Question

How do you solve the system using the elimination method for $3a+4b=2$ and $4a-3b=-14$?

Answer 1

Hint: Multiply required constants with both the equations to make the coefficient of one variable equal. For making the coefficient of ‘a’ equal multiply the first equation by ‘4’ and the second equation by ‘3’. Then subtract the equations to get the value of ‘b’. Again putting that value of ‘b’ in any equation, ‘a’ value can be obtained.

Complete step by step solution:
Elimination method for solving systems: In elimination methods we have to either add or subtract the equations if the coefficients of one variable are the same. If the coefficients of one variable are different, then we have to multiply one equation with a required constant to make the coefficients equal and then we have to add or subtract accordingly.
Now, considering our equations
$3a+4b=2$……….(1)
$4a-3b=-14$……….(2)
To make the coefficients of ‘a’ equal, we have to multiply the equation (1) by 4 and the equation (2) by 3
$eq(1)\times 4\Rightarrow 12a+16b=8$……….(3)
$eq(2)\times 3\Rightarrow 12a-9b=-42$……….(4)
Subtracting equation (4) from equation (3), we get
$\begin{align}
  & eq(3)-eq(4) \\
 & \Rightarrow \left( 12a+16b \right)-\left( 12a-9b \right)=8-\left( -42 \right) \\
 & \Rightarrow 12a+16b-12a+9b=8+42 \\
 & \Rightarrow 25b=50 \\
 & \Rightarrow b=\dfrac{50}{25} \\
 & \Rightarrow b=2 \\
\end{align}$
Putting the value of ‘b’ in equation (2), we get
$\begin{align}
  & 4a-3b=-14 \\
 & \Rightarrow 4a-3\times 2=-14 \\
 & \Rightarrow 4a-6=-14 \\
 & \Rightarrow 4a=-14+6 \\
 & \Rightarrow a=\dfrac{-8}{4} \\
 & \Rightarrow a=-2 \\
\end{align}$

Hence, the solution of the system $3a+4b=2$ and $4a-3b=-14$ is $\left( a,b \right)=\left( -2,2 \right)$.

Note: Coefficients of one variable should be made equal by multiplying suitable constants. After getting one variable, put the value of that variable in any of the equations, to obtain the value of the other variable. The solution can be verified by putting the values of ‘a’ and ‘b’ in one of the given equations.