To create equations in two or more variables that represent relationships between quantities, we typically express one quantity as a function of others. Here’s a breakdown of what you need to know to approach this problem:

Step 1: Understanding Relationships

• Linear relationships: These are typically represented by equations of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. For example, if the number of apples bought depends on the price per apple, you could write $y = 5x$, where $x$ is the number of apples, and $y$ is the total price (assuming 5 dollars per apple).

• Non-linear relationships: These can take various forms, such as quadratic, exponential, or rational equations. A quadratic example is $y = ax^2 + bx + c$, where the relationship between $x$ and $y$ is not linear (it forms a parabola when graphed).

• Systems of equations: Sometimes, multiple relationships need to be expressed together. For example, if $y = 2x + 1$ and $y = 3x - 2$, this can be written as a system of equations. Graphing both on the same coordinate axes helps to find their point of intersection.

Step 2: Creating the Equation

• Define variables: Identify the quantities you want to represent (for example, let $x$ represent the time in hours and $y$ represent the distance traveled).
• Write the equation: Based on the relationship you identified, write an equation involving those variables. For example:
  • Linear relationship: $y = 4x + 2$, where the distance traveled ($y$) increases 4 miles per hour ($x$) plus 2 miles at the start.
  • Quadratic relationship: $y = 2x^2 - 3x + 1$, where the relationship between $x$ and $y$ is more complex, forming a parabola.

Step 3: Graphing the Equation

Once the equation is formed, graph it on a coordinate plane. For example:

• For a linear equation $y = 4x + 2$, plot two points (such as when $x = 0$, $y = 2$, and when $x = 1$, $y = 6$) and draw a straight line through them.
• For a quadratic equation $y = 2x^2 - 3x + 1$, plot points for different values of $x$ (like $x = -1, 0, 1, 2$) and connect them smoothly to form a parabola.

Make sure to:

• Label your axes clearly with $x$ and $y$.
• Choose appropriate scales (for example, if $x$ ranges from -5 to 5, then the $y$-axis might range from -10 to 10 for better visualization).

Example 1: Linear Equation

Let's say a car travels at a constant speed. The relationship between time in hours ($x$) and distance traveled in miles ($y$) can be written as: $y = 60x$ Where 60 is the speed in miles per hour.

Example 2: Quadratic Equation

Suppose the height of an object in free fall is modeled by the equation: $y = -16x^2 + 40x + 5$ Where $x$ is the time in seconds and $y$ is the height in feet.

Step 4: Plot the Graphs

1. Linear graph: Plot points such as (0, 0), (1, 60), (2, 120), and draw a straight line through them.
2. Quadratic graph: Plot points like (0, 5), (1, 29), (2, 37), and plot them to form a parabola opening downward.

---

Would you like to see how a specific equation looks graphed, or would you like to explore more about these equations? Let me know!

---

Follow-Up Questions:

1. How do I graph a quadratic equation like $y = x^2 + 3x - 4$?
2. How can I identify the slope and y-intercept from a graph?
3. How do I solve a system of linear equations graphically?
4. What are the steps to graph a cubic equation, like $y = x^3 + 2x + 1$?
5. How do I find the vertex of a quadratic equation?

Tip: When graphing quadratic functions, always look for the vertex — it tells you the maximum or minimum point on the graph!