Linear Equations

Core Formula

Simultaneous linear equations involve two or more linear equations with the same set of variables. For SSC CGL, you primarily deal with two equations and two variables (x, y).

General form: a₁x + b₁y = c₁ a₂x + b₂y = c₂

Where a₁, b₁, c₁, a₂, b₂, c₂ are constants, and x, y are the variables to be found.

Worked Example 1 (Elimination Method)

Problem: Solve for x and y:

1. 2x + 3y = 13
2. 4x - y = 5

Solution:

• Step 1: Aim to make the coefficients of one variable equal in magnitude. Multiply equation (2) by 3 to make the 'y' coefficients +3 and -3. 3 * (4x - y) = 3 * 5 12x - 3y = 15 (Equation 3)
• Step 2: Add Equation (1) and Equation (3) to eliminate 'y'. (2x + 3y) + (12x - 3y) = 13 + 15 14x = 28 x = 2
• Step 3: Substitute the value of x back into either original equation (e.g., Equation 1). 2(2) + 3y = 13 4 + 3y = 13 3y = 9 y = 3
• Answer: x = 2, y = 3

Worked Example 2 (Substitution Method)

Problem: Solve for x and y:

1. x + 2y = 7
2. 3x - y = 7

Solution:

• Step 1: Express one variable in terms of the other from one equation. From Equation (1): x = 7 - 2y (Equation 3)
• Step 2: Substitute this expression for x into the other equation (Equation 2). 3(7 - 2y) - y = 7 21 - 6y - y = 7 21 - 7y = 7
• Step 3: Solve for y. -7y = 7 - 21 -7y = -14 y = 2
• Step 4: Substitute the value of y back into Equation (3) to find x. x = 7 - 2(2) x = 7 - 4 x = 3
• Answer: x = 3, y = 2

Shortcuts & Tricks

1. Check Options: For objective questions, often the fastest way is to plug in the given options (x, y pairs) into both equations. The pair that satisfies both is the answer.
2. Sum and Difference: If you have equations like x + y = S and x - y = D: x = (S + D) / 2 y = (S - D) / 2 This is incredibly fast for specific problem types. E.g., Sum of two numbers is 45, difference is 11. Larger number (x) = (45+11)/2 = 28. Smaller number (y) = (45-11)/2 = 17.
3. Coefficient Manipulation: Look for opportunities to add or subtract equations directly if coefficients are already similar or easily made similar (e.g., ax + by = c and bx + ay = d). Adding them gives (a+b)(x+y) = c+d, subtracting gives (a-b)(x-y) = c-d. This can simplify finding x+y and x-y first.

Common Mistakes

1. Sign Errors: Careless mistakes when multiplying by negative numbers or transposing terms across the equals sign.
2. Calculation Errors: Simple arithmetic mistakes, especially when dealing with larger numbers or fractions.
3. Not Checking Solutions: Always quickly substitute your derived x and y values back into both original equations to verify. This takes seconds and prevents losing marks.

Core Formula

Linear equations in one variable are equations involving a single variable (usually 'x') raised to the power of 1. Their standard form is:

ax + b = 0

where 'a' and 'b' are constants, and 'a' ≠ 0. Another common form is ax + b = cx + d. The goal is to find the value of the variable 'x' that satisfies the equation.

Worked Example 1

Question: Solve for x: 5x - 8 = 2x + 7

Solution:

1. Group terms with 'x' on one side and constant terms on the other. 5x - 2x = 7 + 8
2. Simplify both sides. 3x = 15
3. Isolate 'x' by dividing both sides by its coefficient. x = 15 / 3 x = 5

Worked Example 2

Question: If 7 is subtracted from three times a number, the result is 11. Find the number.

Solution:

1. Let the unknown number be 'x'.
2. Translate the word problem into an equation: "Three times a number" is 3x. "7 is subtracted from three times a number" is 3x - 7. "The result is 11" means 3x - 7 = 11.
3. Solve the equation: 3x - 7 = 11 3x = 11 + 7 (Transposing -7 to the right side) 3x = 18 x = 18 / 3 x = 6 The number is 6.

Shortcuts & Tricks

1. Transposition Rule: When moving a term from one side of the equation to the other, change its sign. Addition becomes subtraction, multiplication becomes division, and vice-versa. Master this for speed.
2. Mental Math: For simpler equations, try to solve them mentally without writing every step. For example, 2x + 4 = 10 -> 2x = 6 -> x = 3 should be a quick mental calculation.
3. Option Checking (for MCQs): If solving an equation seems complex or time-consuming, especially in word problems, plug in the given options into the equation or the problem statement. The option that satisfies the condition is the answer. This can often be faster than direct solving.

Common Mistakes

1. Sign Errors: The most frequent mistake is incorrect sign changes during transposition (e.g., x - 5 = 10 becomes x = 10 - 5 instead of x = 10 + 5). Always double-check signs.
2. Calculation Errors: Simple arithmetic mistakes (addition, subtraction, multiplication, division) can lead to wrong answers. Practice mental math to minimize these.
3. Misinterpreting Word Problems: Incorrectly translating the verbal statement into a mathematical equation. Read the problem carefully, identify keywords (e.g., "sum," "difference," "product," "quotient," "times," "is"), and form the equation step-by-step.