Pre-U Mathematics CAIE

Algebraic expressions and equations

Algebraic Expressions

• An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply, and divide).
• Variables are symbols used to represent unknown numbers or values. They are typically represented by letters, such as x, y, or z.
• The terms of an algebraic expression are the parts of the expression that are added or subtracted. Each term can be a number, a variable, or a product or quotient of numbers and variables.
• Algebraic expressions can be simplified by combining like terms, which are terms with exactly the same variables and powers.
• The coefficient is the numerical factor in any term. For example, in 5x, the coefficient is 5.
• To evaluate an algebraic expression, you replace the variable with a given number and then perform the operations in the expression.

Algebraic Equations

• An algebraic equation is a statement of equality between two algebraic expressions.
• The solution of an algebraic equation is the numerical value of the variable that makes the equation true.
• An algebraic equation can be solved by isolating the variable, which often involves addition, subtraction, multiplication, or division.
• Sometimes, you'll need to use the distributive property to remove parentheses before isolating the variable.
• Quadratic equations are algebraic equations that have a degree of 2. Their solutions can be found using factoring, completing the square, or using the quadratic formula.
• System of equations is a set of two or more equations that all contain the same set of variables. Systems can be solved by graphing, substitution, or elimination.
• A linear equation forms a straight line when graphed. It has no exponents higher than 1, and its graph is a straight line. You can solve a linear equation by isolating the variable.

Algebraic Formulae

• A formula is a type of equation that shows the relationship between different variables.
• An algebraic formula can be used to calculate measurements, like the area of a circle or rectangle, or the volume of a cylinder or sphere.
• The formula for a linear sequence is an + b, where a is the common difference and b is the first term.
• The formula for a quadratic sequence is an^2+ bn + c, where a, b and c are constants.
• The formula for the sum of an arithmetic series is Sn = n/2 * (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.