Question 1 of 9beginner💡 UnderstandShort Answer3 marks

On comparing the ratios a1a2,b1b2\frac{a_1}{a_2}, \frac{b_1}{b_2} and c1c2\frac{c_1}{c_2}, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i)
5x−4y+8=05x - 4y + 8 = 0 7x+6y−9=07x + 6y - 9 = 0
Answer

Comparing the ratios, we get a1a2=57\frac{a_1}{a_2} = \frac{5}{7} and b1b2=−46=−23\frac{b_1}{b_2} = \frac{-4}{6} = \frac{-2}{3}. Since a1a2≠b1b2\frac{a_1}{a_2} \neq \frac{b_1}{b_2}, the lines intersect at a point.

(ii)
9x+3y+12=09x + 3y + 12 = 0 18x+6y+24=018x + 6y + 24 = 0
Answer

Comparing the ratios, we get a1a2=918=12\frac{a_1}{a_2} = \frac{9}{18} = \frac{1}{2}, b1b2=36=12\frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}, and c1c2=1224=12\frac{c_1}{c_2} = \frac{12}{24} = \frac{1}{2}. Since a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, the lines are coincident.

(iii)
6x−3y+10=06x - 3y + 10 = 0 2x−y+9=02x - y + 9 = 0
Answer

Comparing the ratios, we get a1a2=62=3\frac{a_1}{a_2} = \frac{6}{2} = 3, b1b2=−3−1=3\frac{b_1}{b_2} = \frac{-3}{-1} = 3, and c1c2=109\frac{c_1}{c_2} = \frac{10}{9}. Since a1a2=b1b2≠c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}, the lines are parallel.

Explanation

The nature of the lines representing a pair of linear equations is determined by comparing the ratios of the coefficients a1a2\frac{a_1}{a_2}, b1b2\frac{b_1}{b_2}, and c1c2\frac{c_1}{c_2}. According to the provided context (Table 3.1), if the ratios of aa and bb are not equal, the lines intersect. If all three ratios are equal, the lines are coincident. If the ratios of aa and bb are equal but the ratio of cc is different, the lines are parallel.

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