Problem 89 - College Algebra Essentials

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 easier to use when neither equation can be simply solved for one of the variables but coefficients exist that can let you eliminate a variable through addition or subtraction.

Step 1: Understand the Substitution Method

The substitution method is used when one equation can be solved easily for one variable in terms of the others. It means one equation has a coefficient of 1 or -1 for one of the variables. This process involves re-writing one of the equations so one variable is expressed in terms of the other variable, then substituting this expression into the other equation. For example, if the system is \(x + y = 5\) and \(x - y = 1\), it is easy to solve the second equation for \(x\) to obtain \(x = y + 1\). This is then substituted into the first equation.

Step 2: Understand the Addition Method

The addition (or elimination) method is used when both equations are more complex, unlike in substitution where one can be easily solved. In this method, equations are added or subtracted to eliminate one of the variables, creating a single equation in one variable. This is commonly used when the systems of equations have coefficients that can easily be negated by addition or subtraction. For example, if the system is \(2x + 3y = 6\) and \(3x + 2y = 5\), neither equation can be solved easily for one variable. However, multiplying the two equations by appropriate values can create an equivalent system where the coefficients of y (or x) can be negated to eliminate that variable.

Step 3: Comparing the Two Methods

It is easier to use the addition method rather than the substitution method to solve a system of equations when neither equation can be simply solved for one of the variables. However, if coefficients are such that by simple addition or subtraction, you can eliminate a variable, then the addition method is the favorable method.