Linear Equations

Overview

Linear equations form the backbone of algebra and appear consistently in MP TET Varg-2 mathematics sections. A linear equation is an algebraic equation in which the highest power of the variable is 1—no squares, cubes, or higher powers. These equations represent straight lines when graphed on a coordinate plane.

For the MP TET exam, you must master solving equations in one variable (finding a single unknown value) and in two variables (finding pairs of values or graphing lines). Questions typically test your ability to form equations from word problems, solve them using standard methods, and interpret solutions graphically. This topic connects directly to real-world applications like age problems, profit-loss scenarios, and distance-time relationships commonly asked in the exam.

Understanding linear equations also prepares you for teaching upper-primary students, where building conceptual clarity about variables, constants, and the balance principle of equations is essential.

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Key Concepts

• **Linear equation in one variable**: An equation of the form ax + b = 0, where a ≠ 0. It has exactly one solution (one root).

• **Linear equation in two variables**: An equation of the form ax + by + c = 0, where a and b are not both zero. It has infinitely many solutions, each represented as an ordered pair (x, y).

• **Solution of an equation**: A value (or pair of values) that makes the equation true when substituted for the variable(s).

• **Graph of linear equation in two variables**: Always a straight line. Every point on this line is a solution of the equation.

• **System of linear equations**: Two or more linear equations considered together. The solution is the point(s) where their graphs intersect.

• **Consistent system**: Has at least one solution (lines intersect or coincide). **Inconsistent system**: Has no solution (parallel lines).

• **Transposition rule**: When moving a term from one side of an equation to the other, change its sign.

• **Balance principle**: Whatever operation you perform on one side of an equation, you must perform the same on the other side.

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Standard form (one variable) | ax + b = 0, solution: x = −b/a | | Standard form (two variables) | ax + by + c = 0 | | Slope-intercept form | y = mx + c, where m = slope, c = y-intercept | | Slope from two points | m = (y₂ − y₁)/(x₂ − x₁) | | Condition for parallel lines | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (no solution) | | Condition for coincident lines | a₁/a₂ = b₁/b₂ = c₁/c₂ (infinite solutions) | | Condition for intersecting lines | a₁/a₂ ≠ b₁/b₂ (unique solution) | | Substitution method | Express one variable in terms of the other, then substitute | | Elimination method | Add or subtract equations to eliminate one variable |

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Worked Examples

### Example 1: Linear Equation in One Variable **Problem**: Solve 3x − 7 = 2x + 5

**Solution**:

• Step 1: Bring variable terms to one side → 3x − 2x = 5 + 7
• Step 2: Simplify → x = 12
• **Answer**: x = 12

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### Example 2: Word Problem (One Variable) **Problem**: The sum of two consecutive odd numbers is 36. Find the numbers.

**Solution**:

• Let the first odd number = x
• Next consecutive odd number = x + 2
• Equation: x + (x + 2) = 36
• Simplify: 2x + 2 = 36 → 2x = 34 → x = 17
• **Answer**: The numbers are 17 and 19

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### Example 3: System of Two Variables (Substitution Method) **Problem**: Solve x + y = 10 and 2x − y = 5

**Solution**:

• From equation 1: y = 10 − x
• Substitute in equation 2: 2x − (10 − x) = 5
• Simplify: 2x − 10 + x = 5 → 3x = 15 → x = 5
• Find y: y = 10 − 5 = 5
• **Answer**: x = 5, y = 5

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### Example 4: Checking Consistency **Problem**: Determine if 2x + 3y = 6 and 4x + 6y = 10 have a solution.

**Solution**:

• Compare ratios: a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 6/10 = 3/5
• Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → Lines are parallel
• **Answer**: No solution (inconsistent system)

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Forgetting to change sign when transposing terms → writing 3x = 7 − 2 instead of 3x = 7 + 2 | Always reverse the sign when moving a term across the equals sign | | Assuming linear equation in two variables has only one solution | Remember: it has infinitely many solutions; each point on the line satisfies it | | Confusing "no solution" with "zero solution" → thinking x = 0 means no solution | No solution means the equations are inconsistent (parallel lines); x = 0 is a valid solution | | Making arithmetic errors when eliminating variables → not multiplying all terms | When multiplying an equation by a number, multiply every term including the constant | | Graphing errors → plotting y-intercept on x-axis | y-intercept is where the line crosses the y-axis (x = 0), not the x-axis | | Misapplying consistency conditions → mixing up the ratio comparisons | Memorise: equal ratios of all three (a, b, c) = coincident; first two equal but third different = parallel |

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Quick Reference

• **One variable equation ax + b = 0**: Solution is x = −b/a (always unique)

• **Two variable equation ax + by + c = 0**: Graph is a straight line; infinite solutions

• **Substitution**: Best when one variable has coefficient 1

• **Elimination**: Best when coefficients are easily made equal by multiplication

• **Parallel lines = No solution; Coincident lines = Infinite solutions; Intersecting lines = Unique solution**

• **Always verify your answer by substituting back into the original equation(s)**