Solving Linear Systems and Mathematical Progressions

Number of Solutions in Linear Systems

• a) Intersecting Lines (X): Consistent Independent System (Unique solution).
• b) Coincident Lines (/): Consistent Dependent System (Infinite solutions).
• c) Parallel Lines (//): Inconsistent System (No solutions).

Methods for Solving Systems

Substitution Method

1. Isolate one of the unknowns in the equations: From x - y = 3 to x = 3 + y.
2. Substitute the expression into the other equation: Given 2x - 3y = 4, substitute x: 2(3 + y) - 3y = 4.
3. Solve the equation for the remaining unknown: 6 + 2y - 3y = 4 → -y = 4 - 6 → -y = -2 → y = 2.
4. Calculate the value of the second unknown: Substitute y = 2 into x = 3 + y → x = 3 + 2 → x = 5.
5. Verify the solution: Ensure the resulting values satisfy the system.

Equalization Method

1. Isolate the same unknown in both equations:
   • x - y = 3 → x = 3 + y
   • 2x - 3y = 4 → x = (4 + 3y) / 2
2. Equate the expressions obtained: 3 + y = (4 + 3y) / 2.
3. Solve the equation for the unknown: 2(3 + y) = 4 + 3y → 6 + 2y = 4 + 3y → 6 - 4 = 3y - 2y → y = 2.
4. Calculate the value of the other unknown: Substitute y = 2 into x = 3 + y → x = 3 + 2 → x = 5.
5. Verify the solution: Confirm the solution is correct for the system.

Reduction Method

1. Equalize coefficients by multiplication: Multiply x - y = 3 by 2 → 2x - 2y = 6. Keep 2x - 3y = 4.
2. Subtract or add the equations: Subtracting the equations eliminates x, resulting in y = 2.
3. Solve the equation: In this case, y = 2 is found directly.
4. Calculate the value of the other unknown: Substitute y = 2 into x - y = 3 → x - 2 = 3 → x = 5.
5. Verify the solution: Confirm the resulting values satisfy the system.

Graphing Linear Equations

To graph the system, take both equations and solve for y. Create a table of values for x (e.g., -1, 0, 1) and plot the points. On the coordinate plane, the vertical axis represents y (up is positive, down is negative) and the horizontal axis represents x (right is positive, left is negative). Once plotted, the solution is the intersection point (secant lines).

Arithmetic Progression Formulas

• General Term: a_n = a₁ + (n - 1) · d
• Sum of n Terms: S_n = [(a₁ + a_n) / 2] · n

Geometric Progression Formulas

• General Term: a_n = a₁ · r^n-1
• Sum of n Terms: S_n = (a_n · r - a₁) / (r - 1)
• Infinite Sum (|r| < 1): S_∞ = a₁ / (1 - r)
• Product of n Terms: P_n = √((a₁ · a_n)ⁿ)

Quadratic Equation Formula

For an equation of the form ax² + bx + c = 0, the solution is:

x = [-b ± √(b² - 4ac)] / 2a