Problem 1 - Algebra 1

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. Take for instance a simple system with two variables and two equations. Each equation in the system is true at the same time, representing a specific condition or fact. The solution to a system is the point or points where all equations intersect, meeting all given conditions. This means finding the specific values of the variables that make each equation valid simultaneously.

Linear equations are usually represented in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Solving a system of linear equations aims to find the values of \(x\) and \(y\) that satisfy all equations at once. This can involve several methods, such as graphing, substitution, or elimination, with the linear combinations being a prominent technique.

Elimination Method

The elimination method is a popular approach to solving systems of linear equations. It involves adding or subtracting equations in order to eliminate one of the variables, making it easier to solve for the remaining one. This method requires manipulating the equations so that adding or subtracting them results in one of the variables disappearing.

The goal is to create a situation where one of the numbers in front of a variable (known as the coefficient) becomes zero in the sum or difference of equations. This is done by making the coefficients of the same variable in different equations equal and opposite. When one variable has been eliminated, you only have to solve for one variable, which simplifies the problem significantly.

Variables Elimination

Variables elimination involves the strategic removal of one or more variables from a system of equations. This simplifies the system and allows us to solve for the unknowns more straightforwardly. The process starts by aligning the coefficients of a particular variable across different equations, usually with multiplication.

When coefficients match, it becomes possible to add or subtract the equations to cancel out (eliminate) that variable. Eliminating variables reduces the complexity of the problem significantly, as you temporarily transform a multi-variable system into a simpler single-variable one. After finding one variable's value, substitution can be used to determine the others. This forethought helps efficiently solve complex systems especially when dealing with two or more equations.

Multiplication in Algebra

Multiplication plays a crucial role in the linear combinations method, especially in the elimination process. It is used to adjust the coefficients of variables across different equations so that they can be eliminated efficiently through addition or subtraction.

The purpose of multiplying one or both equations by suitable numbers is to equalize the coefficients of a chosen variable. For example, if an equation in a system has coefficients \(3x\) and another has \(2x\), multiplying the first by 2 and the second by 3 will give both equations a resultant coefficient of 6 for \(x\). This makes it easy to add or subtract the equations to eliminate \(x\).
This crucial step allows simplification of the system, reducing the problem to an equation with only one variable, which heralds the logical progression towards finding the solution to the entire system.