Question

Solve the following pair of linear equations by the substitution method. \[3x-y=3\] and \[9x-3y=9\].

Answer 1

Hint: For a given equations of two lines \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\]and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\], if \[\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}\] then they are parallel to each other. If two parallel lines have an infinite number of points common, then they are coincident lines. Coincident lines represent the same lines.

Complete step-by-step solution:
From the question it is clear that we have to solve a pair of linear equations \[3x-y=3\] and \[9x-3y=9\]. by the substitution method.
Consider
\[\Rightarrow 3x-y=3\]………………(1)
\[\Rightarrow 9x-3y=9\]…………….(2)
Now let us try to compare equation (1) with \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\]
\[{{a}_{1}}=3,{{b}_{1}}=-1,{{c}_{1}}=3\]
Now let us try to compare equation (2) with \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]
\[{{a}_{2}}=9,{{b}_{2}}=-3,{{c}_{2}}=9\]
Now we will check the condition \[\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}\]to verify whether these lines are parallel or not.
\[\Rightarrow \dfrac{3}{9}=\dfrac{-1}{-3}=\dfrac{3}{9}\]
After simplification we get
\[\Rightarrow \dfrac{1}{3}=\dfrac{1}{3}=\dfrac{1}{3}\]
So, we can conclude that these lines are parallel.
Now let us check whether these lines are coincident lines or not.
From the equation \[3x-y=3\] we can write \[y=3x-3\]
Now put \[y=3x-3\] in equation \[9x-3y=9\].
Now we can write
\[\Rightarrow 9x-3y=9\]
\[\Rightarrow 9x-3\left( 3x-3 \right)=9\]
\[\Rightarrow 9x-9x+9=9\]
After simplification we get,
\[\Rightarrow 9=9\]
\[9=9\] is true for any values of \[x\] and \[y\].
So, the lines \[3x-y=3\],\[9x-3x=9\] have infinitely many solutions and these lines are coincident lines.
Now let us try find one solution for both the lines
Now put \[y=0\] in \[3x-y=3\]
\[\Rightarrow 3x-y=3\]
\[\Rightarrow 3x-0=3\]
\[\Rightarrow 3x=3\]
After simplification we get
\[\Rightarrow x=1\]
At \[y=0\],\[x=1\]
\[\left( 1,0 \right)\] is one of the solutions of lines \[3x-y=3\],\[9x-3x=9\].

Since they are coincident lines, they represent the same line.

Note:Since\[\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}\], many students may think that given lines are parallel so they do not have any solutions in common. But in some situations parallel lines can also be coincident lines.
 In this case we have to check whether these lines are coincident lines. Otherwise students will do this question wrong.