Question

Solve the linear equation \[10x - 5 - 7x = 5x + 15 - 8\] .

Answer 1

Hint: If we see the given linear equation, we can see that there are like terms on the Left-hand side of the equation and two constant terms on the right-hand side of the equation. We first apply the mathematical operations on them and then we apply the transposition method to simplify it further.

Complete step-by-step solution:
Given, \[10x - 5 - 7x = 5x + 15 - 8\].
Now subtracting the like terms in the left-hand side of the equation, we have
\[3x - 5 = 5x + 15 - 8\]
Now subtracting the constants on the right-hand side of the equation
\[3x - 5 = 5x + 7\]
We transpose \[ - 5\] which is present in the left-hand side of the equation to the right-hand side of the equation by adding \[5\]on the right-hand side of the equation.
\[3x = 5x + 7 + 5\]
Similarly, we transpose $5x$ to the left-hand side by subtracting $5x$ on left-hand side of the equation.
\[3x - 5x = 7 + 5\]
We can see that the variable ‘x’ and the constants are separated, then
\[ - 2x = 12\]
Now divide the equation by $-2$
\[x = \dfrac{{12}}{{ - 2}}\]
\[ \Rightarrow x = - 6\]. This is the required result.

Note: We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘x’ in the given problem.
\[10x - 5 - 7x = 5x + 15 - 8\]
\[\Rightarrow 10( - 6) - 5 - 7( - 6) = 5( - 6) + 15 - 8\]
\[ \Rightarrow - 60 - 5 + 42 = - 30 + 15 - 8\]
\[\Rightarrow - 65 + 42 = - 38 + 15\]
\[ \Rightarrow - 23 = - 23\]
Hence the obtained answer is correct. We know that the product of two negative numbers results in a positive number.
In the above, we did the transpose of addition and subtraction. Similarly, if we have multiplication, we use division to transpose. If we have division, we use multiplication to transpose. Follow the same procedure for these kinds of problems.