Let's go through an example to create equations in two variables, representing relationships between quantities. We’ll set up the equations, interpret them, and graph them with labeled axes and scales.

Example Problem:

Suppose you have a budget for a party and are deciding how many cupcakes and drinks you can buy with a total budget. Let’s assume:

• Cupcake cost: $2 each
• Drink cost: $1 each
• Total budget: $20

We want to set up equations to show how many cupcakes $x$ and drinks $y$ you can buy within this budget.

Step 1: Set up the equation

The total cost of cupcakes and drinks should be less than or equal to the budget. We can write this as:

$2x + y = 20$

where:

• $x$ represents the number of cupcakes,
• $y$ represents the number of drinks.

This is a linear equation in two variables that describes the relationship between cupcakes and drinks within a $20 budget.

Step 2: Solve for $y$ in terms of $x$

To make it easier to graph, we’ll solve for $y$:

$y = 20 - 2x$

Step 3: Choose values for $x$ and calculate $y$

Let's choose different values for $x$ and calculate the corresponding values of $y$:

$x$ (Cupcakes)	$y = 20 - 2x$ (Drinks)
0	20
5	10
10	0

Step 4: Graph the equation

1. Draw a coordinate plane with $x$ representing cupcakes on the horizontal axis and $y$ representing drinks on the vertical axis.

2. Label each axis and choose a scale. For example:

   • Horizontal axis (cupcakes): scale by 1 unit per cupcake.
   • Vertical axis (drinks): scale by 1 unit per drink.

3. Plot the points from the table:

   • (0, 20), (5, 10), and (10, 0).

4. Draw a line through the points. This line represents all combinations of cupcakes and drinks that exactly use up the $20 budget.

Explanation of the Graph

• Every point on the line is a possible combination of cupcakes and drinks that uses up the entire budget.
• Points below the line represent combinations that cost less than $20.

Would you like more details on the process or the graph?

Related Questions:

1. How would the equation change if each cupcake cost $3 instead?
2. What if the total budget were increased to $30?
3. How would you set up an inequality if you wanted to spend less than $20?
4. What is the slope of this line, and what does it represent in this context?
5. How would you interpret the intercepts of this line?

Tip:

When creating equations in multiple variables, always start by defining what each variable represents and make sure to align it with a real-world interpretation for easier visualization and graphing.