At a price of $580$ there is demand for $924$ items and a supply of $560$ items. At a price of $\$140$ there is demand for $564$ items and a supply of $980$ items. Assuming supply and demand are linear, find the equilibrium price and quantity.

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Grade 8

Interpreting Linear Expressions

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Q. At a price of

580

there is demand for

924

items and a supply of

560

items. At a price of

\$140

there is demand for

564

items and a supply of

980

items. Assuming supply and demand are linear, find the equilibrium price and quantity.

Define Functions: Let's denote the demand function as $D(p)$ and the supply function as $S(p)$ , where $p$ is the price. We have two points for each function: $\newline$ For demand: $(580, 924)$ and $(140, 564)$ $\newline$ For supply: $(580, 560)$ and $(140, 980)$
Find Demand Slope: First, let's find the slope $m$ of the demand function using the two points: $\newline$ $m_{\text{demand}} = \frac{564 - 924}{140 - 580}$ $\newline$ $m_{\text{demand}} = \frac{-360}{-440}$ $\newline$ $m_{\text{demand}} = 0.8182$
Find Supply Slope: Now, let's find the slope $m$ of the supply function using the two points: $\newline$ $m_{\text{supply}} = \frac{980 - 560}{140 - 580}$ $\newline$ $m_{\text{supply}} = \frac{420}{-440}$ $\newline$ $m_{\text{supply}} = -0.9545$
Find Demand Intercept: Next, we'll find the $y$ -intercept ( $b$ ) of the demand function using one of the points and the slope we just found: $\newline$ $924 = 0.8182 \times 580 + b_{\text{demand}}$ $\newline$ $b_{\text{demand}} = 924 - (0.8182 \times 580)$ $\newline$ $b_{\text{demand}} = 924 - 474.556$ $\newline$ $b_{\text{demand}} = 449.444$
Find Supply Intercept: Now, we'll find the $y$ -intercept ( $b$ ) of the supply function using one of the points and the slope we just found: $\newline$ $560 = -0.9545 \times 580 + b_{\text{supply}}$ $\newline$ $b_{\text{supply}} = 560 + (0.9545 \times 580)$ $\newline$ $b_{\text{supply}} = 560 + 553.61$ $\newline$ $b_{\text{supply}} = 1113.61$
Establish Equilibrium: We have the demand and supply functions now: $\newline$ $D(p) = 0.8182p + 449.444$ $\newline$ $S(p) = -0.9545p + 1113.61$ $\newline$ To find the equilibrium, we set $D(p)$ equal to $S(p)$ : $\newline$ $0.8182p + 449.444 = -0.9545p + 1113.61$
Establish Equilibrium: We have the demand and supply functions now: $\newline$ $D(p) = 0.8182p + 449.444$ $\newline$ $S(p) = -0.9545p + 1113.61$ $\newline$ To find the equilibrium, we set $D(p)$ equal to $S(p)$ : $\newline$ $0.8182p + 449.444 = -0.9545p + 1113.61$ Solve for $p$ : $\newline$ $0.8182p + 0.9545p = 1113.61 - 449.444$ $\newline$ $1.7727p = 664.166$ $\newline$ $p = \frac{664.166}{1.7727}$ $\newline$ $p = 374.49$