Question

How do you solve the system $ 5a+3b=17 $ and $ 4a-5b=21 $ ?

Answer 1

Hint: There are 2 unknown variables and 2 equations in the given question. This is a linear equation in 2 variables. We can solve the equation by eliminating any one of the 2 variables.

Complete step by step answer:
The given 2 equations in the question
 $ 5a+3b=17 $ …..eq1
 $ 4a-5b=21 $ ……eq2
Let’s solve the question by eliminating b.
In eq2 we can multiply 3 in both LHS and RHS
The equation will be $ 12a-15b=63 $ ….eq3
In eq1 we can multiply 5 in both LHS and RHS
The equation will be $ 25a+15b=85 $ …..eq4
Now adding eq3 and eq4
 $ 37a=148 $
So $ a=\dfrac{148}{37} $
 $ \Rightarrow a=4 $
Now we can get the value of b by substituting the value of a in any one of the equation.
Let’s put the value of a in eq1
 $ 5\times 4+3b=17 $
 $ \Rightarrow 20+3b=17 $
 $ \Rightarrow b=-1 $
Now we can verify the answers by putting the value of a and b in both the equations.

Note:
Another method is to solve by determinant method
Suppose there are 2 linear equation $ {{a}_{1}}x+{{b}_{1}}y={{c}_{1}} $ and $ {{a}_{2}}x+{{b}_{2}}y={{c}_{2}} $ the solution to this problem let’s take
D= $ \left| \begin{matrix}
   {{a}_{1}} & {{b}_{1}} \\
   {{a}_{2}} & {{b}_{2}} \\
\end{matrix} \right| $
A= $ \left| \begin{matrix}
   {{b}_{1}} & {{c}_{1}} \\
   {{b}_{2}} & {{c}_{2}} \\
\end{matrix} \right| $
B= $ \left| \begin{matrix}
   {{a}_{1}} & {{c}_{1}} \\
   {{a}_{2}} & {{c}_{2}} \\
\end{matrix} \right| $
 $ x=-\dfrac{A}{D} $ and $ y=\dfrac{B}{D} $ Where $ D\ne 0 $
If $ D=0 $ and any one of A and B is not equal to 0 then there will be no solution for the system of equation
If $ D=A=B=0 $ there will be infinitely many solution of the system of equation.
If $ D\ne 0 $ there will be one solution for the system of equation.
In this case $ {{a}_{1}}=5,{{b}_{1}}=3,{{c}_{1}}=17 $ and $ {{a}_{2}}=4,{{b}_{2}}=-5,{{c}_{2}}=21 $
Solving the determinant $ D=-37,A=148 $ and $ B=37 $
So $ a=-\dfrac{A}{D} $
 $ a=-\dfrac{148}{-37} $
=4
And $ b=\dfrac{B}{D} $ =-1
While solving the system of linear equation we can imagine the equation having 2 unknown as equation of straight in 2-D Cartesian plane so 2 equations means 2 lines will intersect at 1 point unless both lines are parallel or both lines are overlapping. Same goes with 3 unknowns we can imagine the equations as equation of plane in 3D geometry and then solve for the unknown variables.