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The demand equation in economics helps us understand the relationship between the price of a product and the quantity demanded by consumers. It usually shows that as prices decrease, the quantity demanded increases, reflecting consumer willingness to buy more when items cost less. In this exercise, the demand equation is given by \( p = 140 - 0.00002x \), where \( p \) is the price and \( x \) is the quantity demanded. This is a linear equation with a negative slope, meaning the demand decreases as price increases. The minus sign before \( 0.00002x \) suggests a decrease in price results in an increase in demand. Understanding the structure of this equation is crucial for predicting how changes in price affect consumer behavior.

The supply equation depicts the relationship between the price of a product and the quantity that producers are willing and able to supply. Typically, as prices increase, the quantity supplied also increases, reflecting producers' readiness to sell more at higher prices. In our exercise, we have the supply equation \( p = 80 + 0.00001x \). Here, \( p \) denotes price, and \( x \) represents the quantity supplied. The positive slope of \( 0.00001x \) indicates that as the price goes up, so does the supply. This equation helps sellers and businesses anticipate how their supply will change with variations in market price. A solid grasp of this equation's implications assists in making informed production and pricing decisions.

Linear equations are a fundamental concept in algebra that describe a straight-line relationship between two variables. Both our demand and supply equations are linear, meaning they can be graphically represented as straight lines. A linear equation is typically in the form \( y = mx + b \) where \( m \) is the slope, and \( b \) is the y-intercept. For the demand equation \( p = 140 - 0.00002x \), the slope is \(-0.00002\), and the intercept is 140. For the supply equation \( p = 80 + 0.00001x \), the slope is \(0.00001\), and the intercept is 80. The slope dictates the steepness and direction of the line, while the intercept determines where the line crosses the vertical axis. Understanding these helps to predict changes and intersections which are key to finding equilibrium.

Price equilibrium is a vital concept in economics where the quantity demanded by consumers equals the quantity supplied by producers. This equilibrium point is where the demand and supply curves intersect. In this exercise, the equilibrium price is found by setting the demand equation equal to the supply equation, \( 140 - 0.00002x = 80 + 0.00001x \), and solving for \( x \). This represents the market-clearing price, meaning there is no excess supply or demand. Price equilibrium helps stabilize markets, ensuring that the goods produced are exactly what consumers want to buy, at given prices. It's the point where neither a surplus nor a shortage exists.

Solving for the equilibrium involves a simple linear equation calculation, starting by equating the demand and supply equations. The first step is performing operations to collect like terms: \( 140 - 0.00002x = 80 + 0.00001x \). Simplify to \( 0.00003x = 60 \) by adding \( 0.00002x \) to both sides and subtracting 80 from both. Next, solve for \( x \) by dividing both sides by \( 0.00003 \), resulting in the equilibrium quantity. Substituting \( x \) back into either the demand or supply equation will yield the equilibrium price. These steps simplify complex market theories into manageable mathematical operations, illustrating how equations dictate real-world financial dynamics. Understanding and practicing these steps can sharpen analytical skills and provide clarity on economic balance.