Class 10 Maths Chapter 3 Introduction Part 1 Pair of Linear Equations in Two Variables NCERT

Pair of Linear Equations in Two Variables: Graphical and Algebraic Solutions

A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero. Its graph is always a straight line on the coordinate plane. In many real-world situations, we encounter conditions that lead to two linear equations simultaneously — for example, finding two numbers whose sum is 10 and whose difference is 4. This chapter in the CBSE Class 10 Mathematics syllabus teaches students to represent such pairs of equations graphically, determine the nature of their solutions, and solve them using algebraic methods including substitution, elimination, and cross-multiplication.

A pair of linear equations in two variables, a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, can be solved graphically by drawing both lines on the same coordinate plane. The point where the two lines intersect gives the solution (x, y). There are three possible cases depending on the relationship between the ratios of the coefficients. If a₁/a₂ ≠ b₁/b₂, the lines intersect at exactly one point — there is a unique solution and the system is consistent. If a₁/a₂ = b₁/b₂ = c₁/c₂, the two lines are coincident (they overlap completely) — there are infinitely many solutions and the system is consistent and dependent. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and never intersect — there is no solution and the system is inconsistent. These three cases correspond to the three geometric possibilities for two straight lines: intersecting, coincident, and parallel.

The substitution method involves expressing one variable in terms of the other from one equation and substituting this expression into the second equation, thereby reducing two equations in two variables to a single equation in one variable. For example, if x + 2y = 6, then x = 6 − 2y, and substituting this into a second equation gives a single equation in y alone. The elimination method involves multiplying one or both equations by suitable numbers so that adding or subtracting them eliminates one variable entirely. This method is often faster when the coefficients share a convenient relationship. The cross-multiplication method provides a direct formula: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), giving x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁) and y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁). Equations reducible to linear form are those that appear non-linear at first but can be transformed by substituting 1/x and 1/y (or similar) as new variables. Board examination word problems are abundant and include: finding the cost of individual items given total costs, determining speeds of vehicles given distance-time relationships, finding fractions where digits satisfy certain conditions, and solving age-related problems. The key strategy is to identify the two unknown quantities, assign variables, form two equations from the given conditions, and solve using any algebraic method.

• A pair of linear equations a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0 has a unique solution if a₁/a₂ ≠ b₁/b₂, infinitely many if all three ratios are equal, and no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
• Graphical method: plot both lines — intersecting (one solution), coincident (infinite solutions), parallel (no solution).
• Algebraic methods: substitution (express one variable, plug into other equation) and elimination (add/subtract to cancel a variable).
• Cross-multiplication gives a direct formula for both x and y from the coefficients.
• Word problems: identify two unknowns, form two equations from conditions, solve — common types include cost, speed, age, and digit problems.