Key Rules for Solving Equations

When solving for a variable (like $x$), the goal is to isolate it on one side of the equation. To do this, we move numbers to the other side by performing the opposite operation. This is called transposition.

1. Addition and Subtraction

• If a number is added, it becomes subtraction on the other side:$x + a = b \Rightarrow x = b – a$
• If a number is subtracted, it becomes addition on the other side:$x – a = b \Rightarrow x = b + a$

2. Multiplication and Division

• If a number multiplies $x$, it divides the other side:$ax = b \Rightarrow x = \frac{b}{a}$
• If a number divides $x$, it multiplies the other side:$\frac{x}{a} = b \Rightarrow x = ab$

3. Handling Fractions and Cross-Multiplication

When dealing with multiple fractions, these variations are helpful:

• $\frac{ax}{b} = \frac{c}{d} \Rightarrow x = \frac{c}{d} \times \frac{b}{a}$

• $\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$ (Cross-multiplication)

4. Distributive Property

To remove parentheses, multiply the term outside by every term inside:

• $a(b + c) = ab + ac$

• $a(b – c) = ab – ac$

Practical Examples

Let’s apply these rules to solve some equations involving rational numbers.

Example 1: Solving with Addition

Problem: Solve $x + \frac{1}{2} = \frac{5}{2}$

Solution:

1. Transpose the fraction: Move $+\frac{1}{2}$ to the right side. It becomes $-\frac{1}{2}$.$x = \frac{5}{2} – \frac{1}{2}$
2. Subtract the numerators: Since the denominators are the same, subtract directly.$x = \frac{5 – 1}{2}$$x = \frac{4}{2}$
3. Simplify:$x = 2$

Example 2: Solving with Subtraction

Problem: Solve $x – \frac{5}{2} = \frac{3}{8}$

Solution:

1. Transpose the fraction: Move $-\frac{5}{2}$ to the right side. It becomes $+\frac{5}{2}$.$x = \frac{3}{8} + \frac{5}{2}$
2. Find a common denominator: To add these, convert $\frac{5}{2}$ to a fraction with a denominator of 8 (multiply top and bottom by 4).$x = \frac{3 + 20}{8}$
3. Calculate the final result:$x = \frac{23}{8}$

Advanced Step-by-Step Examples

Here are three common scenarios you will encounter when solving for $x$:

Example 3: Combining Fractions with Different Denominators

Problem: Solve $2x + \frac{5}{4} = -\frac{1}{7}$

Solution:

1. Isolate the variable term: Move $+\frac{5}{4}$ to the right side by subtracting it.$2x = -\frac{1}{7} – \frac{5}{4}$
2. Find a common denominator: The common denominator for 7 and 4 is 28.$2x = \frac{-4 – 35}{28} \Rightarrow 2x = \frac{-39}{28}$
3. Solve for $x$: Move the 2 (multiplication) to the denominator on the right (division).$x = \frac{-39}{28 \times 2}$$x = -\frac{39}{56}$

Example 4: Dealing with Variable Fractions

Problem: Solve $\frac{x}{2} + \frac{3}{4} = \frac{3}{8}$

Solution:

1. Transpose the constant: Move $+\frac{3}{4}$ to the right.$\frac{x}{2} = \frac{3}{8} – \frac{3}{4}$
2. Calculate the right side: Use 8 as the common denominator.$\frac{x}{2} = \frac{3 – 6}{8} \Rightarrow \frac{x}{2} = -\frac{3}{8}$
3. Multiply to isolate $x$: Multiply the right side by 2 and simplify the fraction.$x = -\frac{3}{8} \times 2$$x = -\frac{3}{4}$

Example 5: Multi-Step Fraction Transposition

Problem: Solve $\frac{5x}{6} + \frac{3}{4} = \frac{5}{6}$

Solution:

1. Transpose the fraction: Move $+\frac{3}{4}$ to the right side.$\frac{5x}{6} = \frac{5}{6} – \frac{3}{4}$
2. Subtract using a common denominator: The common denominator for 6 and 4 is 12.$\frac{5x}{6} = \frac{10 – 9}{12} \Rightarrow \frac{5x}{6} = \frac{1}{12}$
3. Isolate $x$ using the reciprocal: Multiply $\frac{1}{12}$ by the reciprocal of $\frac{5}{6}$ (which is $\frac{6}{5}$).$x = \frac{1}{12} \times \frac{6}{5}$
4. Simplify:$x = \frac{1}{2} \times \frac{1}{5}$$x = \frac{1}{10}$

Key Takeaways

• Step 1: Always move the constants (the numbers without $x$) to one side first.

• Step 2: Ensure you find a common denominator before adding or subtracting fractions.

• Step 3: To move a fraction attached to $x$ (like $\frac{5}{6}x$), multiply the other side by its flipped version (the reciprocal).

Intermediate Examples

Example 3: Handling Multiple Transpositions

Problem: Solve $2x + \frac{5}{4} = -\frac{1}{7}$

Solution:

1. Transpose the constant: Move $+\frac{5}{4}$ to the right side as $-\frac{5}{4}$.
   $$2x = -frac{1}{7} – frac{5}{4}$$
2. Find a common denominator: For 7 and 4, the common denominator is 28.
   $$2x = frac{-4 – 35}{28} Rightarrow 2x = frac{-39}{28}$$
3. Divide to isolate $x$:
   $$x = frac{-39}{28 times 2} Rightarrow x = -frac{39}{56}$$

Example 4: Variable Fractions

Problem: Solve $\frac{x}{2} + \frac{3}{4} = \frac{3}{8}$

Solution:

1. Transpose: $\frac{x}{2} = \frac{3}{8} – \frac{3}{4}$

2. Combine fractions: $\frac{x}{2} = \frac{3 – 6}{8} \Rightarrow \frac{x}{2} = -\frac{3}{8}$

3. Solve: $x = -\frac{3}{8} \times 2 \Rightarrow x = -\frac{3}{4}$

Example 5: Using Reciprocals

Problem: Solve $\frac{5x}{6} + \frac{3}{4} = \frac{5}{6}$

Solution:

1. Transpose: $\frac{5x}{6} = \frac{5}{6} – \frac{3}{4}$

2. Common denominator (12): $\frac{5x}{6} = \frac{10 – 9}{12} \Rightarrow \frac{5x}{6} = \frac{1}{12}$

3. Isolate $x$: $x = \frac{1}{12} \times \frac{6}{5} \Rightarrow x = \frac{1}{10}$

Advanced: Variables on Both Sides

When $x$ appears on both sides, group all $x$ terms together first.

Example 6: Grouping Variable Terms

Problem: Solve $\frac{x}{2} – \frac{3}{5} = \frac{x}{3} + \frac{1}{2}$

Solution:

1. Group terms: Move $\frac{x}{3}$ to the left and $-\frac{3}{5}$ to the right.
   $$frac{x}{2} – frac{x}{3} = frac{1}{2} + frac{3}{5}$$
2. Find common denominators: Use 6 for the left side and 10 for the right.
   $$frac{3x – 2x}{6} = frac{5 + 6}{10} Rightarrow frac{x}{6} = frac{11}{10}$$

3. Final Step: $x = \frac{11}{10} \times 6 \Rightarrow x = \frac{33}{5}$

Example 7: Complex Multi-Variable Equations

Problem: Solve $\frac{x}{2} – \frac{x}{5} + \frac{1}{2} = \frac{x}{4} + 3$

Solution:

1. Group all $x$ terms: $\frac{x}{2} – \frac{x}{5} – \frac{x}{4} = 3 – \frac{1}{2}$

2. Simplify with common denominators:
   $$frac{10x – 4x – 5x}{20} = frac{6 – 1}{2} Rightarrow frac{10x – 9x}{20} = frac{5}{2}$$

3. Solve: $\frac{x}{20} = \frac{5}{2} \Rightarrow x = \frac{5 \times 20}{2} \Rightarrow x = 50$

Pro-Tips for Success

• Identify the LCD: Finding the Least Common Denominator (LCD) early makes subtraction much easier.

• Balance the Equation: Whatever operation you do to one side (like multiplying by 20), you must do to the other.

Handling Multiple Transpositions

Problem: Solve $2x + \frac{5}{4} = -\frac{1}{7}$

Solution:

1. Transpose the constant: Move $+\frac{5}{4}$ to the right side as $-\frac{5}{4}$.
   $$2x = -\frac{1}{7} – \frac{5}{4}$$
2. Find a common denominator: For 7 and 4, the common denominator is 28.
   $$2x = \frac{-4 – 35}{28} \Rightarrow 2x = \frac{-39}{28}$$
3. Divide to isolate $x$:
   $$x = \frac{-39}{28 \times 2} \Rightarrow x = -\frac{39}{56}$$

Using Reciprocals and Simplification

Problem: Solve $\frac{5x}{6} + \frac{3}{4} = \frac{5}{6}$

Solution:

1. Transpose: $\frac{5x}{6} = \frac{5}{6} – \frac{3}{4}$

2. Common denominator (12): $\frac{5x}{6} = \frac{10 – 9}{12} \Rightarrow \frac{5x}{6} = \frac{1}{12}$

3. Isolate $x$: $x = \frac{1}{12} \times \frac{6}{5}$.

4. Final result: $x = \frac{1}{10}$.

Variables on Both Sides

When the variable appears on both sides of the equals sign, group all variable terms together and all constant terms together.

Grouping Variable Terms

Problem: Solve $\frac{x}{2} – \frac{3}{5} = \frac{x}{3} + \frac{1}{2}$

Solution:

1. Group terms: Move $\frac{x}{3}$ to the left and $-\frac{3}{5}$ to the right.
   $$\frac{x}{2} – \frac{x}{3} = \frac{1}{2} + \frac{3}{5}$$
2. Simplify with common denominators: Use 6 for the left side and 10 for the right.
   $$\frac{3x – 2x}{6} = \frac{5 + 6}{10} \Rightarrow \frac{x}{6} = \frac{11}{10}$$

3. Final Step: $x = \frac{11}{10} \times 6 \Rightarrow x = \frac{33}{5}$.

Handling Distributive Property

Problem: Solve $3(x – 3) = 5(2x + 1)$

Solution:

1. Distribute: $3x – 9 = 10x + 5$.

2. Group variable terms: $3x – 10x = 5 + 9$.

3. Simplify: $-7x = 14$.

4. Solve: $x = \frac{14}{-7} \Rightarrow x = -2$.

Complex Multi-Variable Equations

Problem: Solve $15(y – 4) – 2(y – 9) + 5(y + 6) = 0$

Solution:

1. Expand the terms: $15y – 60 – 2y + 18 + 5y + 30 = 0$.

2. Combine like terms: $18y – 12 = 0$.

3. Transpose: $18y = 12$.

4. Simplify: $y = \frac{12}{18} \Rightarrow y = \frac{2}{3}$.

Pro-Tips for Success

• Distribute First: If there are parentheses, multiply them out before trying to move terms.

• LCD is Key: Finding the Least Common Denominator (LCD) makes adding or subtracting fractional terms much cleaner.

• Balance the Equation: Whatever operation you perform on the left side, you must perform on the right side to keep it balanced.

Leveling Up: Solving Multi-Step Equations with Rational Numbers

Now that you have the basics down, let’s look at more complex scenarios. Solving equations often involves gathering variable terms on one side and constants on the other before simplifying.

Intermediate Step-by-Step Examples

Handling Multiple Transpositions

When an equation has both multiplication and addition/subtraction, start by moving the constant terms.

Example 3: Solve $2x + \frac{5}{4} = -\frac{1}{7}$

1. Transpose the constant: Move $+\frac{5}{4}$ to the right side as $-\frac{5}{4}$.$2x = -\frac{1}{7} – \frac{5}{4}$
2. Find a common denominator: For 7 and 4, the common denominator is 28.$2x = \frac{-4 – 35}{28} \Rightarrow 2x = \frac{-39}{28}$
3. Divide to isolate $x$: Move the 2 to the denominator.$x = \frac{-39}{28 \times 2} \Rightarrow x = -\frac{39}{56}$

Example 5: Solve $\frac{5x}{6} + \frac{3}{4} = \frac{5}{6}$

1. Transpose: $\frac{5x}{6} = \frac{5}{6} – \frac{3}{4}$

2. Subtract with LCD (12): $\frac{5x}{6} = \frac{10 – 9}{12} \Rightarrow \frac{5x}{6} = \frac{1}{12}$

3. Isolate $x$: Multiply by the reciprocal $\frac{6}{5}$.$x = \frac{1}{12} \times \frac{6}{5} \Rightarrow x = \frac{1}{10}$

Variables on Both Sides

When the variable appears on both sides, group all variable terms together and all constant terms together.

Example 6: Grouping Variable Terms

Solve: $\frac{x}{2} – \frac{3}{5} = \frac{x}{3} + \frac{1}{2}$

1. Group terms: Move $\frac{x}{3}$ to the left and $-\frac{3}{5}$ to the right.$\frac{x}{2} – \frac{x}{3} = \frac{1}{2} + \frac{3}{5}$
2. Simplify with common denominators: Use 6 for the left and 10 for the right.$\frac{3x – 2x}{6} = \frac{5 + 6}{10} \Rightarrow \frac{x}{6} = \frac{11}{10}$

3. Final Step: $x = \frac{11}{10} \times 6 \Rightarrow x = \frac{33}{5}$

Using the Distributive Property and Cross-Multiplication

For more advanced problems, you must expand parentheses or cross-multiply to remove fractions entirely.

Example 8: Distributive Property

Solve: $3(x – 3) = 5(2x + 1)$

1. Distribute: $3x – 9 = 10x + 5$

2. Group variables: $3x – 10x = 5 + 9 \Rightarrow -7x = 14$

3. Solve: $x = \frac{14}{-7} \Rightarrow x = -2$

Example 10: Cross-Multiplication

Solve: $\frac{x – 5}{3} = \frac{x – 3}{5}$

1. Cross-multiply: Multiply the numerator of one side by the denominator of the other.$5(x – 5) = 3(x – 3)$

2. Expand and group: $5x – 25 = 3x – 9 \Rightarrow 5x – 3x = -9 + 25$

3. Simplify and solve: $2x = 16 \Rightarrow x = 8$

Pro-Tips for Success

• Expand First: If there are parentheses, multiply them out immediately.

• Check Your Signs: Remember that moving a term across the equals sign flips its sign (e.g., $-5x$ becomes $+5x$).

• Cross-Multiply: If you have one fraction equal to another, cross-multiplication is usually the fastest way to a solution.

Handling Multiple Transpositions

When an equation has both multiplication and addition/subtraction, start by moving the constant terms.

Example 3: Solve $2x + \frac{5}{4} = -\frac{1}{7}$

1. Transpose the constant: Move $+\frac{5}{4}$ to the right side as $-\frac{5}{4}$.$2x = -\frac{1}{7} – \frac{5}{4}$
2. Find a common denominator: For 7 and 4, the common denominator is 28.$2x = \frac{-4 – 35}{28} \Rightarrow 2x = \frac{-39}{28}$
3. Divide to isolate $x$: Move the 2 to the denominator.$x = \frac{-39}{28 \times 2} \Rightarrow x = -\frac{39}{56}$

Variables on Both Sides

When the variable appears on both sides of the equals sign, group all variable terms together and all constant terms together.

Example 6: Grouping Variable Terms

Solve: $\frac{x}{2} – \frac{3}{5} = \frac{x}{3} + \frac{1}{2}$

1. Group terms: Move $\frac{x}{3}$ to the left and $-\frac{3}{5}$ to the right.$\frac{x}{2} – \frac{x}{3} = \frac{1}{2} + \frac{3}{5}$
2. Simplify with common denominators: Use 6 for the left side and 10 for the right.$\frac{3x – 2x}{6} = \frac{5 + 6}{10} \Rightarrow \frac{x}{6} = \frac{11}{10}$
3. Final Step: Multiply by 6 and simplify.$x = \frac{11}{10} \times 6 \Rightarrow x = \frac{33}{5}$

Advanced Techniques: Expansion and Cross-Multiplication

For more advanced problems, you must expand parentheses or cross-multiply to remove fractions entirely.

Example 8: The Distributive Property

Solve: $3(x – 3) = 5(2x + 1)$

1. Distribute: Multiply the number outside the parentheses by every term inside.$3x – 9 = 10x + 5$

2. Group variables: $3x – 10x = 5 + 9 \Rightarrow -7x = 14$

3. Solve: $x = \frac{14}{-7} \Rightarrow x = -2$

Example 10: Cross-Multiplication

Solve: $\frac{x – 5}{3} = \frac{x – 3}{5}$

1. Cross-multiply: Multiply the numerator of one side by the denominator of the other.$5(x – 5) = 3(x – 3)$

2. Expand and group: $5x – 25 = 3x – 9 \Rightarrow 5x – 3x = -9 + 25$

3. Simplify and solve: $2x = 16 \Rightarrow x = 8$

Pro-Tips for Success

• Expand First: If there are parentheses, multiply them out immediately to simplify the expression.

• Check Your Signs: Remember that moving a term across the equals sign flips its sign (e.g., $-25$ becomes $+25$).

• Cross-Multiply: If you have one fraction equal to another, cross-multiplication is usually the fastest way to a solution.