Question

A pair of linear equations \[x + 2y - 4 = 0,\,2x + 4y - 12 = 0\] has \[\_\_\] solutions.

Answer 1

Hint: We are given a pair of linear equations. We have to find the number of solutions they have. We first check if they are parallel or not, and to do so we find the ratios of corresponding coefficients. If the ratios of the coefficients of the two variables are equal but different from the ratio of constants, we say that the pair of linear equations is parallel. We will use this condition here to solve this question.
Formula used: For pair of equations\[ax + by - c = 0,\,dx + ey - f = 0\], the condition of them to be parallel is,
\[\dfrac{a}{d} = \dfrac{b}{e} \ne \dfrac{c}{f}\]

Complete step-by-step solution:
We are given a pair of linear equations as,
\[x + 2y - 4 = 0,\,2x + 4y - 12 = 0\]
We have to find the number of solutions they have. To do so we first check if they are parallel or not. We do it by finding the ratios of coefficients of corresponding variables and also the ratio of constants. For pair of equations\[ax + by - c = 0,\,dx + ey - f = 0\], the condition of them to be parallel is,
\[\dfrac{a}{d} = \dfrac{b}{e} \ne \dfrac{c}{f}\]
Using this, we find the ratio for given pair as,
\[
  \dfrac{1}{2} = \dfrac{2}{4} \ne \dfrac{{ - 4}}{{ - 12}} \\
   \Rightarrow \dfrac{1}{2} = \dfrac{1}{2} \ne \dfrac{1}{3} \\
 \]
Hence the condition of parallelism is satisfied. Thus the pair of linear equations is parallel.
Since, we know that pair of parallel linear equations has no solution, so we fill in the blank as
A pair of linear equations \[x + 2y - 4 = 0,\,2x + 4y - 12 = 0\] has a \[0\] solution.

Note: While trying to find the number of solutions to the pair of linear equations, we at first hand always see if they are parallel or perpendicular or not. Then we move ahead to other methods. Number solution means the number of times both the lines cut each in a Cartesian plane.