Question

Solve the following system of equations by the method of substitution:
\[3x - 4y = 10\], \[4x + 3y = 5\]

Answer 1

Hint: Here we will evaluate the value of any one variable out of the two variables from one of the equations and put it in another equation and then evaluate the value of the variable from the equation so formed. Then we will substitute the value obtained back in the first equation to get the value of another variable.

Complete step-by-step answer:
The given equations are:-
\[3x - 4y = 10\]…………………………. (1)
\[4x + 3y = 5\]……………………………. (2)
Now we will evaluate the value of y from the equation.
Hence, from equation 1 we get:-
\[4y = 3x - 10\]
Dividing whole equation by 4 we get:-
\[y = \dfrac{{3x - 10}}{4}\]
Now putting this value in equation 2 we get:-
\[4x + 3\left( {\dfrac{{3x - 10}}{4}} \right) = 5\]
Taking the LCM we get:-
\[\dfrac{{4\left( {4x} \right) + 3\left( {3x - 10} \right)}}{4} = 5\]
Solving it further we get:-
\[16x + 9x - 30 = 4 \times 5\]
Simplifying it we get:-
\[25x - 30 = 20\]
Solving for x we get:-
\[25x = 50\]
\[ \Rightarrow x = 2\]
Now putting this value in equation 1 we get:-
\[3\left( 2 \right) - 4y = 10\]
Simplifying it we get:-
\[6 - 4y = 10\]
Solving it for y we get:-
\[4y = - 4\]
\[ \Rightarrow y = - 1\]

Therefore, the value of x is 2 and the value of y is -1.

Note: Students can verify their answer by solving the given equations by elimination method.
The given equations are:-
\[3x - 4y = 10\]…………………………. (1)
\[4x + 3y = 5\]……………………………. (2)
Multiplying equation 1 by 4 we get:-
\[4\left( {3x} \right) - 4\left( {4y} \right) = 4\left( {10} \right)\]
Simplifying it we get:-
\[12x - 16y = 40\]…………………………… (3)
Now multiplying equation 2 by 3 we get:-
\[3\left( {4x} \right) + 3\left( {3y} \right) = 3 \times 5\]
Simplifying it further we get:-
\[12x + 9y = 15\]………………………………. (4)
Now subtracting equation 4 from equation 3 we get:-
\[
  {\text{ }}12x - 16y = 40 \\
  \underline { - \left( {12x + 9y = 15} \right)} \\
  {\text{ }} - 25y = 25 \\
 \]
Therefore, \[y = - 1\]
Putting this value in equation 1 we get:-
\[3x - 4\left( { - 1} \right) = 10\]
Simplifying it we get:-
\[3x + 4 = 10\]
Solving for x we get:-
\[3x = 6\]
\[ \Rightarrow x = 2\]