Question

How do you solve the system of equations $3x + y = 23$ and $4x - y = 19$?

Answer 1

It is given as we have to solve the given system of simultaneous equations by the elimination method. So, add both equations. After addition, you will find that y is eliminated and you will be left with x and then simplify and get the value of x and then substitute this value of x in any of the equations. It will give you the value of x.

Complete step-by-step answer:
The two simultaneous equations given in the question are:
$\Rightarrow 3x + y = 23$ ….. (1)
$\Rightarrow 4x - y = 19$ ….. (2)
We are going to solve the above simultaneous equations by elimination method in which we add both the equations and then y will be eliminated and we get the value of x.
Now, add both equations,
$\begin{array}{l} 3x + y = 23\\\ \underline {4x - y = 19} \\\ 7x\,\,\,\,\,\,\,\,\, = 42 \end{array}$
Now, divide, both sides by 7,
$\Rightarrow x = 6$
Plugging this value of x in equation (1) we get,
$\Rightarrow 3 \times 6 + y = 23$
Multiply the terms,
$\Rightarrow 18 + y = 23$
Subtract 18 from both sides,
$\Rightarrow 18 + y - 18 = 23 - 18$
Simplify the terms,
$\Rightarrow y = 5$

Hence, the value of x is 6 and y is 5.

Note:
This question can be done in another way also.
The two simultaneous equations given in the question are:
$\Rightarrow 3x + y = 23$ ….. (1)
$\Rightarrow 4x - y = 19$ ….. (2)
We are going to solve the above simultaneous equations by substitution method in which find the value of y in terms of x from equation (2) and then substitute it in equation (2) to find the value of x.
The value of y in terms of x in equation (2)
$\Rightarrow 4x - y = 19$
Simplify the terms,
$\Rightarrow y = 4x - 19$ ….. (3)
Substitute the value in equation (1),
$\Rightarrow 3x + \left( {4x - 19} \right) = 23$
Simplify the terms,
$\Rightarrow 3x + 4x = 23 + 19$
Add the like terms,
$\Rightarrow 7x = 42$
Now, divide, both sides by 7,
$\Rightarrow x = 6$
Plugging this value of x in equation (3) we get,
$\Rightarrow y = 4 \times 6 - 19$
Multiply the terms,
$\Rightarrow y = 24 - 19$
Subtract the terms,
$\Rightarrow y = 5$
Hence, the value of x is 6 and y is 5.