Question

which equation represents a linear function?

\\(y - 2 = -5(x - 2)\\)
\\(x + 7 = -4(x + 8)\\)
\\(y - 3 = y(x + 4)\\)
\\(y + 9 = x(x - 1)\\)

Explanation:

Define a linear function

A linear function in two variables \(x\) and \(y\) can be written in the standard form \(Ax + By = C\) or slope-intercept form \(y = mx + b\), where the variables \(x\) and \(y\) are raised only to the first power and are not multiplied together.

Analyze each option

• \(y - 2 = -5(x - 2)\): Simplifying gives \(y - 2 = -5x + 10 \implies y = -5x + 12\). This is a linear function in slope-intercept form.
• \(x + 7 = -4(x + 8)\): This equation contains only the variable \(x\) and simplifies to a single value for \(x\), which represents a vertical line, not a function of \(x\).
• \(y - 3 = y(x + 4)\): Simplifying gives \(y - 3 = xy + 4y\). The term \(xy\) is a product of two variables, making it non-linear.
• \(y + 9 = x(x - 1)\): Simplifying gives \(y + 9 = x^2 - x\). The term \(x^2\) makes this a quadratic equation, not linear.

Identify the correct equation

The only equation that represents a linear function of the form \(y = mx + b\) is \(y - 2 = -5(x - 2)\).

Answer:

• (A) \(y - 2 = -5(x - 2)\) (Correct answer)
• (B) \(x + 7 = -4(x + 8)\)
• (C) \(y - 3 = y(x + 4)\)
• (D) \(y + 9 = x(x - 1)\)