.3 Substitution in formulae

**Step 1. Understand the Concept** Substitution in formulae means replacing the variables in an expression or equation with specific values. Once you substitute, you perform the arithmetic operations to simplify or solve the formula. --- **Step 2. Identify the Formula and the Given Values** For example, consider the formula for the area of a circle: \[ A = \pi r^2 \] Suppose we need to find the area when the radius is given as \( r = 4 \). --- **Step 3. Substitute the Value into the Formula** Replace \( r \) with 4 in the formula: \[ A = \pi (4)^2 \] --- **Step 4. Simplify the Expression** First compute the square of 4: \[ (4)^2 = 16 \] Then, substitute back into the formula: \[ A = \pi \times 16 \] \[ A = 16\pi \] --- **Step 5. Interpret the Result** The area of the circle is \( 16\pi \) square units. --- **Another Example** Consider the linear equation: \[ y = 3x + 5 \] Suppose we want to find \( y \) when \( x = 2 \). **Substitute \( x = 2 \) into the equation:** \[ y = 3(2) + 5 \] **Simplify the expression:** \[ 3(2) = 6 \] \[ y = 6 + 5 = 11 \] The resulting value is \( y = 11 \). --- **Summary** - Identify the formula and the variable values you need to substitute. - Replace the variable in the formula with the given number. - Simplify the resulting expression step by step. - Interpret the final result. This process, known as substitution, is widely used in algebra to evaluate expressions for specific variable values.