Recall how each method works and think of when one of them can be more useful.
See solution.
Practice makes perfect
We are asked to explain the similarities between the Substitution and the Elimination Method, as well as tell when it's preferable to use one or the other. Let's start by recalling each method, and after that we can make our conclusions about their similarities and possible advantages for different cases.
Substitution Method
This method consists of isolating one of the variables in one of the equations.
Then, we can substitute it in the other equation, effectively reducing the two-equation system into a one-variable equation.
2y=8x +10 3x + 3y = 30 &⇒ y=4x + 5 3x + 3y = 30
y=4x + 5 3x + 3y = 30 &⇒ y=4x + 5 3x + 3( 4x + 5) = 30
y=4x + 5 3x + 3( 4x + 5) = 30 &⇒ y=4x + 5 3x + 12x + 15 = 30
We can solve for the remaining variable to then substitute it back into the other equation and solve for the originally isolated variable.
This method consists of eliminating one of the variables by cross multiplying its coefficients. It creates one opposite signs, so that they can be added and canceled mutually.
2y=8x +10 3x + 3y = 30 ⇒ - 3 (2y=8x + 10 ) 2(3x + 3y = 30 )
-6 y = -24 x -30 6x + 6y = 60 ⇒ 6x = -24 x + 30
Then we solve for the remaining variable and substitute its value into any of the original equations to solve for the other variable.
6x = -24 x + 30 &⇒ x = 1
x = 1 &⇒ 2y=8( 1) + 10 3x + 3y = 30
2y= 8 + 10 3x + 3y = 30 &⇒ y = 9 3x + 3y = 30
Conclusions
Similarities
They both are well-structured procedures to solve a system of equations by reducing it to a one-variable equation. However, how they do it is different for each.
When to Use Which Method?
The way in which the equation is written is what determines which method would work best. For a system where one of the variables is already isolated, the substitution method would be the way to go. This is since the first step would be already done.
y=8x +10 3x + 2y = 58
On the other hand, if none of the variables are isolated but one variable has coefficients that are opposites or are the same, the Elimination Method can be applied directly by adding or subtracting the equations as required.
2y=8x +10 3x -2y = - 15
If none of the above conditions are met, no method is preferred over the other. We can use the one we like best.
