Question

Solve the linear equation $5y-2y=3y+2$ ?

Answer 1

Hint: We add and subtract suitable values on both sides of a given linear equation so that we can collect variable terms at one side of the equation and the constant terms at the other side. We divide the coefficient of the variable to find the value of the variable. If we get constant terms on both sides then we have no solution to the equation.

Complete step-by-step solution:
We know from algebra that the linear equation in one variable $x$ is given by
\[ax=b\]
Here $a$ and $b$ are real numbers called constant of the equation where $a$ cannot be zero. The side left to the sign of equality ${}^{'}{{=}^{'}}$ is called the left hand side and side right to the sign of equality is called right hand side of the equation. We also know that if we add, subtract, multiply or divide the same number on both sides of the equation, equality holds. It is called balancing the equation. It means for some real number $c$ we have,
\[\begin{align}
  & ax+c=b+c \\
 & ax-c=b-c \\
 & ax\times x=b\times c \\
 & \dfrac{ax}{c}=\dfrac{b}{c} \\
\end{align}\]
When we are asked to solve an equation we find the values of unknown variables $x$. If we are given variable terms on both sides for example $ax+b=cx+d$ then we collect the like terms (variable terms and constant terms) at two different sides.
 We are given in the following linear equation to solve
\[5y-2y=3y+2\]
We see that there is one variable $y$ and it is present in both sides of the equation. We solve $5y-2y$ in the left hand side by taking $y$ common to have;
\[\begin{align}
  & \Rightarrow y\left( 5-2 \right)=3y+2 \\
 & \Rightarrow y\times 3=3y+2 \\
 & \Rightarrow 3y=3y+2 \\
\end{align}\]
We subtract $3y$ from both sides of the above equation to have;
\[\begin{align}
  & \Rightarrow 3y-3y=3y-3y+2 \\
 & \Rightarrow 0=2 \\
\end{align}\]
We see that we get two constants at two sides of the equation. Here the result $0=2$ is a contradiction since 0 can never be equal to 2. So there is no solution of the equation.

Note: We note that a linear equation in one variable can have one unique solution no solution at all. A linear equation two variables which is given by $ax+by=c$ can have unique solutions, no solution or infinite solution. We need only one equation to solve linear equations in one variable but we need 2 equations to solve linear equations in two variables.