Question

How do you solve \[x-2y=3\] and \[4x-8y=12\]using substitution?

Answer 1

Hint: Any two linear equations can be solved to get a common point lying on their graphs. We solve these equations by addition and subtraction methods. We can solve these equations by substituting one variable after making required modifications such that any one of the variables remains in an equation. Then, we substitute the rearranged variable into the second equation and find the value of each variable.

Complete step by step answer:
As per the given question we need to solve \[x-2y=3\] and \[4x-8y=12\] to get a common point called a solution of these equations.
Let \[x-2y=3\] \[---\left( 1 \right)\]
      \[4x-8y=12\] \[---\left( 2 \right)\]
Let us consider equation \[\left( 1 \right)\]
Now we add \[2y\] on both sides to the equation \[\left( 1 \right)\]. Then the equation becomes
\[\Rightarrow x-2y+2y=3+2y\]
\[\Rightarrow x=3+2y\] \[---\left( 3 \right)\]
Now we substitute equation 3 in equation 2. Then the equation becomes
\[\begin{align}
  & \Rightarrow 4x-8y=12 \\
 & \Rightarrow 4\left( 3+2y \right)-8y=12 \\
\end{align}\]
 Now we use distributive property to expand the equation.
\[\begin{align}
  & \Rightarrow 4x-8y=12 \\
 & \Rightarrow 4\times 3+4\times 2y-8y=12 \\
 & \Rightarrow 12+8y-8y=12 \\
 & \Rightarrow 12=12 \\
\end{align}\]
Here the left hand side and right hand side have the same value. It implies that both the straight lines have an infinite number of solutions. Since the two straight lines are the same lines.
Therefore, both the straight lines have an infinite number of solutions.

Note:
In order to solve these types of problems, we need to have knowledge of straight lines and basic arithmetic properties. We can solve for x and y of two equations by adding them after making required modifications such that any one of the variables gets canceled. Then, we get the value of one variable. Using this we get the other variable value. We can solve these two straight lines by taking 4 commons in equation 2 then modifying the equation gives the value.