Evaluating Algebraic Expressions

1. Basic Substitution

Replacing a variable with a given number

Evaluate f(x) when $$x = a$$

Example: $$x + b$$ where $$x = a$$

Replacing a single variable with a given numerical value and simplifying.

Example

Evaluate $$3x + 7$$ when $$x = 4$$

Show solution

Step 1: Substitute $$x = 4$$ into expression: $$3(4) + 7$$.

Step 2: Multiply: $$12 + 7 = 19$$.

Result: 19.

Practice problems

Problem 1
Evaluate $$5x - 3$$ when $$x = 4$$
Show solution
Answer: 17

Step-by-step substitution & calculation:
- Evaluated Value: 17
=> Correct Option: B (17).
Problem 2
Evaluate $$2 - 4x$$ when $$x = -3$$
Show solution
Answer: 14

Step-by-step substitution & calculation:
- Evaluated Value: 14
=> Correct Option: A (14).
Problem 3
Evaluate $$7 - 3(2x + 1)$$ when $$x = -2$$
Show solution
Answer: 16

Step-by-step substitution & calculation:
- Evaluated Value: 16
=> Correct Option: B (16).

Remember

Replace every instance of the variable with the number (use parentheses for negative values), then simplify using basic arithmetic.

2. Multiple Variables

Expressions with two or more variables

Evaluate f(x, y) when $$x = a$$, $$y = b$$

Example: ax + by

Substituting given values for two or more distinct variables in an expression.

Example

Evaluate $$2x + 3y$$ when $$x = 5$$, $$y = -2$$

Show solution

Step 1: Substitute $$x = 5$$ and $$y = -2$$: $$2(5) + 3(-2)$$.

Step 2: Multiply terms: $$10 + (-6)$$.

Step 3: Add: $$10 - 6 = 4$$.

Result: 4.

Practice problems

Problem 4
Evaluate $$3x + 2y$$ when $$x = 4$$ and $$y = 5$$
Show solution
Answer: 22

Step-by-step substitution & calculation:
- Evaluated Value: 22
=> Correct Option: A (22).
Problem 5
Evaluate $$4x - 5y + 3$$ when $$x = -2$$ and $$y = 3$$
Show solution
Answer: -20

Step-by-step substitution & calculation:
- Evaluated Value: -20
=> Correct Option: B ($$-20$$)
Problem 6
Evaluate $$\frac{(3x + y)}{(2x - y)}$$ when $$x = 4$$ and $$y = -2$$
Show solution
Answer: 1

Step-by-step substitution & calculation:
- Evaluated Value: 1
=> Correct Option: B (1).

Remember

Substitute each specific given value for its corresponding variable carefully. Keep track of negative signs between terms.

3. Expressions with Powers

Handling variables with exponents

Evaluate $$x^n$$ or $$a \times x^n$$ when $$x = b$$

Handling variables raised to exponents during evaluation.

Example

Evaluate $$-2x^2 + 5$$ when $$x = -3$$

Show solution

Step 1: Substitute $$x = -3$$: $$-2(-3)^2 + 5$$.

Step 2: Exponent first: $$(-3)^2 = 9$$.

Step 3: Multiply: $$-2 \times 9 = -18$$.

Step 4: Add: $$-18 + 5 = -13$$.

Result: -13.

Practice problems

Problem 7
Evaluate $$x^2 + 4$$ when $$x = -3$$
Show solution
Answer: 13

Step-by-step substitution & calculation:
- Evaluated Value: 13
=> Correct Option: B (13).
Problem 8
Evaluate $$3x^2 - 2x + 1$$ when $$x = -2$$
Show solution
Answer: 17

Step-by-step substitution & calculation:
- Evaluated Value: 17
=> Correct Option: B (17).
Problem 9
Evaluate $$-x^3 + 2x^2 - 5x$$ when $$x = -2$$
Show solution
Answer: 26

Step-by-step substitution & calculation:
- Evaluated Value: 26
=> Correct Option: A (26).

Remember

Crucial: Always wrap negative base values in parentheses before applying exponents! Note that $$(-3)^2 = 9$$, whereas $$-3^2 = -9$$

4. Using PEMDAS

Following order of operations after substitution

Evaluate complex expression f(x, y, z) following PEMDAS

Following strict order of operations (PEMDAS) after substituting all variables.

Example

Evaluate $$(x + y)^2 \times z$$ when $$x = 1$$, $$y = 2$$, $$z = 3$$

Show solution

Step 1: Substitute: $$(1 + 2)^2 \times 3$$.

Step 2: Parentheses: $$(3)^2 \times 3$$.

Step 3: Exponent: $$9 \times 3$$.

Step 4: Multiply: 27.

Result: 27.

Practice problems

Problem 10
Evaluate $$(x + y) \times z$$ when $$x = 2$$, $$y = 3$$, $$z = 4$$
Show solution
Answer: 20

Step-by-step substitution & calculation:
- Evaluated Value: 20
=> Correct Option: A (20).
Problem 11
Evaluate $$(2x - y)^2 - 3z$$ when $$x = 3$$, $$y = 4$$, $$z = 2$$
Show solution
Answer: -2

Step-by-step substitution & calculation:
- Evaluated Value: -2
=> Correct Option: B (-2).
Problem 12
Evaluate $$[x^2 + (y - z)^3] / 2$$ when $$x = 4$$, $$y = 1$$, $$z = 3$$
Show solution
Answer: 4

Step-by-step substitution & calculation:
- Evaluated Value: 4
=> Correct Option: B (4).

Remember

First substitute all variables. Then follow PEMDAS rigorously: Parentheses, Exponents, Multiplication & Division (left-to-right), Addition & Subtraction (left-to-right).