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Question

Given the linear equation $$2x + 3y - 8 = 0$$, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) intersecting lines        (ii) parallel lines             (iii) coincident lines


Solution


a) Intersecting lines
Solution: For intersecting line, the linear equations should meet following condition:
$$\displaystyle \frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}$$

For getting another equation to meet this criterion, multiply the coefficient of x with any number and multiply the coefficient of y with any other number. A possible equation can be as follows:
$$4x + 9y - 8 = 0$$

(b) Parallel lines
Solution: For parallel lines, the linear equations should meet following condition:
$$\displaystyle \frac{a_{1}}{a_{2}}=  \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}$$

For getting another equation to meet this criterion, multiply the coefficients of x and y with the same number and multiply the constant term with any other number. A possible equation can be as follows:
$$4x + 6y – 24 = 0$$

(c) Coincident lines
Solution: For getting coincident lines, the equations should meet following condition;
$$\displaystyle \frac{a_{1}}{a_{2}}=  \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}$$

For getting another equation to meet this criterion, multiply the whole equation with any number. A possible equation can be as follows:
$$4x + 6y – 16 = 0$$

Mathematics
RS Agarwal
Standard X

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