Equations Revision Notes

Algebraic expressions represent relationships between quantities using variables and operations. For example, to express the product of two consecutive even numbers, define the first even number as 2n (since even numbers are multiples of 2). The next consecutive even number is then 2n + 2. The product of these two numbers is expressed as (2n)(2n + 2).

Creating equations involves setting expressions equal to each other or to known values, often based on problem conditions. For instance, if the product of two consecutive even numbers is 48, the equation is (2n)(2n + 2) = 48.

• Define variables clearly, e.g., n represents an integer.
• Use algebraic expressions to represent quantities.
• Translate problem statements into equations by equating expressions to values or each other.

Linear equations have the form ax + b = c, where a, b, and c are constants and x is the variable. The goal is to isolate x on one side to find its value.

Example: Solve 3x + 4 = 10. Subtract 4 from both sides: 3x = 6. Divide both sides by 3: x = 2. Check by substituting back: 3(2) + 4 = 6 + 4 = 10, which is true.

When variables appear on both sides, collect like terms first. Example: Solve 5 - 2x = 3(x + 7). Expand right side: 5 - 2x = 3x + 21. Add 2x to both sides: 5 = 5x + 21. Subtract 21: -16 = 5x. Divide by 5: x = -16/5.

• Isolate the variable by inverse operations.
• Expand brackets before collecting like terms.
• Always check solutions by substitution.

Fractional equations contain variables in denominators. To solve, multiply both sides by the least common denominator (LCD) to clear fractions.

Example 1: Solve x/2 = 4. Multiply both sides by 2: x = 8.

Example 2: Solve (2x + 1)/3 + x/2 = 1. The LCD of 3 and 2 is 6. Multiply every term by 6: 6 × (2x + 1)/3 + 6 × x/2 = 6 × 1, which simplifies to 2(2x + 1) + 3x = 6. Expand: 4x + 2 + 3x = 6. Combine like terms: 7x + 2 = 6. Subtract 2: 7x = 4. Divide by 7: x = 4/7.

• Identify the least common denominator (LCD).
• Multiply entire equation by LCD to eliminate fractions.
• Simplify and solve the resulting linear equation.
• Check for values that make denominators zero (excluded values).

Simultaneous equations involve two or more equations with multiple variables. The solution is the set of variable values satisfying all equations simultaneously.

For two linear equations in two variables, use substitution or elimination.

Example: Solve 1) x + y = 7 2) 2x - y = 4 Add equations: (x + y) + (2x - y) = 7 + 4 → 3x = 11 → x = 11/3. Substitute into (1): (11/3) + y = 7 → y = 7 - 11/3 = (21/3) - (11/3) = 10/3. Solution: x = 11/3, y = 10/3.

• Use substitution: express one variable from one equation and substitute into the other.
• Use elimination: add or subtract equations to eliminate one variable.
• Check solutions by substituting into both equations.

When one equation is linear and the other nonlinear (e.g., quadratic), substitute the linear expression into the nonlinear equation to reduce to one variable.

Example: Solve 1) y = 2x + 1 (linear) 2) y^2 = 4x + 3 (nonlinear) Substitute y from (1) into (2): (2x + 1)^2 = 4x + 3. Expand left side: 4x^2 + 4x + 1 = 4x + 3. Subtract 4x + 3 from both sides: 4x^2 + 4x + 1 - 4x - 3 = 0 → 4x^2 - 2 = 0. Solve: 4x^2 = 2 → x^2 = 1/2 → x = ±1/√2. Then find y using (1).

• Substitute the linear equation into the nonlinear one.
• Simplify to a quadratic or polynomial equation in one variable.
• Solve using quadratic methods.
• Find corresponding values of the other variable.

Quadratic equations have the form ax^2 + bx + c = 0. Methods to solve include:

1. Factorization: Express the quadratic as a product of two binomials, then set each factor to zero.

Example: x^2 + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0, so x = -2 or x = -3. Check by substitution: (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0, true.

• Factorize when possible.
• Set each factor equal to zero to find solutions.