Solve each problem. Suppose that it costs 5000 to start up...

Solve each problem. Suppose that it costs $\$ 5000$ to start up a business selling snow cones. Furthermore, it costs $\$ 0.50$ per cone in labor, ice, syrup, and overhead. Then the cost to make $x$ snow cones is given by $y$ dollars, where $$ y=0.50 x+5000 $$ Express each of the following as an ordered pair. (a) When 100 snow cones are made, the cost is $\$ 5050$. (b) When the cost is $\$ 6000$, the number of snow cones made is 2000 .

Solve each problem.
Suppose that it costs $\$ 5000$ to start up a business selling snow cones. Furthermore, it costs $\$ 0.50$ per cone in labor, ice, syrup, and overhead. Then the cost to make $x$ snow cones is given by $y$ dollars, where
$$
y=0.50 x+5000
$$
Express each of the following as an ordered pair.
(a) When 100 snow cones are made, the cost is $\$ 5050$.
(b) When the cost is $\$ 6000$, the number of snow cones made is 2000 .

Key Concepts

Linear Functions

A linear function models a relationship between two variables using a constant rate of change. It is typically represented by the equation y = mx + b, where m is the slope and b is the y-intercept. This concept is foundational because it allows us to predict one quantity based on another using a straight-line relationship.

Slope-Intercept Form

The slope-intercept form of a linear equation, given by y = mx + b, explicitly shows how changes in the independent variable affect the dependent variable. The slope (m) reflects the rate at which the dependent variable changes, while the y-intercept (b) represents the initial value when the independent variable is zero.

Ordered Pairs

Ordered pairs, written in the form (x, y), represent the relationship between two variables in a coordinate system. They allow us to pair a specific input with its corresponding output, making it easier to understand and visualize the function's behavior.

Function Evaluation

Function evaluation involves substituting a specific value into the function to determine the corresponding output. This process is essential for finding the value of the dependent variable for a given independent variable, which in turn helps in understanding and solving real-world problems.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that satisfies the equation. In practical applications, this often means rearranging the linear equation to isolate the variable and determine its value based on a given condition for the other variable.

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