Equation Solving Formulas and Methods

Linear Equations

A linear equation in one variable can be written in the form:

To solve for x:

1. Move all terms with x to one side of the equation
2. Move all constant terms to the other side
3. Divide both sides by the coefficient of x

Example: 2x + 3 = 7

• Step 1: 2x = 7 - 3
• Step 2: 2x = 4
• Step 3: x = 4/2 = 2

Quadratic Equations

A quadratic equation can be written in the form:

The solutions can be found using the quadratic formula:

Where the discriminant (b² - 4ac) determines the number and type of solutions:

• If b² - 4ac > 0, there are two distinct real solutions
• If b² - 4ac = 0, there is one repeated real solution
• If b² - 4ac < 0, there are two complex solutions

Example: x² - 3x - 4 = 0

• a = 1, b = -3, c = -4
• Discriminant = (-3)² - 4(1)(-4) = 9 + 16 = 25
• x = (3 ± √25) / 2 = (3 ± 5) / 2
• x₁ = (3 + 5) / 2 = 4
• x₂ = (3 - 5) / 2 = -1

Systems of Linear Equations

A system of two linear equations in two variables can be written as:

There are several methods to solve such systems:

Substitution Method

1. Solve one equation for one variable in terms of the other
2. Substitute this expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the value of the first variable

Elimination Method

1. Multiply one or both equations by constants to make the coefficients of one variable the same (but opposite in sign)
2. Add the equations to eliminate one variable
3. Solve for the remaining variable
4. Substitute back into either original equation to find the other variable

Cramer's Rule

For a system of two equations:

Where (a₁b₂ - a₂b₁) is the determinant of the coefficient matrix.

Example: 2x + y = 5, x - y = 1

Using elimination:

• Add the equations: 3x = 6
• Solve for x: x = 2
• Substitute back: 2(2) + y = 5
• Solve for y: 4 + y = 5, y = 1
• Solution: x = 2, y = 1