Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(a) $3x - 4xy = 6$	$\circ$	$\circ$
(b) $7x = 4 - y$	$\circ$	$\circ$
(c) $\frac{5}{x} - 3y = 9$	$\circ$	$\circ$
(d) $4y^3 - y = x$	$\circ$	$\circ$

Explanation:

Step1: Recall linear equation definition

A linear equation in two variables \(x\) and \(y\) is of the form \(Ax + By = C\), where \(A\), \(B\), \(C\) are constants, and the highest power of \(x\) and \(y\) is 1, and there are no products of variables (like \(xy\)), no variable in the denominator, and no variable with exponent greater than 1.

Step2: Analyze equation (a) \(3x - 4xy = 6\)

The term \(-4xy\) is a product of \(x\) and \(y\), so it's not in the form \(Ax + By = C\). So answer: No.

Step3: Analyze equation (b) \(7x = 4 - y\)

Rearrange to \(7x + y = 4\), which is \(Ax + By = C\) with \(A = 7\), \(B = 1\), \(C = 4\). So answer: Yes.

Step4: Analyze equation (c) \(\frac{5}{x}-3y = 9\)

The term \(\frac{5}{x}\) is \(5x^{-1}\), so \(x\) is in the denominator (exponent -1), not linear. Answer: No.

Step5: Analyze equation (d) \(4y^{3}-y = x\)

The term \(4y^{3}\) has \(y\) with exponent 3, greater than 1. Not linear. Answer: No.

Answer:

(a) No
(b) Yes
(c) No
(d) No