Question Statement

The equations for supply and demand are given by two linear equations:

• Supply equation: $S(x)=2x+10$
• Demand equation: $D(x)=-3x+40$

Where $x$ is the quantity and $S(x)$ or $D(x)$ represents the price. Find the equilibrium point where the price of supply equals the price of demand by drawing the graphs of both equations.

---

Background and Explanation

In economics, the equilibrium point is the specific quantity and price where the supply and demand curves intersect. At this point, the amount of goods supplied matches the amount of goods demanded, and the market is in balance. To find this graphically, we plot both linear functions and identify their shared coordinate.

---

Solution

To find the equilibrium point graphically, we first need to determine at least two points for each line to plot them accurately.

Step 1: Calculate coordinates for the Supply equation

Using the equation $S(x)=2x+10$:

• If $x=0$: $S(0)=2(0)+10=10$. This gives us the point $(0,10)$.
• If $x=5$: $S(5)=2(5)+10=20$. This gives us the point $(5,20)$.

Step 2: Calculate coordinates for the Demand equation

Using the equation $D(x)=-3x+40$:

• If $x=0$: $D(0)=-3(0)+40=40$. This gives us the point $(0,40)$.
• If $x=5$: $D(5)=-3(5)+40=25$. This gives us the point $(5,25)$.

Step 3: Tabulate the values

We can summarize these points in a table to make graphing easier:

Step 4: Draw the graph

Plot the points from the table on a coordinate plane where the horizontal axis is quantity ($x$) and the vertical axis is price ($y$).

Step 5: Identify the Equilibrium Point

By observing the graph, we look for the point where the supply line (red) and the demand line (blue) cross.

From the graph, the two lines intersect at $(6,22)$. This means that at a quantity of 6 units, the equilibrium price is 22.

---

Key Formulas or Methods Used

• Linear Function Plotting: Finding $(x,y)$ coordinates by substituting values into the equation.
• Equilibrium Condition: The point where $S(x)=D(x)$.
• Graphical Analysis: Identifying the intersection point of two lines on a Cartesian plane.

---

Summary of Steps

1. Select two values for $x$ (e.g., 0 and 5) to find corresponding prices for both supply and demand.
2. Create a table of values for $S(x)$ and $D(x)$.
3. Plot the points on a graph and draw the lines for both equations.
4. Locate the intersection point $(x,y)$ on the graph.
5. Conclude that the intersection $(6,22)$ is the equilibrium point.