Question

What is the connection between linear systems and linear functions?

Explain.

Answer 1

A linear function of one variable is one whose graph is a straight line. In general, a linear function can be a function of one or more variables. Each term in a linear function is a polynomial of degree one in one of the variables, or a constant. So ()=3+2
f
(
x
)
=
3
x
+
2
is a linear function.

A linear equation has an equals sign, with linear functions on both sides. You can always move everything to one side of the equation, so solving a linear equation is essentially finding the zero of some linear function. 3+2=0
3
x
+
2
=
0
is a linear equation. A little algebra yields =−23.
x
=
−
2
3
.


It does get murky because the notation is overloaded. Consider =3+2.
y
=
3
x
+
2.
We normally think of that as a function of a single variable
x
, pretty much just an abbreviation for ()=3+2
f
(
x
)
=
3
x
+
2
. But it’s also a linear equation in two variables
x
and .
y
.
That equation’s solutions are the zeros of the linear function of two variables (,)=3−+2
g
(
x
,
y
)
=
3
x
−
y
+
2
.

Answer 2

Answer: While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).

Step-by-step explanation: Hope this helps :)