Problem 28 - Algebra 1: Concepts and Skills

Question

Match the situation with the corresponding linear system. You buy 5 pairs of socks for 19 dollar. The wool socks cost 5 dollar per pair and the cotton socks cost 3 dollar per pair.

Step-by-Step Solution

Answer

The corresponding system is \(5x + 3y = 19\) and \(x + y = 5\). The solution is \(x = 2\) and \(y = 3\). These results imply that you bought 2 pairs of wool socks and 3 pairs of cotton socks.

Step 1: Define Variables

Let's define the variables to the problem as follows: x is the quantity of wool socks and y is the quantity of cotton socks.

Step 2: Setup the Equations

First, the cost equation will be defined as \(5x + 3y = 19\). This equation represents the total cost of the socks bought by the type of socks and their respective prices. The second, the quantity equation is represented as \(x + y = 5\). This equation represents the total quantity of socks purchased, which is 5 pairs.

Step 3: Solve the System

First solve the quantity equation for y: \(y = 5 - x\). Substitute y in the cost equation and solve for x: \(5x + 3(5 - x) = 19\) so that you get \(5x + 15 - 3x = 19\) which simplifies to \(2x = 4\), and gives \(x = 2\). Now plug x=2 into the first equation: \(2 + y = 5\) to get \(y = 3\).

Step 4: Interpret the Results

The solution in context of the problem implies that 2 pairs of socks are wool and 3 pairs are cotton.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two algebraic expressions. These expressions can include numbers, variables (which represent unknown values), and various operations like addition, subtraction, multiplication, and division. In the context of linear systems word problems, algebraic equations are used to translate real-world scenarios into mathematical language that can be analyzed and solved.

For instance, in the given exercise regarding sock purchases, we can create an algebraic equation to represent the total cost of the socks. The equation, \(5x + 3y = 19\), translates the problem's details into a mathematical statement where \(x\) and \(y\) are the quantities of wool and cotton socks respectively, and their associated costs per pair are 5 and 3 dollars. This equation sets the foundation for solving the word problem using algebraic methods.

System of Equations

A system of equations is a set of two or more equations that share the same variables. The goal is to find the values of these variables that make all the equations true simultaneously. Systems can be solved using various methods, including graphing, substitution, elimination, and matrix operations, among others.

In our exercise, we have a system composed of two equations: the cost equation \(5x + 3y = 19\) and the quantity equation \(x + y = 5\). Solving this system will give us the quantity of wool and cotton socks purchased. The concept is crucial in many fields, from simple budgeting problems to complex engineering and scientific calculations, underpinning the interconnected nature of multiple conditions or constraints.

Variable Substitution

Variable substitution is a method to solve systems of equations where one equation is solved in terms of one variable, and this expression is substituted into the other equations. This method simplifies the system into a single variable equation that can be solved, and then the solution is used to find the remaining variable(s).

For the sock-buying problem, we first express \(y\) in terms of \(x\) using the quantity equation: \(y = 5 - x\). This expression for \(y\) is then substituted into the cost equation, resulting in an equation with a single variable \(x\). This process demonstrates how variable substitution can help simplify and solve an otherwise complicated system of equations. The strategy is particularly useful when one of the equations in the system is easily solvable for one of the variables.

Linear Equations

Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. These equations represent straight lines when graphed on a coordinate plane. A linear system consists of two or more linear equations with the same variables. The solutions to a linear system are the points where the lines intersect.

The given exercise involves two linear equations. The 'cost' equation, \(5x + 3y = 19\), and the 'quantity' equation, \(x + y = 5\), form a linear system that, when graphed, intersect at the point corresponding to the number of wool and cotton socks. The solution \(x = 2\), \(y = 3\) indicates that the lines intersect at the point (2, 3), which aligns with the context of the problem. Understanding linear equations and how to manipulate them is essential for solving many practical problems in mathematics and related disciplines.