Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Supply and Demand, find the point of equilibrium for the pair of demand and supply equations. Supply \(p=80+0.00001 x\) Demand $$p=140-0.00002 x$$

Step-by-Step Solution

Answer

The equilibrium point where the quantity supplied equals the quantity demanded occurs at quantity \(x = 2,000,000\) and price \(p = 100.\)

Step 1: Set Up the Equations

To find the equilibrium point, we set up the equation by setting the supply and demand equations equal to each other. So, the equation becomes \(80 + 0.00001x = 140 - 0.00002x\).

Step 2: Simplify the Equation

Rearranging the equation to isolate x, we add \(0.00002x\) to both sides and subtract \(80\) from both sides, the equation becomes \(0.00003x = 60\).

Step 3: Solve for x (Quantity)

x can be found by dividing both sides of the equation by \(0.00003\). This yields \(x = 60/0.00003 = 2,000,000\). Therefore, the quantity (x) at equilibrium is \(2,000,000.\)

Step 4: Solve for p (Price)

Substitute \(x = 2,000,000\) into the supply or demand equation to find p. Here the supply equation is used. So, \(p = 80 + 0.00001*2,000,000 = 100\). Therefore, the price (p) at equilibrium is \(100.\)

Key Concepts

Supply and Demand

Supply and demand are key economic concepts that describe the quantities of goods or services available and desired, respectively. The supply curve typically illustrates the relationship between the price and quantity of goods suppliers are willing and able to offer. Usually, this curve slopes upwards, indicating that suppliers are inclined to provide more of a product as its price increases. On the flip side, the demand curve shows how much consumers are ready to purchase at different price points. This curve typically slopes downwards, reflecting that consumers generally buy more of a product when the price drops.

To find the equilibrium point, which is the price and quantity at which the market balances, the supply and demand curves intersect. In the equilibrium state, the quantity supplied equals the quantity demanded, ensuring there is neither a surplus nor a shortage of goods. This balance is crucial for efficient market operations and helps determine fair pricing.

Linear Equations

A linear equation is an algebraic expression of a straight line. It describes a relationship between two variables using a straight path on a graph. The standard form of a linear equation is often written as:

• y = mx + b

where m is the slope of the line, b is the y-intercept, and x and y are the variables.

In the context of supply and demand, each equation represents a straight line where p (price) is the dependent variable affected by x (quantity). When solving these equations to find equilibrium, you'll be setting the two linear equations equal to each other and solving for the variable x. This gives the equilibrium quantity. Understanding how these equations interact graphically can provide a visual representation of the equilibrium point, making it easier to grasp how changes in supply or demand influence market outcomes.

Algebra Problem Solving

Algebra problem solving involves utilizing mathematical operations to find unknown variables. Breaking down equations into simple steps helps clarify the problem. In the exercise, each of the supply and demand equations was set as a linear relationship with variables p and x.

To solve for the equilibrium, the goal was to identify when these two expressions of p — the price in terms of quantity — are equal. By setting them equal, the problem utilizes systematic algebra techniques such as:

• Isolating variables
• Rearranging terms
• Equating expressions

At step 2, we particularly rearranged the terms to isolate the **Quantity** term on one side. Process such as these is crucial in simplifying complex equations into solvable algebraic forms. Finally, once the value of **x** (quantity) is ascertained, substitution into either original equation enables finding **p** (price), completing the solution to the algebra problem. Engaging with these algebraic skills fortifies overall problem-solving ability and enhances understanding of economic principles.