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: