Question

How do you solve the system of equations \[3x + 4y = - 23\] and \[2y - x = - 19\] ?

Answer 1

Hint: We can solve this using substitution method or by elimination method or by cross product method. There are two linear systems of equations with two variables. We solve this using a substitution method. Using the first equation we find the value of ‘x’ in terms ‘y’. Then we substitute the ‘x’ value in the second equation. By doing this we will get the value of ‘y’. to find the value of ‘x’ we substitute the value of ‘y’ in any one of the equations.

Complete step-by-step answer:
We have,
\[3x + 4y = - 23{\text{ }} - - - (1)\]
\[2y - x = - 19{\text{ }} - - - - (2)\]
Consider equation (1)
\[3x + 4y = - 23\] . We find the value of ‘x’ in terms of ‘y’.
Subtract \[4y\] on both sides we have,
\[3x = - 23 - 4y\]
Divide the whole equation by ‘3’.
\[x = \dfrac{{ - 23 - 4y}}{3}\]
Now substituting this in the equation (2). We have,
\[2y - x = - 19\]
\[2y - \left( {\dfrac{{ - 23 - 4y}}{3}} \right) = - 19\]
Multiply each term by ‘3’ we get,
\[6y - \left( { - 23 - 4y} \right) = - 19 \times 3\]
\[6y + 23 + 4y = - 57\]
Subtract 23 on both side we have,
\[6y + 4y = - 57 - 23\]
Taking ‘y’ common, we have:
\[y(6 + 4) = - 57 - 23\]
\[10y = - 80\]
Divide the whole equation by 10 we get,
\[y = - \dfrac{{80}}{{10}}\]
\[y = - 8\] .
Now to find the value of ‘x’. Substitute the value of ‘y’ in any one of the equations.
Let’s substitute \[y = - 8\] in equation (2). We have,
\[2( - 8) - x = - 19\]
\[ - 16 - x = - 19\]
\[ - x = - 19 + 16\]
\[ - x = - 3\]
Multiply by \[ - 1\] on both sides,
\[x = 3\] .
Thus we have \[x = 3\] and \[y = - 8\] .
So, the correct answer is “\[x = 3\] and \[y = - 8\] ”.

Note: We can check whether the obtained equation is correct or not by substituting the values of ‘x’ and ‘y’ in the given problem.
Substitute in equation (1).
\[3(3) + 4( - 8) = - 23\]
\[9 - 32 = - 23\]
\[ - 23 = - 23\]
That is, the left hand and right hand side values are equal. If we substitute in equation (2) we will have, \[ - 19 = - 19\] .
Also know that the product of two negative numbers gives us a positive number.