Question

Solve the following equation without transposing and check your result.
\[5y + 10 = 4y - 10\]

Answer 1

Hint: Here we are asked to solve the given equation without transposing, that is we are not allowed to use the transposition method. We will solve this equation by systematic method. This method is like balancing the weight on the traditional weighing scale. We will keep adding, subtracting, dividing, and multiplying on both sides of the equation until we get the unknown variable alone on one side of the equation. On the other hand, we will get the value of the unknown variable.

Complete step by step answer:
It is given that \[5y + 10 = 4y - 10\], we aim to solve this equation without using the transposition method. So, we will use the systematic method.
In the systematic method, the ultimate goal is to get the unknown variable alone on one side of the equation to find its value. For that, we will be adding, subtracting, dividing, and multiplying on both sides of the equation to eliminate the terms that are along with the unknown variable.
Considering the given equation \[5y + 10 = 4y - 10\] we can see that the term \[10\] is along with the term containing an unknown variable on the left-hand side, to eliminate it we need to subtract the left-hand side by ten. Since it is an equation, we need to do this operation on both sides to balance the equation.
\[ \Rightarrow 5y + 10 - 10 = 4y - 10 - 10\]
On simplifying the above, we get
\[ \Rightarrow 5y = 4y - 20\]
Now we can see that a term containing the unknown variable is on the right-hand side to get it on the left-hand side we need to eliminate it on the right-hand side by subtracting itself. And to balance the equation we have to do it on the left-hand side also.
\[ \Rightarrow 5y - 4y = 4y - 20 - 4y\]
On simplifying the above, we get
\[ \Rightarrow 5y - 4y = - 20\]
Now we can see that we have grouped the terms containing the unknown variable on the left-hand side and other terms on the right-hand side. Now let us simplify it to get the value of the unknown variable.
\[ \Rightarrow y = - 20\]
Thus, we have got the value of the unknown variable \[y\] as \[ - 20\].
Now let us check if the result is true or not.
Let us substitute the value we found on the given equation.
\[5\left( { - 20} \right) + 10 = 4\left( { - 20} \right) - 10\]
On simplifying this we get
\[ - 100 + 10 = - 80 - 10\]
\[ - 90 = - 90\]
Since it satisfies the given equation, the value we found is correct.

Note:
 In this problem, they have mentioned that we are not supposed to use the transposition method so we have used the systematic method although the transposition method is more time-efficient than the systematic method. And we are also asked to check for the result so it is necessary to give steps on checking the value we have found in our calculation.