Solving Equations Involving Multiplication & division

Solving Equations Involving Multiplication

Introduction

• The process for solving equations that involve multiplication is parallel to the methods used in addition and subtraction.

• The primary objective is to isolate the variable in the equation.

Steps to Isolate the Variable

1. Identify the Coefficient: Recognize the coefficient (the multiplier in front of the variable).

   • Example: In the expression three x, the coefficient is three.

2. Find the Reciprocal: The reciprocal of a number is defined as one divided by that number.

   • Example: The reciprocal of three is one third (or $rac{1}{3}$).

3. Multiply by the Reciprocal: Multiply both sides of the equation by the reciprocal of the coefficient, which effectively cancels the coefficient out and leaves the variable isolated.

Example: Solving the Equation 5x = 7

• Begin with the equation:
  $5x = 7$

Step 1: Multiply by the Reciprocal of the Coefficient

• The coefficient of x is 5. Its reciprocal is one fifth (or $rac{1}{5}$).

• Multiply both sides of the equation by one fifth:

$5x imes rac{1}{5} = 7 imes rac{1}{5}$

Step 2: Simplify both sides

• The left side simplifies as follows:

  • $5 imes rac{1}{5} = 1$

  • Therefore, the left side becomes x.

• The right side simplifies to:

  • $7 imes rac{1}{5} = rac{7}{5}$

• Resulting equation:
  $x = rac{7}{5}$

Verification of the Solution

• To ensure the solution is correct, substitute $rac{7}{5}$ back into the original equation:

1. Calculate:

   • $5 imes rac{7}{5} = 7$

2. Simplification yields:

   • $7 = 7$, which is a true statement.

Conclusion

• The process for isolating the variable in an equation involving multiplication entails multiplying by the reciprocal of the coefficient, allowing for straightforward solutions of linear equations.