## Which of the following equations are linear?

Answer 1:

Yes, the correct answer is a.\(\text { a and c. }\)

Explanation:

A linear equation in \(x\) has constant multiples of \(x\) and additive constants only. There can be no cross term in \(x y\), or terms in \(x^{2}, y^{2}\) etc. When graphed, a linear function represents a \(\text { straight line. }\)

Thus it takes the form:

\(y=A x+B\), Or equivalently: \(a y+b x+c=0\)

Hence we categorize the various options as follows:

\(\text { a. } y=2 x-5 \quad \text { linear }\)

\(\text { b. } y=x^{2} \quad \text { non-linear }\)

\(\text { c. } x+2 y=7 \quad \text { linear }\)

\(\text { d. } x^{2}+y^{2}=25 \text { non-linear }\)

\(\text { e. } y=x^{3} \quad \text { non-linear }\)

Thus the correct answer is a. \(\text { a and c. }\)

Answer 2:

\(A \text { and } C\)

Explanation:

Linear equations have either 2 or 3 of the following:

- an \(x\)-term
- a \(y\)-term
- a number term

Terms may NOT be in the form:

\(x^{2}, x^{3}, y^{2}, y^{3} \text { etc }\)

\(\sqrt{x}, \sqrt{y}\)

\(\frac{1}{x} \text { or } \frac{1}{y}\)

\(xy\)

The following are all linear equations:

\(3 x=4\)

\(y=11\)

\(5 y=2 x\)

\(4 x-3 y=10\)

\(x-2 y+5=0\)

\(y=\frac{1}{2} x-7\)

Answer 3:

a and c.

Explanation: