Solving Linear Equations Step-by-Step

Example 1: Algebraic Solution of the Linear Equation

• Initial Equation: $2x + 5 = \frac{1}{2}x(2 + 4)$

• Step 1: Isolate the Variable Term on the Left Side

  • Subtract $5$ from both sides of the equation: $2x + 5 - 5 = \frac{1}{2}x(2 + 4) - 5$

  • Result after subtraction: $2x = \frac{1}{2}x(2 + 4) - 5$

• Step 2: Simplify Terms Inside the Parentheses

  • Evaluate the arithmetic addition inside the parentheses: $2 + 4 = 6$

  • Substitute $6$ back into the expression: $2x = \frac{1}{2}x(6) - 5$

• Step 3: Simplify the Coefficient Product

  • Multiply the fraction $\frac{1}{2}$ by the constant factor $6$: $\frac{1}{2}x(6) = \frac{6}{2}x = 3x$

  • Substitute $3x$ back into the equation: $2x = 3x - 5$

• Step 4: Apply the Symmetric Property of Equality

  • Rearrange the equation by swapping the left and right sides: $3x - 5 = 2x$

• Step 5: Isolate Constant Terms

  • Add $5$ to both sides of the equation: $3x = 2x + 5$

• Step 6: Combine Like Variable Terms

  • Subtract $2x$ from both sides of the equation: $3x - 2x = 2x + 5 - 2x$

• Step 7: Final Computation

  • Subtract coefficients on the left-hand side: $3x - 2x = 1x = x$

  • Obtain the final value for $x$: $x = 5$

Example 2: Linear Equation with Fractional Coefficient and Constant Terms

• Initial Equation: $4x - 3 = \frac{1}{3}x(1 + 8) + 2$

• Step 1: Simplify Terms Inside Parentheses

  • Evaluate addition inside parentheses: $1 + 8 = 9$

  • Substitute $9$ back into the expression: $4x - 3 = \frac{1}{3}x(9) + 2$

• Step 2: Simplify Coefficient Product

  • Multiply fraction by factor: $\frac{1}{3}x(9) = 3x$

  • Substitute $3x$ back into equation: $4x - 3 = 3x + 2$

• Step 3: Combine Like Variable Terms

  • Subtract $3x$ from both sides: $4x - 3x - 3 = 3x - 3x + 2$

  • Result after subtraction: $x - 3 = 2$

• Step 4: Isolate the Variable

  • Add $3$ to both sides: $x - 3 + 3 = 2 + 3$

  • Obtain final value: $x = 5$

Example 3: Linear Equation with Rearrangement

• Initial Equation: $\frac{2}{5}x(3 + 2) + 7 = 3x - 2$

• Step 1: Simplify Terms Inside Parentheses

  • Evaluate parentheses: $3 + 2 = 5$

  • Equation becomes: $\frac{2}{5}x(5) + 7 = 3x - 2$

• Step 2: Simplify Coefficient Product

  • Multiply coefficient and factor: $\frac{2}{5} \times 5x = 2x$

  • Equation simplifies to: $2x + 7 = 3x - 2$

• Step 3: Apply Symmetric Property of Equality

  • Rearrange sides: $3x - 2 = 2x + 7$

• Step 4: Isolate Constant Terms

  • Add $2$ to both sides: $3x = 2x + 9$

• Step 5: Combine Like Variable Terms

  • Subtract $2x$ from both sides: $3x - 2x = 2x - 2x + 9$

  • Final result: $x = 9$

Core Mathematical Principles

• Addition and Subtraction Properties of Equality:

  • Adding or subtracting the same quantity from both sides of an equation preserves equality: $\text{If } A = B \text{ then } A - C = B - C$

  • Used when isolating terms across the equal sign.

• Order of Operations (PEMDAS):

  • Perform operations inside parentheses prior to multiplying external coefficients.

• Multiplication of Rational Numbers and Variables:

  • Scaling a variable expression by a fraction: $\frac{1}{2} \times 6x = \frac{6}{2}x = 3x$

• Symmetric Property of Equality:

  • If $A = B$, then $B = A$. This allows equivalent swapping of left and right expressions.