Problem 1 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The purpose of using multiplication as a first step in linear combinations to solve a linear system is to adjust the coefficients of the variables in a way to eliminate or simplify one or more variables, making the system of equations easier to solve.

Step 1: Understanding Linear System and Linear Combination

A linear system is a set of two or more linear equations with the same variables. A linear combination involves adding or subtracting multiples of these equations with the aim of eliminating one or more variables, making the linear system easier to solve.

Step 2: Role of Multiplication in Linear Combination

The role of multiplication in a linear combination is to scale or modify the equations to simplify them. Multiplication can adjust the coefficients of the variables in the equation such that one or more variables can be eliminated.

Step 3: Why Multiplication is Used as the First Step in Linear Combination

Multiplication is used as the first step in a linear combination to is to match the coefficients of one variable in two equations. This is typically done so that one of the variables can be eliminated in the subsequent addition/subtraction step, leading the system of equations to be more easily solvable.

Key Concepts

Linear Combination

Imagine you're trying to find the intersection of two paths represented by equations. A linear combination is like using a map to redraw these paths so they cross at simpler, more easily identifiable points. Specifically, a linear combination in algebra involves adding or subtracting equations after they've been multiplied by some value to eliminate one variable, like erasing confusing paths on our map. This technique simplifies the process of solving for the remaining variables.

For instance, given two equations, Equation 1: \( 2x + 3y = 6 \) and Equation 2: \( x - y = 4 \), we can manipulate these equations to eliminate either \( x \) or \( y \) and solve the system. If our goal is to eliminate \( y \), we could multiply Equation 1 by a number that would create a coefficient for \( y \) that is the negation of the coefficient in Equation 2. By doing so, we are setting up the equations to cancel out \( y \) when they're added together.

System of Linear Equations

A system of linear equations consists of two or more linear equations that share common solutions. Think of these as different paths crossing at a single meeting point, and our job is to find that point. These equations represent lines in a two-dimensional plane, and the solution(s) to the system correspond to points where the lines intersect.

In mathematical terms, if we have a system with equations, Equation A: \( a_1x + b_1y = c_1 \) and Equation B: \( a_2x + b_2y = c_2 \) where \( a_1, a_2, b_1, b_2, c_1, \) and \( c_2 \) are constants, we are looking for the values of \( x \) and \( y \) that satisfy both equations. Solving these systems can be done using various methods, such as graphing, substitution, elimination (including linear combination), or matrix operations. The goal is always to find the value(s) of the variables that work for all equations simultaneously.

Multiplication in Linear Systems

When we come across a system of linear equations, sometimes the coefficients of the variables are not lined up for easy elimination. That's where multiplication comes into play. It's like resizing one puzzle piece to fit perfectly with another.

By multiplying one or both equations by appropriate numbers, we can create matching coefficients for a specific variable in two different equations. Once the coefficients are matched, adding or subtracting the equations results in cancelling out that variable. It simplifies our system into a single equation with one variable, which is a straightforward solve.

Let's adopt the earlier example: for Equation 1: \( 2x + 3y = 6 \) and Equation 2: \( x - y = 4 \), multiplying Equation 2 by 3 transforms it into \( 3x - 3y = 12 \). Now, by adding this new equation to Equation 1, \( 3y \) and \( -3y \) cancel out, leaving us with a single-variable equation: \( 5x = 18 \) which can be solved to find \( x \). This is a practical illustration of why multiplication is often the necessary first step in solving linear systems through linear combinations.