Question

which of the following equations is an example of inverse variation between variables x and y? a. $y = x + 7$ b. $y = 7x$ c. $y = \frac{7}{x}$ d. $y = \frac{x}{7}$

Explanation:

Step1: Recall inverse variation definition

Inverse variation between \(x\) and \(y\) is of the form \(y=\frac{k}{x}\) (where \(k\) is a non - zero constant), meaning \(xy = k\) (the product of \(x\) and \(y\) is constant).

Step2: Analyze Option A

For \(y=x + 7\), this is a linear equation in the form of a linear function \(y=mx + b\) (here \(m = 1\), \(b = 7\)). It represents a direct linear relationship (not inverse variation) as \(y\) increases by 1 when \(x\) increases by 1 (with a constant added).

Step3: Analyze Option B

For \(y = 7x\), this is a direct variation equation of the form \(y=kx\) (where \(k = 7\)). In direct variation, \(y\) is directly proportional to \(x\) (as \(x\) increases, \(y\) increases proportionally), not inverse variation.

Step4: Analyze Option C

For \(y=\frac{7}{x}\), this is in the form of inverse variation \(y=\frac{k}{x}\) with \(k = 7\). If we rewrite it as \(xy=7\), we can see that the product of \(x\) and \(y\) is a constant (7), which satisfies the definition of inverse variation.

Step5: Analyze Option D

For \(y=\frac{x}{7}\), this can be rewritten as \(y=\frac{1}{7}x\), which is a direct variation equation of the form \(y = kx\) (where \(k=\frac{1}{7}\)). It represents direct variation (as \(x\) increases, \(y\) increases proportionally), not inverse variation.

Answer:

C. \(y=\frac{7}{x}\)