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Chapter 3

Pair of Linear Equations in Two Variables

NCERT solutions and explanations for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables — covering Graphical and algebraic methods for pairs of linear equations, consistency, substitution, elimination, and cross-multiplication.

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Questions

6
Q1

What is a pair of linear equations in two variables? Write its general form and explain what a solution means.

A pair of linear equations in two variables has the form a1x+b1y+c1=0a_1x+b_1y+c_1=0 and a2x+b2y+c2=0a_2x+b_2y+c_2=0. A solution is an ordered pair (x,y)(x,y) satisfying both equations.
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Q2

Explain the three graphical possibilities for a pair of linear equations.

The two lines may intersect at one point, be parallel, or coincide. Intersecting lines give one solution; parallel lines give no solution; coincident lines give infinitely many solutions.
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Q3

State the ratio test for consistency of a1x+b1y+c1=0a_1x+b_1y+c_1=0 and a2x+b2y+c2=0a_2x+b_2y+c_2=0.

If a1a2≠b1b2\frac{a_1}{a_2}\ne\frac{b_1}{b_2}, there is one solution. If a1a2=b1b2≠c1c2\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}, there is no solution. If all three ratios are equal, there are infinitely many solutions.
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Q4

Solve x−2y=0x-2y=0 and 3x+4y=203x+4y=20 by substitution.

From x−2y=0x-2y=0, x=2yx=2y. Substitute in 3x+4y=203x+4y=20: 3(2y)+4y=203(2y)+4y=20, so 10y=2010y=20, y=2y=2. Hence x=4x=4.
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Q5

Solve 2x+3y=112x+3y=11 and 2x−y=32x-y=3 by elimination.

Subtract the second equation from the first: (2x+3y)−(2x−y)=11−3(2x+3y)-(2x-y)=11-3, so 4y=84y=8, y=2y=2. Substitute in 2x−y=32x-y=3: 2x−2=32x-2=3, so x=52x=\frac52.
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Q6

State the cross-multiplication formula for solving a pair of linear equations.

For a1x+b1y+c1=0a_1x+b_1y+c_1=0 and a2x+b2y+c2=0a_2x+b_2y+c_2=0, xb1c2−b2c1=yc1a2−c2a1=1a1b2−a2b1\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}, provided a1b2−a2b1≠0a_1b_2-a_2b_1\ne0.
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