Problem 89 - Algebra and Trigonometry

Question

When is it easier to use the addition method rather than the substitution method to solve a system of equations?

Step-by-Step Solution

Answer

The addition method is typically easier to use than the substitution method when the equations are arranged such that the coefficients of one variable are the same in both equations or negatives of each other. This allows for straightforward elimination of that variable. However, overall the choice between methods really depends on the specific system of equations that is given. In cases where one variable is easily isolated in one of the equations, the substitution method may be simpler.

Step 1: Understand the Addition Method

The addition method, also known as the elimination method, involves adding or subtracting the equations in order to eliminate one of the variables. This method is quite straightforward when the coefficients of one of the variables are the same in both equations, because simply adding or subtracting the equations will eliminate that variable. For instance, consider two equations: equation 1, \(2x + 3y = 12\) and equation 2, \(2x - y = 2\). Adding these two equations results in \(4x + 2y = 14\), and \(y\) can be easily isolated to solve for \(x\).

Step 2: Understand the Substitution Method

The substitution method involves isolating one variable in one equation and then substituting that expression into the other equation. This method is typically easier when one of the variables in one of the equations is already isolated, or can be isolated with just one operation. For example, given the equation 1: \(y = 3x -1\) and equation 2: \(2x + y = 4\), \(y\) is already isolated in equation 1, so it is easier to substitute \(3x -1\) into equation 2 for \(y\).

Step 3: Compare the Methods

Looking at both methods, it can be concluded that the addition (elimination) method can be significantly easier to use than the substitution method when the coefficients of one of the variables in the equations are the same or negatives of each other. Otherwise, if the coefficients are not easily manageable or one of the variables cannot be easily eliminated, it may be simpler to use the substitution method, particularly when one of the variables in one of the equations is already isolated or can be isolated with one operation.