This section is dedicated to the foundational skill of constructing algebraic expressions and equations, pivotal for solving a wide range of algebraic problems.

Constructing Algebraic Expressions

Algebraic expressions represent quantities and their relationships without an equality sign. The construction process involves identifying variables and applying operations as described in scenarios or word problems.

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Example 1: Constructing an Expression for a Rectangle's Length

Given: A rectangle's length is 5 units more than its width, $w$.

Construct the expression:

• Length expression: $w + 5$

Example 2: Expression from a Scenario

Given: Triple a number decreased by 7.

Construct the expression:

• Expression: $3x - 7$

Constructing Algebraic Equations

Equations set two expressions equal to each other, allowing for the representation of scenarios where quantities are related through an exact balance.

Example 1: Equating an Expression to a Value

Given: The sum of three times a number and 11 equals 44.

Construct the equation:

• Equation: $3x + 11 = 44$

Example 2: Word Problem to Equation

Given: Twice the difference of a number and 5 equals the number increased by 3.

Construct the equation:

• Equation: $2(x - 5) = x + 3$

Practice Problems

Problem 1: Expression Construction

Given: Five less than four times a number.

Construct the expression:

• Expression: $4x - 5$

Problem 2: Equation Construction

Given: The product of a number and 7 equals 21.

Construct the equation:

• Equation: $7x = 21$

Problem 3: From Word Problem to Equation

Given: The perimeter of a rectangle is twice the sum of its length $(l)$ and width $(w)$. Express the perimeter $(P)$ in terms of $l$ and $w$.

Construct the equation:

• Perimeter equation: $P = 2(l + w)$