A supply function for widgets is modeled by P(q)=a q+b,...

A supply function for widgets is modeled by $P(q)=a q+b,$ where $q$ is the number of widgets supplied and $P(q)$ is the total price of $q$ widgets, in dollars. It is known that 200 widgets can be supplied for $\$ 40$ and 100 widgets can be supplied for $\$ 25 .$ Use a system of linear equations to find the constants $a$ and $b$ in the expression for the supply function.

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Key Concepts

Linear Functions

A linear function represents a straight-line relationship between two variables, where one variable can be expressed as a linear combination of the other. Its standard form is often written as y = mx + c, where m is the slope (representing the rate of change) and c is the y-intercept (the value when x equals 0). This concept is central in modeling relationships that change at a constant rate.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find a common solution that satisfies all equations simultaneously. Methods to solve such systems include substitution and elimination, which are essential techniques in algebra for finding the values of unknown constants in linear models.

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