Algebraic Expressions: Simplification, Evaluation, Order of Operations

12 Questions

What is the result of simplifying the expression $3x + (2x - 4)$?

$5x - 4$

In the expression $4a - (a + 7)$, what is the simplified form?

$3a - 7$

If $x = -2$, what is the value of $2x^2 + 3x - 1$?

$-5$

When evaluating $5(y - 2)^2$ for $y = 4$, what is the result?

$44$

Following the order of operations, what is the value of $5(7 + 3) - 2^2$?

$25$

What is the combined form of $3x^2 + x + x^2 + 2$?

$4x^2 + x + 2$

In the expression $5(9 + 2) - 4^2$, which operation should be performed first according to the order of operations?

Addition

If we substitute $x = 3$ and $y = 5$ into the expression $2x + 3y - y$, what is the result?

$12$

When evaluating $4(a + 2) - 3$ for $a = 5$, what is the resulting value?

$17$

Which property of algebra allows us to combine terms like $2x^2$, $-5x^2$, and $7x^2$ into a simplified form?

Associative Property

What is the value of $x$ in the expression $10x - 3x + 8 = 15$?

$4$

When simplifying the expression $3(x + 5) - x + 2$, which part should be calculated first?

$3(x + 5)$

Study Notes

Algebraic Expressions: Simplification, Evaluation, Order of Operations, Substitution, Combining Like Terms

Algebraic expressions are mathematical equations consisting of terms connected by algebraic operators such as addition, subtraction, multiplication, division, exponentiation, modulo, and more. These expressions can be manipulated using various methods to achieve different results. This article will explore five fundamental aspects of working with algebraic expressions: simplifying expressions, evaluating with variables, order of operations, substitution, and combining like terms.

Simplifying Expressions

Simplifying algebraic expressions involves rewriting them in their most basic form while maintaining their intrinsic properties. Common methods include removing unnecessary parentheses, combining like terms, and applying known arithmetic properties. For example, we could transform (a + b) - (c + d) into ac - bd, which is the sum of ac and -bd.

Evaluating With Variables

Evaluating algebraic expressions involves finding the value of a specific expression based on given values for its variables. For instance, if the equation is x^2 = 9, we would evaluate it by setting x = 3 and obtaining 81, which is the square of 3. To solve this problem, we must first determine whether both sides of the equation are equal; in this case, they are. Then, we substitute the expression's variable x with the given value 3 and compute the result.

Order Of Operations

The order of operations specifies the sequence in which mathematical symbols are applied when solving complex expressions. It is commonly remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Applying these rules consistently helps ensure accurate calculation of complex algebraic expressions.

Substitution

Substitution refers to replacing one term with another in an algebraic expression. For example, the Pythagorean theorem states that in a right triangle, the squares of the lengths of the two legs are equal to the square of the length of the hypotenuse. Mathematically, this can be written as a² + b² = c², where a, b, and c represent the lengths of the respective sides. If we want to find the value of a in terms of b and c, we need to solve for a in the equation, resulting in a = sqrt(c² - b²).

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable raised to the same power, as per the distributive property of algebra. For example, consider the expression 3x³ + x³ + 4x³. Since all three terms involve x³, we can combine them by adding their coefficients, giving us 3x³ + x³ + 4x³ = 8x³. By simplifying further, the expression becomes 2x³, representing the combination of the original terms in the correct order.