Question

When you use the linear combinations method to solve a linear system, what is the purpose of using multiplication as the first step?

Short answer

The purpose of using multiplication as the first step in the linear combinations method for solving a linear system is to manipulate the system such that addition or subtraction will result in the elimination of a variable, simplifying the system to a single variable equation.

Step-by-step solution

Understanding linear combinations method

Firstly, it's helpful to understand what the linear combinations method is. This method is used to solve a system of linear equations by combining the equations in such a way that eliminates one or more of the variables.

Explaining the role of multiplication

Multiplication is used in the first step of the linear combinations method as a means to manipulate one or both of the equations in a system, so that when they are added or subtracted, one variable will cancel out, resulting in an equation with only one variable.

Purpose of multiplication

The goal of this first multiplication step is to make it possible to eliminate one of the variables via addition or subtraction, simplifying the system to a single-variable equation that can be easily solved. After finding the value of this variable, it's substituted back into one of the original equations to find the value of the other variable.