‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

When does a pair of linear equations have no solution?

When \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\) — the lines are parallel, so the system is inconsistent.

Name the algebraic methods of solving a pair of linear equations.

The substitution method, the elimination method and the cross-multiplication method.

When does a pair of linear equations have a unique solution?

When \(a_1/a_2 \ne b_1/b_2\); the lines intersect at one point and the system is consistent.

When does a pair have no solution and when infinitely many?

No solution when \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\) (parallel lines, inconsistent); infinitely many when \(a_1/a_2 = b_1/b_2 = c_1/c_2\) (coincident lines, dependent).

Solve \(x + y = 14\) and \(x - y = 4\) by elimination.

Adding the equations gives \(2x = 18\), so \(x = 9\); then \(y = 14 - 9 = 5\).

Solve \(2x + y = 8\) and \(x = y + 1\) by substitution.

Substituting \(x = y + 1\): \(2(y + 1) + y = 8\), so \(3y = 6\), \(y = 2\) and \(x = 3\).

Is the pair \(3x + 2y = 5\) and \(6x + 4y = 10\) consistent? Why?

Here \(a_1/a_2 = b_1/b_2 = c_1/c_2 = 1/2\), so the lines are coincident and the system is consistent with infinitely many solutions.

The sum of two numbers is 20 and their difference is 4. Frame the equations and find them.

Let the numbers be \(x\) and \(y\): \(x + y = 20\) and \(x - y = 4\); solving gives \(x = 12\) and \(y = 8\).

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Chapter 3: Pair of Linear Equations in Two Variables Class 10 Mathematics — summary, notes, extra questions & NCERT solutions

Q: Name three algebraic methods to solve a pair of linear equations.

Substitution, elimination, and cross-multiplication.

Q: What does a pair of linear equations in two variables represent graphically?

Two straight lines in a plane; their solution is the point (or points) common to both lines (refer to the graphs in your NCERT textbook).

Try one from this chapter

The graph of a linear equation in two variables is always a:

Summary

A pair of linear equations in two variables represents two straight lines, and solving the pair means finding the point(s) common to both. The chapter covers graphical and algebraic methods — substitution, elimination and cross-multiplication — to find solutions. The nature of the solution depends on the ratios of the coefficients: the lines may intersect at one point (a unique solution, consistent), be parallel (no solution, inconsistent), or coincide (infinitely many solutions, dependent). Word problems translate real situations into such equations, making this a powerful tool for modelling.

Graphical methodSubstitution methodElimination methodConditions for consistencyWord problems

Key terms

Consistent system

A pair of equations that has at least one solution.

Inconsistent system

A pair of equations with no common solution (parallel lines).

Substitution method

Solving by expressing one variable in terms of the other.

Elimination method

Adding/subtracting equations to remove one variable.

Extra questions & answers

EXERCISE 3.1

EXERCISE 3.2

EXERCISE 3.3

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Solve a pair of linear equations graphically

x +y =line 1

x +y =line 2

Intersection (3, 2) — consistent, unique solution.

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Consistent system

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A pair of equations that has at least one solution.

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