Question

How do you solve for $r$ in $4\left( r+3 \right)=t$ ? \[\]

Answer 1

Hint: We recall the linear equations and basic properties of linear equations. We recall that when we are asked for a variable we have to treat all other terms as constant. So we treat here as consonant and use necessary arithmetic operations on both sides to keep only $r$ at the left hand side of the equation. \[\]

Complete step by step answer:
We know from algebra that the linear equation in one variable $x$ and constants $a\ne 0,b$ is given by
\[ax=b\]
The term with which the variable is multiplied is called variable term and with whom not multiplied is called a constant term. 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 term $c$ we have,
\[\begin{align}
  & ax+c=b+c \\
 & ax-c=b-c \\
 & ax\times c=b\times c \\
 & \dfrac{ax}{c}=\dfrac{b}{c} \\
\end{align}\]
When we are asked to solve for $x$ in an equation it means we have to find the value or values of $x$ for which the equation satisfies. We are given the following linear equation
\[4\left( r+3 \right)=t\]
 We are asked to solve for $r$ which means we have to find the value of $r$. We have also an unknown $t$ here which we have to treat as constant. So let us open the bracket using distributive property of multiplication to have;
\[\begin{align}
  & \Rightarrow 4\times r+4\times 3=t \\
 & \Rightarrow 4r+12=t \\
\end{align}\]
We subtract both sides of above equation by 12 to have;
\[\begin{align}
  & \Rightarrow 4r+12-12=t-12 \\
 & \Rightarrow 4r=t-12 \\
\end{align}\]
We divide both sides of above equation by 4 to have;
\[\begin{align}
  & \Rightarrow \dfrac{4r}{4}=\dfrac{t-12}{4} \\
 & \Rightarrow r=\dfrac{t-12}{4} \\
\end{align}\]
So the solution for $r$ is $r=\dfrac{t-12}{4}$.\[\]

Note:
We note that distributive law of multiplication over division is given by $a\left( b+c \right)=ab+ac$.We also note if we treat both $r,t$ as variables the given equation $4\left( r+3 \right)=t$ is linear equation two variables which has infinite solutions. If we want a unique solution we require another equation in $r,t$ such that coefficients of $r,t$ not in proportion.