Problem 3 - Algebra 2

Question

Find the whole number solutions of each system using tables. $$ \left\\{\begin{array}{l}{x+y<8} \\ {3 x \leq y+6}\end{array}\right. $$

Step-by-Step Solution

Answer

The common whole number solutions are the pairs of (x, y) that satisfy both x + y < 8 and 3x ≤ y + 6.

Step 1: Create a Table for the First Inequality

List pairs of whole numbers (x, y) that satisfy the first inequality, x + y < 8. Start with x = 0 and find the corresponding y values that make the inequality true. Then increase x by 1 and find new y values until no more whole number pairs can be found that satisfy the inequality.

Step 2: Create a Table for the Second Inequality

For the second inequality, 3x ≤ y + 6, rearrange the inequality to y ≥ 3x - 6 to create a table. List pairs of whole numbers (x, y) that satisfy this inequality. Begin with x = 0 and increase x by 1, finding corresponding y values that fulfill the inequality, until no more whole number pairs can be found.

Step 3: Find Common Pairs

Compare the tables from Step 1 and Step 2 to identify common pairs of (x, y) that satisfy both inequalities. The pairs that appear in both tables are the whole number solutions to the system.

Key Concepts

Understanding Inequality Solutions

Inequalities are like equations, but instead of an equals sign, they use symbols to show that one side is less than or greater than the other. Solving systems of inequalities means finding all the possible values (or solutions) that satisfy all the inequalities at the same time.

For example, consider the inequality system \(x + y < 8\) and \(3x \leq y + 6\). To solve this, you create a set of ordered pairs \(x, y\) that satisfy both inequalities. It's crucial to remember that for whole number solutions, we're only interested in non-negative integer values for \(x\) and \(y\), as fractions and decimals are not considered whole numbers.

Finding whole number solutions can help in understanding patterns and relationships between variables, which can be applied to real-world problems, such as budgeting or planning. The use of a table to list out values is a systematic method that ensures you consider all possible combinations without missing any potential solutions.

Whole Number Solutions

Whole number solutions refer to solutions of an inequality that are non-negative integers (0, 1, 2, 3, ...). To find these solutions, start by listing easy-to-find pairs and progressively move toward more complex ones. When given the inequality \(x + y < 8\), you might start with \(x = 0\) and list out values for \(y\) until the inequality is no longer true. Then increment \(x\) by 1 and repeat the process.

In the context of our exercise, a systematic approach ensures that you explore every possible combination of \(x\) and \(y\) up to the boundary where the inequality no longer holds. After completing the tables for each inequality, the next step is comparing these to find common pairs. The intersection of these sets gives us the whole number solutions for the system.

Graphical Solution Method

Graphical solution methods involve plotting each inequality on a coordinate system. To graph an inequality, first turn the inequality into an equation by replacing the inequality symbol with an equals sign and draw the line or curve for that equation. Then, based on the inequality symbol, you shade the region of the graph where the inequality holds true.

For instance, with \(x + y < 8\), you would graph the line \(x + y = 8\) then shade the region below the line because we're looking for points where \(y\) is less than \(8 - x\). Similarly, for \(3x \leq y + 6\), graph \(y = 3x - 6\) and shade the region above the line since \(y\) is greater than or equal to \(3x - 6\). The feasible region -- where the shaded areas overlap -- represents the set of all possible solutions. If you focus only on the points with whole numbers in this overlap, those are your whole number solutions.