Question

How do you solve this system of equations:
\[5x - 2y = 0\] and \[ - 4x + 3y = 7\]?

Answer 1

Hint: We will multiply both the equations with some constants and then we will subtract or add these two equations to solve for \[x,y\]. On doing some simplification we get the required answer.

Formula used:
If two equations are given in the below format:
\[Ax + By = C\]
\[Mx + Ny = D\]
Then we can equate both the equations using the value of \[x\].
To get the value of \[x\] from the first equation, we can perform the following steps:
\[Ax + By = C\]
\[ \Rightarrow Ax = C - By\]
\[ \Rightarrow x = \dfrac{{C - By}}{A}\]
Again, we can do the same operation for the second equation also:
\[Mx + Ny = D\]
\[ \Rightarrow Mx = D - Ny\]
\[ \Rightarrow x = \dfrac{{D - Ny}}{M}\]
Now, if we equate these two above equations, we can find the value of \[y\]from the above equations.
Like,
\[\dfrac{{D - Ny}}{M} = \dfrac{{C - By}}{A}\]
\[ \Rightarrow AD - ANy = CM - BMy\]
\[ \Rightarrow y(BM - AN) = (CM - AD)\]
\[ \Rightarrow y = \dfrac{{(CM - AD)}}{{(BM - AN)}}.\]
Now, putting these values of \[y\], in any of the above equations, we can find the value of \[x\].

Complete step by step answer:
The given two equations are as following:
\[5x - 2y = 0................(1)\] and,
\[ - 4x + 3y = 7...............(2)\]
Now, multiply the equation \[(1)\] with \[4\] on the both sides of the equation, we get:
\[ \Rightarrow 5 \times 4x - 2 \times 4y = 0 \times 4\].
By doing the simplification, we get:
\[ \Rightarrow 20x - 8y = 0................(3)\]
Now, multiply the equation \[(2)\] with \[5\] on the both sides of the equation, we get:
\[ \Rightarrow - 4 \times 5 \times x + 3 \times 5 \times y = 7 \times 5\].
By doing further simplification, we get:
\[ \Rightarrow - 20x + 15y = 35...............(4)\]
Now by adding the equation \[(3)\]and \[(4)\], we get:
\[ \Rightarrow (20x - 8y) + ( - 20x + 15y) = 0 + 35\].
Now, add the \[x\]term and \[y\]term separately, we get:
\[ \Rightarrow (20x - 20x) + ( - 8y + 15y) = 35\].
Now, performing addition and subtraction, we get:
\[ \Rightarrow 7y = 35\].
Now, divide both the sides by \[7\], we get:
\[ \Rightarrow y = \dfrac{{35}}{7} = 5\].
So, if we put the value of \[y\]in the equation \[(1)\], we get:
\[ \Rightarrow 5x - 2 \times 5 = 0\].
Now, doing the multiplication, we get:
\[ \Rightarrow 5x - 10 = 0\].
Now, take the constant terms on the right hand side, we get:
\[ \Rightarrow 5x = 10\].
Now, divide both the sides by \[5\], we get:
\[ \Rightarrow x = \dfrac{{10}}{5} = 2\].

Therefore, the solution of the above question is \[x = 2\] and \[y = 5\].

Note: Alternative method:
We can take all the terms except \[x\] on the right hand side and then we can equate both the equations to find the value of \[y\].
Given equations are:
\[5x - 2y = 0................(1)\] and,
\[ - 4x + 3y = 7...............(2)\]
So, if we take the ‘\[2y\]’ on the R.H.S, we get:
\[ \Rightarrow 5x = 0 + 2y\].
Now, divide both sides by \[5\], we get:
\[ \Rightarrow x = \dfrac{{2y}}{5}................(3)\]
Again, if we take the ‘\[3y\]’ on the R.H.S, we get:
\[ \Rightarrow - 4x = 7 - 3y\].
Now, divide both the sides by \[ - 4\], we get:
\[ \Rightarrow x = \dfrac{{7 - 3y}}{{ - 4}}................(4)\]
Now, the value of \[x\] in equation \[(3)\] and \[(4)\] shall always be equal.
So, we can write the following equation:
\[ \Rightarrow \dfrac{{2y}}{5} = \dfrac{{7 - 3y}}{{ - 4}}\].
Now, by doing the cross multiplication, we get:
\[ \Rightarrow - 4 \times 2y = 5 \times (7 - 3y)\].
Now, by doing multiplication, we get:
\[ \Rightarrow - 8y = 35 - 15y\].
Now, taking the variable term on the L.H.S, we get:
\[ \Rightarrow - 8y + 15y = 35\].
Now, by adding the variable terms, we get:
\[ \Rightarrow 7y = 35\].
Now, divide both the sides by \[7\], we get:
\[ \Rightarrow y = 5\].
So, if we put the value of \[y\] in the equation \[(1)\], we get:
\[ \Rightarrow 5x - 2 \times 5 = 0\].
Now, doing the multiplication, we get:
\[ \Rightarrow 5x - 10 = 0\].
Now, take the constant terms on the right hand side, we get:
\[ \Rightarrow 5x = 10\].
Now, divide both the sides by \[5\], we get:
\[ \Rightarrow x = \dfrac{{10}}{5} = 2\].
Therefore, the solution of the above question is \[x = 2\] and \[y = 5\].