Question

Find x if \[x + y = 5,x - y = 7\]

Answer 1

Hint: We use the substitution method to solve two linear equations given in the question. We find the value of x from the first equation in terms of y and substitute in the second equation which becomes an equation in y entirely. Solve for the value of y and substitute back the value of y to obtain the value of x.

Complete step-by-step answer:
We have two linear equations \[x + y = 5\]and \[x - y = 7\]
Let us solve the first equation to obtain the value of x in terms of y.
We have \[x + y = 5\]
Shift the value of y to RHS of the equation.
\[ \Rightarrow x = 5 - y\] … (1)
Now we substitute the value of \[x = 5 - y\]from equation (1) in the second linear equation.
Substitute \[x = 5 - y\]in \[x - y = 7\]
\[ \Rightarrow (5 - y) - y = 7\]
Open the bracket in LHS of the equation
\[ \Rightarrow 5 - y - y = 7\]
Add like terms in LHS of the equation
\[ \Rightarrow 5 - 2y = 7\]
Shift the constant values to one side of the equation.
\[ \Rightarrow - 2y = 7 - 5\]
\[ \Rightarrow - 2y = 2\]
Divide both sides by -2
\[ \Rightarrow \dfrac{{ - 2y}}{{ - 2}} = \dfrac{2}{{ - 2}}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow y = - 1\]
Now substitute the value of \[y = - 1\]in equation (1) to get the value of x.
\[ \Rightarrow x = 5 - ( - 1)\]
Open the bracket in RHS of the equation.
\[ \Rightarrow x = 5 + 1\]
\[ \Rightarrow x = 6\]

So, the value of x is 6.

Note: Alternate method:
We are given the equations \[x + y = 5\]and \[x - y = 7\]
We find the value of y in terms of x from the first equation
\[ \Rightarrow x + y = 5\]
Shift the value of y to RHS of the equation.
\[ \Rightarrow y = 5 - x\] … (2)
Substitute the value of x in equation \[x - y = 7\]
\[ \Rightarrow x - (5 - x) = 7\]
Open the bracket in LHS of the equation
\[ \Rightarrow x - 5 + x = 7\]
Shift constant values to one side of the equation.
\[ \Rightarrow 2x = 7 + 5\]
\[ \Rightarrow 2x = 12\]
Divide both sides by 2
\[ \Rightarrow \dfrac{{2x}}{2} = \dfrac{{12}}{2}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow x = 6\]
So, the value of x is 6.