Question

How do you solve \[y\, = \,4x\] and $x\, + \,y\, = \,5?$

Answer 1

Hint:We take either \[x\,or\,y\] in terms of each other and substitute the value in the second equation. Solving the equation, we get the values of both \[x\,and\,y\].

Complete step by step answer:Here it is given that \[y\, = \,4x\] so substituting this value of y which is in terms of x in the other equation that is $x\, + \,y\, = \,5$ .
So, we can write the equation $x\, + \,y\, = \,5$ in terms of \[x\,\] as shown below:
\[x\, + \,4x\, = \,5\]
This can be further simplified by adding the \[x\,\]terms, we can have
\[\,5x\, = \,5\]
On dividing by \[5\] on both sides of the equation, we get
\[\,\dfrac{{5x}}{5}\, = \,\dfrac{5}{5}\]
We just cancelled the same terms from the numerator and the denominators and after
Simplifying it we get
\[\,x\, = \,1\]
Now, as we get the value of \[x\,\], we can find the value of \[y\]
Since, \[y\, = \,4x\]
Substituting the value of \[x\,\] to find the value of \[y\], we can get
\[y\, = \,4 \times \,1\]
Therefore, \[1\,and\,\,4\]
Hence, the values of \[x\,and\,y\] are \[1\,\,and\,\,4\] respectively.

Note:
We can solve the question using other methods as well. We can start with second equation, find \[x\,\] in terms of \[y\] and then substitute the value of \[y\] to get the value of \[x\,\] as shown below:
$x\, + \,y\, = \,5$
Transposing \[y\] on other side of the equation and finding \[x\,\] in terms of \[y\]
Hence we get
\[x\, = \,5\, - \,y\]
No substituting this value of \[x\,\] in the equation\[y\, = \,4x\], we get
\[y\, = \,4 \times (5\, - \,y)\]
Solving this equation, we get
\[y\, = \,20 - 4y\]
Transposing \[4y\] on the left side we get
\[y\, + 4y = \,20\]
On solving this linear equation, we get the value of \[y\].
\[5y = \,20\]
We are just dividing, $y = \dfrac{{20}}{5}$
Hence we get,
\[y\, = \,4\]
Now, putting this value of \[y\] in the equation \[y\, = \,4x\] we get,
\[x\, = \,\dfrac{y}{4}\]
Therefore \[x\, = \,\dfrac{4}{4}\]
\[x\, = \,1\]
Therefore, we get the solution as \[(x,y)\, = \,(1,4)\]