Which equation represents a linear function?

[tex]y(x-1)=9[/tex]
[tex]y-5=x(-x+2)[/tex]
[tex]y(y-1)=x+25[/tex]
[tex]y-5=x-20[/tex]

The problem asks to identify a linear function from a set of equations.
A linear function has the form y = m x + b .
Analyze each equation and rewrite it to match the linear form.
The equation y − 5 = x − 20 is the only one that can be written in the form y = x − 15 , thus it is a linear function. y − 5 = x − 20 ​

Explanation

Problem Analysis We are given four equations and we need to determine which one represents a linear function. A linear function can be written in the form y = m x + b , where m and b are constants. We will rewrite each equation to see if it fits this form.

Analyzing Each Equation Let's analyze each equation:

y ( x − 1 ) = 9 . We can rewrite this as y = x − 1 9 ​ . This is not a linear function because x is in the denominator.

y − 5 = x ( − x + 2 ) . We can rewrite this as y = − x 2 + 2 x + 5 . This is not a linear function because of the x 2 term.

y ( y − 1 ) = x + 25 . We can rewrite this as y 2 − y = x + 25 , or y 2 − y − ( x + 25 ) = 0 . This is a quadratic equation in y , so it is not a linear function.

y − 5 = x − 20 . We can rewrite this as y = x − 15 . This is a linear function with m = 1 and b = − 15 .

Conclusion Therefore, the equation that represents a linear function is y − 5 = x − 20 , which simplifies to y = x − 15 .


Examples
Linear functions are incredibly useful in everyday life. For example, imagine you are saving money. If you save a fixed amount each week, the total amount you've saved can be modeled by a linear function. If you save 10 p er w ee k , t h ee q u a t i o n w o u l d b e y = 10x , w h ere y i s t h e t o t a l s a v in g s an d x$ is the number of weeks. This helps you predict how much you'll have saved after a certain amount of time.

Answered by GinnyAnswer | 2025-07-08