Chapter 2 — Linear Equations In One Variable

What is a linear equation in one variable?

A linear equation in one variable is an equation that can be written in the form ax + b = 0, where x is the unknown (variable), and a and b are constants with a ≠ 0. The variable has power 1 (degree 1).

Key characteristics

• Only one variable appears (for example, x).
• The highest power of the variable is 1.
• It has a unique solution when a ≠ 0, or may have no solution or infinitely many solutions in special cases.

How to solve a linear equation — basic steps

1. Simplify both sides (expand brackets, combine like terms).
2. Use the properties of equality to isolate the variable: you can add/subtract the same number from both sides or multiply/divide both sides by the same non-zero number.
3. If the variable appears on both sides, bring all terms with the variable to one side and constants to the other.
4. Solve for the variable and always check the solution by substituting it into the original equation.

Properties of equality (allowed operations)

• Add or subtract the same number on both sides.
• Multiply or divide both sides by the same non-zero number.
• For equations with fractions, multiply both sides by the LCM of denominators to clear fractions.

Special cases

• If after simplification you get a false statement like 0 = 5, the equation has no solution.
• If you get a true identity like 0 = 0 (all terms cancel), the equation has infinitely many solutions.

Meaning on a graph

Rewrite the equation as y = ax + b and plot that line. The solution of ax + b = 0 is the x-intercept of the line (point where y = 0). For equations in one variable you can also represent the final solution on a number line.

What is equality? An equality is a statement that two expressions have the same value, written as a = b. The properties of equality tell us what operations we may perform on both sides of an equation without changing its truth.

Basic properties

• Addition property: If a = b, then a + c = b + c for any number c. Adding the same number to both sides preserves equality.
• Subtraction property: If a = b, then a − c = b − c for any number c. Subtracting the same number from both sides preserves equality.
• Multiplication property: If a = b, then a·c = b·c for any number c. Multiplying both sides by the same number preserves equality.
• Division property: If a = b and c ≠ 0, then a/c = b/c. Dividing both sides by the same nonzero number preserves equality.
• Transitive property: If a = b and b = c, then a = c.
• Substitution principle: If a = b, then one can replace a by b in any expression or equation.

Why these hold (intuition)
Think of an equality as a balanced scale. If both pans balance, adding/removing the same weight, or scaling both pans by the same factor (except zero) keeps the balance. That is the intuition behind these properties.

Using properties to solve linear equations
To solve an equation for a variable, perform inverse operations in a way that keeps both sides equal: (1) remove/add constants (use addition/subtraction property), (2) remove coefficients (use multiplication/division property). Always do the same operation to both sides and check the solution by substitution.

Important cautions

• Never divide by 0. Division property requires the divisor ≠ 0.
• When multiplying or dividing by a negative number, the equality remains true; for inequalities the direction changes, but that does not apply here.
• Operations must be applied to both sides of the equation to preserve equality.

What is a linear equation in one variable? A linear equation in one variable is an equation that can be written in the form ax + b = c, where x is the unknown, and a, b, c are known numbers with a ≠ 0. The highest power of x is 1.

Goal: Find the value of x that makes the equation true.

Basic idea (balancing method): An equation is like a balanced scale. Whatever you do to one side must be done to the other side to keep it balanced. Use inverse operations to isolate x:

• Add or subtract the same number from both sides to remove constant terms.
• Multiply or divide both sides by the same nonzero number to remove coefficients of x.

Steps to solve ax + b = c (standard procedure):

1. Subtract b from both sides: ax = c − b.
2. Divide both sides by a: x = (c − b)/a.

Other common cases:

• If variable appears on both sides, collect x-terms on one side: ax + b = dx + e ⇒ (a − d)x = e − b ⇒ x = (e − b)/(a − d), provided a ≠ d.
• If there are fractions, multiply both sides by the least common denominator (LCD) first to remove denominators.
• Special outcomes: If simplification leads to a true statement like 0 = 0, the equation is an identity (infinitely many solutions). If it leads to a false statement like 0 = 5, there is no solution (inconsistent).

Checking: Always substitute the found value of x back into the original equation to verify it satisfies the equation.

What it means: An equation with the variable on both sides is a linear equation where the unknown (usually x) appears in terms on both the left and the right side of the equals sign, for example 2x + 3 = x + 7. The goal is to find the value(s) of x that make the equality true.

General form and short method: A general linear equation with variable on both sides can be written as ax + b = cx + d. To solve, collect all x-terms on one side and constants on the other:

1. Subtract (or add) terms to get all x-terms together: (a - c)x = d - b.
2. If a - c ≠ 0, divide both sides by a - c to get x = (d - b)/(a - c).

Special cases:

• If a - c = 0 and d - b = 0, then both sides are identical for every x; there are infinitely many solutions (identity).
• If a - c = 0 but d - b ≠ 0, then no value of x will satisfy the equation; there is no solution (contradiction).

Steps to solve (practical):

1. Simplify both sides (remove parentheses, combine like terms).
2. Move variable terms to one side (use +/− same term on both sides).
3. Move constants to the opposite side.
4. Combine like terms and divide by the coefficient of x (if nonzero).
5. Check by substituting back into the original equation.

Checking and pitfalls: Always check the solution in the original equation. Do not divide by an expression that could be zero. Keep operations balanced by doing same operation on both sides.

An equation involving parentheses (brackets) is a linear equation in which one or more terms are grouped inside parentheses or other brackets, for example 3(x + 2) = 15 or 2[3x - (x + 4)] = 10. To solve such equations we use the distributive property to remove brackets, collect like terms, and then isolate the variable.

• Types of brackets: (), [], {} — they all indicate grouping. Work from the innermost brackets outward when brackets are nested.
• Distributive property (key tool): a(b + c) = ab + ac and a(b - c) = ab - ac. Use this to expand expressions inside brackets.
• Steps to solve:
  1. Remove brackets by expansion using the distributive law (or by factoring if convenient).
  2. Simplify each side: combine like terms and clear any parentheses left from negatives.
  3. Bring variable terms to one side and constants to the other (use addition/subtraction).
  4. Divide (or multiply) to get the variable alone.
  5. Check the solution by substituting it back into the original equation.
• Common mistakes: forgetting to distribute a negative sign, distributing incorrectly when there are nested brackets, or failing to check the solution at the end.

Example strategy for nested brackets: first simplify the innermost parentheses, then expand outward. When the same expression appears on both sides, you may factor it to simplify the equation.

What this topic means

An equation involving fractions and/or decimals is a linear equation in one variable where terms contain fractional numbers (like 2/3) or decimals (like 0.75). The goal is the same as any linear equation: isolate the variable and find its numerical value.

General strategy — fractions

1. Write the equation clearly and identify all denominators.
2. Find the least common multiple (LCM) of the denominators.
3. Multiply every term on both sides of the equation by the LCM to clear denominators; this gives an equivalent equation with integers.
4. Solve the resulting linear equation (bring variable terms together, isolate the variable).
5. Check the solution by substituting it back into the original equation.

General strategy — decimals

1. Count the maximum number of decimal places (say n) among all terms.
2. Multiply every term on both sides by 10^n to convert decimals to integers.
3. Solve the resulting integer linear equation and check the solution in the original equation (or after converting back to decimals).

Mixed fractions and decimals

Either convert decimals to fractions (or fractions to decimals), or use a combined approach: clear denominators (using LCM) and remove decimals (multiply by 10^n). Always perform the same multiplication on both sides to keep the equation equivalent.

Important checks and cautions

• Never divide or multiply by an expression that could be zero unless you check that case separately (for linear one-variable equations this typically means avoid dividing by the variable if it could be zero).
• Always substitute your solution into the original equation to verify (especially when clearing denominators).

What it means
An equation is called reducible to linear form when it is not written as ax + b = 0 at first, but can be transformed into this standard linear form by simple algebraic operations (expanding brackets, clearing fractions, cross‑multiplication or substitution). Once reduced, it can be solved by the usual method for linear equations.

General steps to reduce and solve

1. Simplify both sides: expand brackets and collect like terms.
2. If there are fractions, multiply both sides by the least common denominator (LCD) to remove denominators. (Be careful about values that make denominators zero.)
3. If you have a proportion or a simple rational equation, apply cross‑multiplication when appropriate.
4. Bring all x‑terms to one side and constants to the other, factor if needed, and reduce to ax + b = 0.
5. Solve x = -b/a (provided a ≠ 0) and check any domain restrictions (e.g., denominators not zero).

Important caution: If you multiply both sides by an expression that may be zero for some x (for example, a denominator containing x), mention and exclude those values before finalizing the solution. Multiplying can introduce extraneous solutions.

Short example (illustration)
Solve 3(2x - 4) = 5 - x.
Expand: 6x - 12 = 5 - x. Collect x: 6x + x = 5 + 12 = 17. So 7x = 17 and x = 17/7.

What the topic is about
When we solve linear equations in one variable, we usually get one value for the variable. But two special outcomes are possible after simplifying an equation:

• No solution (inconsistent): The equation simplifies to a contradiction such as 0 = non-zero (for example, 0 = 5). This means there is no value of the variable that can make the equation true.
• Infinitely many solutions (identity): The equation simplifies to a true statement like 0 = 0. This means every value of the variable satisfies the equation.

How to detect these cases (general approach)
1. Expand brackets and simplify both sides. 2. Collect the terms containing the variable on one side and constants on the other. 3. You obtain an equation of the form A x = B.

• If A ≠ 0, there is exactly one solution x = B/A.
• If A = 0 and B ≠ 0, we get 0 = B (B ≠ 0) → no solution.
• If A = 0 and B = 0, we get 0 = 0 → infinitely many solutions.

Equivalent condition for two linear expressions
For equations of the form a x + b = c x + d, bring x-terms together to get (a - c)x = d - b. Then apply the same test on coefficient (a - c) and constant (d - b).

Interpretation on graphs
Treat the left and right sides as two straight lines y = a x + b and y = c x + d in the xy-plane. If the lines are:

• Intersecting at one point → one solution (x-coordinate of intersection).
• Parallel (same slope, different intercepts) → no solution.
• Coincident (exactly the same line) → infinitely many solutions.

What it means
Verification and checking of solutions means confirming that a number obtained after solving an equation really satisfies the original equation. For a linear equation in one variable, after finding x = value, we substitute this value into the original equation and see whether the left-hand side (LHS) equals the right-hand side (RHS).

Why it is important
Mistakes in algebraic manipulation (sign errors, wrong simplification, illegal division by zero) can give wrong answers. Verification catches these errors. It also helps handle special cases like no solution or infinitely many solutions.

Step-by-step checking procedure

• 1. Solve the equation to get a candidate solution(s).
• 2. Substitute the candidate value into the original equation exactly as given (do not use the rearranged form only).
• 3. Simplify both sides. If LHS = RHS, the value is a valid solution.
• 4. If any algebraic step used division, make sure you did not divide by zero; check coefficients that could be zero (0x = 0 or 0x = c).

Special cases

• If substitution gives a true statement like 0 = 0, the equation is an identity and has infinitely many solutions (every real number is a solution).
• If substitution gives a false statement like 0 = 5, the equation has no solution.

Quick tips

• Always check by substituting into the original equation, not a changed form that may hide domain restrictions.
• Use the balance idea: every operation performed on one side must be performed on the other to preserve equality.

Worked verification example (illustrative)
Solve and verify: 2(x + 3) = x + 11.
Solution steps: 2x + 6 = x + 11 → 2x - x = 11 - 6 → x = 5. Verification: Substitute x = 5 into original: 2(5 + 3) = 5 + 11 → 2(8) = 16 → 16 = 16, so x = 5 is correct.

What it means
Formulation and solution of word problems means translating a real-life situation into a linear equation in one variable, solving the equation, and interpreting the result in the context of the problem.

Steps to solve

1. Read carefully: Understand what is asked and the quantities given.
2. Choose a variable: Let one unknown be x (or any letter). State clearly what x represents (for example, x = Raj's present age).
3. Translate to an equation: Use relationships given in words to form an equation. Replace phrases like 'sum', 'difference', 'product', 'increased by', 'decreased by', 'times', 'per' with +, −, ×, ÷, etc.
4. Solve the equation: Use algebraic operations (collect like terms, transpose, multiply/divide) to find x.
5. Check and interpret: Substitute back into the original statement to check; give the answer with units and context.

Common modelling hints
- Use n, n+1, n+2 for consecutive integers; n, n+2 for consecutive even/odd integers.
- For ages, phrases like '5 years hence' mean add 5: (x+5). '5 years ago' means (x−5).
- For money problems, keep currency units consistent.
- For distance-speed-time with constant speed, use distance = speed × time and form linear equations when one variable remains unknown.

Checking and units
Always check the solution in the original wording and ensure the result is reasonable (ages non-negative, integers when required, etc.).

What this topic covers
Problem-solving strategies for linear equations in one variable teach how to convert word problems into an equation of the form ax + b = 0 (or more generally, expressions with x on both sides), solve for x, and check the answer. The aim is to develop systematic steps so you can solve any one-variable linear problem correctly and quickly.

Step-by-step strategy

• 1. Read and understand: Read the problem twice. Identify what is asked and what information is given. Look for keywords (total, more than, less than, twice, thrice, difference, per).
• 2. Choose a variable: Let x denote the unknown quantity. Write exactly what x represents (e.g., let x = the number of pens).
• 3. Translate to an equation: Convert sentences to algebraic expressions and form an equation using equality or relationships described in the problem.
• 4. Simplify both sides: Expand brackets, combine like terms, and bring like terms together (collect x-terms on one side and constants on the other).
• 5. Solve for x: Use inverse operations (add/subtract, multiply/divide) to isolate x. Keep the equation balanced: perform the same operation on both sides.
• 6. Check and interpret: Substitute the obtained value back into the original statement to verify. Make sure the solution makes sense in context (e.g., non-negative ages).

Helpful tips and shortcuts

• Estimate first: Quickly estimate the answer to spot calculation errors later.
• Clear fractions early: Multiply both sides by the LCM of denominators to avoid fractional arithmetic.
• Use the balance model: Visualize the equation as a balance scale — whatever you do to one side, do to the other.
• Move x-terms together: If x appears on both sides, subtract or add to move all x terms to one side before simplifying.
• Watch for special cases: If after simplification you get 0x = c (c ≠ 0) → no solution; if 0x = 0 → infinitely many solutions.
• Clear parentheses carefully: Distribute negative signs correctly when removing brackets.
• Check domain constraints: Some problems require integer or positive solutions only; discard extraneous answers that violate these.

Visual aids
Use diagrams (bar models), number lines, and simple graphs to understand and check solutions. A straight-line graph of y = ax + b helps find the x-intercept (where y = 0) which is the solution of ax + b = 0.