Problem 2 - Algebra 2

Question

Find the whole number solutions of each system using tables. $$ \left\\{\begin{array}{l}{x=y+3} \\ {x+y \leq 12}\end{array}\right. $$

Step-by-Step Solution

Answer

(3, 0), (4, 1), (5, 2), (6, 3), (7, 4), and (8, 5) are the whole number solutions to the system.

Step 1: Create a table for the first equation

Begin by creating a table of values for the first equation, where you choose values for y and then find corresponding values for x using the equation x=y+3. Record the pairs (x, y) that are whole numbers.

Step 2: Identify the constraints

Recognize that the second equation, x + y eq 12 represents a constraint on the possible solutions. Solutions must satisfy both the first equation and the inequality.

Step 3: Create a table for the inequality

Using the same y-values as from the first table, find corresponding x-values that satisfy the inequality x+y eq 12. If the sum of x and y is less than or equal to 12 and both x and y are whole numbers, record the pairs (x, y).

Step 4: Find the common solutions

Compare the tables from Step 1 and Step 3 to find the pairs (x, y) that appear in both tables. These pairs satisfy both the equation and the inequality and are the whole number solutions to the system.

Key Concepts

System of Linear Equations

When we talk about a system of linear equations, we're looking at a set of equations that all need to be true at the same time. Our goal is to find values for the variables that make all the equations work together. Imagine it's like finding the perfect balance where every condition is satisfied.

In the given exercise, we have the system \[\left\{\begin{array}{l}{x=y+3} \ {x+y \leq 12}\end{array}\right.\] Here, we are looking for where these two relationships intersect in term of x and y values that are whole numbers. The first equation \(x=y+3\) establishes a direct link between x and y, meaning for every y, we can find one exact x. However, the system is completed by the second condition, an inequality that restricts the x and y values to a certain range, specifically, x and y must add up to 12 or less.

Inequalities

In mathematics, inequalities are like the open-minded siblings of equations. They are less about exactness and more about ranges and boundaries. An inequality tells us how quantities relate to each other, whether one is bigger, smaller, or within a certain distance from another.

Understanding Inequalities

For instance, the inequality in our exercise \(x+y \leq 12\) doesn't ask for a precise answer but rather a set of answers where the sum of x and y is less than or equal to 12. This includes a variety of combinations, and our job is to find all the pairs of whole numbers that fit this bill. It's worth noting that the symbol '\(\leq\)' means 'less than or equal to', indicating that the sum could be 12, but not more.

Tables of Values

A table of values is a simple yet powerful tool in mathematics. It helps us organize information and see patterns that might not be immediately obvious. Think of it like keeping score in a game - it's much easier to see who's winning and how the game is going when you have all the scores neatly written down.

Using Tables to Solve Systems

When it comes to solving our system of linear equations, we create two separate tables of values: one for each equation or inequality. In these tables, we systematically choose values for one variable and then determine the corresponding value of the other variable. By comparing these tables, we can spot which pairs of (x, y) satisfy both the equation and the inequality, leading us to the whole number solutions.

The process of table-making allows students to visually cross-reference the solutions and verify their accuracy. Not only does it aid in finding the correct answers, but it also helps in understanding the relationship between the variables in a tangible way.