Question

How do you solve \[a+3b=7\ and \ 2a=b-7\]?

Answer 1

Hint:In the given question, we have been asked to solve a system of equation i.e. \[a+3b=7\ and\ 2a=b-7\]. In order to solve the equations we will use elimination methods. We need to either add both the equations or subtract both the equations to get the equation in one variable. Then we solve the equation in one variable in a way we solve the general linear equation.

Complete step by step solution:
We have given that
\[\Rightarrow a+3b=7\]------ (1)
\[\Rightarrow 2a=b-7\]
We will rewrite the equation as
\[\Rightarrow 2a-b=-7\]----- (2)
Multiply the equation (1) by 2 we get
\[\Rightarrow 2a+6b=14\]----- (3)
Subtracting equation (2) and equation (3), we get
\[\Rightarrow 2a-b-2a-6b=-7-14\]
Combining the like terms, we get
\[\Rightarrow -7b=-21\]
Dividing both the sides of the equation by -7, we get
\[\Rightarrow b=3\]
Substitute the value of \[b=3\] in equation (1), we get
\[\Rightarrow a+3b=7\]
\[\Rightarrow a+3\times 3=7\]
\[\Rightarrow a+9=7\]
Subtracting \[6\] from both the sides of the equation, we get
\[\Rightarrow a=7-9=-2\]
\[\Rightarrow a=-2\]
Therefore, we get
\[\Rightarrow \left( a,b \right)=\left( -2,3 \right)\]
Therefore, the possible value of ‘a’ and ‘b’ is\[\left( a,b \right)=\left( -2,3 \right)\].
It is the required solution.

Note: In an elimination method for solving a system of equations we will either add both the equations or subtract both the equations to get the equation in one variable. To eliminate the variable, you will add the two given equations if the coefficients of one variable are opposites and you will subtract the two equations if the coefficient of one variable is exactly the same. If both the situations do not satisfy then we will need to multiply one of the equations by a number that will lead to the same leading coefficient.