=14\frac{1}{5} \left( \frac{x}{2} + \frac{3}{4} \right) = \frac{1}{4}

  1. First, simplify the terms inside the parentheses: x2+34=2x4+34=2x+34\frac{x}{2} + \frac{3}{4} = \frac{2x}{4} + \frac{3}{4} = \frac{2x + 3}{4} So the equation becomes: 152x+34=14\frac{1}{5} \cdot \frac{2x + 3}{4} = \frac{1}{4}
  2. Multiply both sides by 5 to eliminate the 15\frac{1}{5}: 2x+34=54\frac{2x + 3}{4} = \frac{5}{4}
  3. Multiply both sides by 4 to eliminate the denominator: 2x+3=52x + 3 = 5
  4. Subtract 3 from both sides: 2x=53=22x = 5 - 3 = 2
  5. Solve for xx: x=22=1x = \frac{2}{2} = 1

d) x+32+(x1)=45\frac{x + 3}{2} + (x - 1) = \frac{4}{5}

  1. Multiply the entire equation by 10 to eliminate the fractions: 10(x+32)+10(x1)=104510 \cdot \left( \frac{x + 3}{2} \right) + 10 \cdot (x - 1) = 10 \cdot \frac{4}{5} 5(x+3)+10(x1)=85(x + 3) + 10(x - 1) = 8
  2. Distribute the terms: 5x+15+10x10=85x + 15 + 10x - 10 = 8 Simplify: 15x+5=815x + 5 = 8
  3. Subtract 5 from both sides: 15x=315x = 3
  4. Solve for xx: x=315=15x = \frac{3}{15} = \frac{1}{5}

Would you like further explanations on any step? Here are some related questions for practice:

  1. How would you simplify equations that have variables on both sides?
  2. What is the process of eliminating fractions in multi-step equations?
  3. How can you check your solution for fractional equations?
  4. How do you factor equations that involve fractions?
  5. How would you approach solving an equation if all terms are fractions?

Tip: Always try to eliminate fractions early by multiplying through by the least common denominator.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Fractions
Multi-Step Equations

Formulas

Basic operations with fractions
Solving linear equations

Theorems

Properties of equality
Distributive property

Suitable Grade Level

Grades 8-10