The percent P of companies that are still in business t...

The percent $P$ of companies that are still in business $t$ years after the fifth year in operation can be represented by the equation $$ P=-3 t+50 $$ a. What percentage of companies are still in business after 1 year in operation? b. What percentage of companies are still in business after 25 years in operation? c. After how many years are there only $35 \%$ of companies still in business?

The percent $P$ of companies that are still in business $t$ years after the fifth year in operation can be represented by the equation
$$
P=-3 t+50
$$
a. What percentage of companies are still in business after 1 year in operation?
b. What percentage of companies are still in business after 25 years in operation?
c. After how many years are there only $35 \%$ of companies still in business?

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Key Concepts

Linear Functions

A linear function is a mathematical expression that describes a straight-line relationship between two variables. It is typically written in the form y = mx + b, where m represents the slope and b represents the y-intercept. This concept is key for understanding how one quantity changes at a constant rate with respect to another.

Substitution Method in Functions

The substitution method involves replacing the variable in a function with a specific value to calculate the corresponding output. This is fundamental when analyzing how changes in the input affect the output of a function, as demonstrated by evaluating the function for different years.

Solving Linear Equations

Solving linear equations involves isolating the unknown variable on one side of the equation using algebraic manipulations such as addition, subtraction, multiplication, and division. This concept is important for determining the input value that produces a given output in a linear function.

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