[tex]\[
\begin{array}{l}
f(x) = \frac{4}{3}x - 3 \\
h(x) = ax + 2
\end{array}
\][/tex]

The functions [tex]\( f \)[/tex] and [tex]\( h \)[/tex] are defined above, where [tex]\( a \)[/tex] is a constant. In the [tex]\( xy \)[/tex]-plane, the graph of function [tex]\( f \)[/tex] is moved up [tex]\( k \)[/tex] units to obtain the graph of function [tex]\( g \)[/tex]. The system constituting the functions [tex]\( g \)[/tex] and [tex]\( h \)[/tex] has infinitely many solutions. What is the value of [tex]\( k \)[/tex]?

Your answer: [tex]\(\square\)[/tex]



Answer :

To find the value of [tex]\( k \)[/tex] such that the system of equations formed by the functions [tex]\( g \)[/tex] and [tex]\( h \)[/tex] has infinitely many solutions, follow these steps:

1. Define the given functions:
[tex]\[ f(x) = \frac{4}{3} x - 3 \][/tex]
[tex]\[ h(x) = ax + 2 \][/tex]

2. Define the function [tex]\( g \)[/tex], which is [tex]\( f(x) \)[/tex] moved up by [tex]\( k \)[/tex] units:
[tex]\[ g(x) = f(x) + k = \frac{4}{3} x - 3 + k \][/tex]

3. For the system constituted by [tex]\( g \)[/tex] and [tex]\( h \)[/tex] to have infinitely many solutions, [tex]\( g \)[/tex] must be identical to [tex]\( h \)[/tex]. This means that both the slopes and the y-intercepts of [tex]\( g \)[/tex] and [tex]\( h \)[/tex] must be equal.

4. Compare the slopes:
The slope of [tex]\( f(x) \)[/tex] is [tex]\( \frac{4}{3} \)[/tex].
The slope of [tex]\( g(x) \)[/tex] is also [tex]\( \frac{4}{3} \)[/tex] because it is simply [tex]\( f(x) \)[/tex] shifted vertically.
The slope of [tex]\( h(x) \)[/tex] is [tex]\( a \)[/tex].
For [tex]\( g \)[/tex] and [tex]\( h \)[/tex] to have the same slope:
[tex]\[ \frac{4}{3} = a \implies a = \frac{4}{3} \][/tex]

5. Compare the y-intercepts:
The y-intercept of [tex]\( f(x) \)[/tex] is [tex]\( -3 \)[/tex].
The y-intercept of [tex]\( g(x) \)[/tex], after moving up [tex]\( k \)[/tex] units, is:
[tex]\[ -3 + k \][/tex]
The y-intercept of [tex]\( h(x) \)[/tex] is [tex]\( 2 \)[/tex].
For [tex]\( g \)[/tex] and [tex]\( h \)[/tex] to have the same y-intercept:
[tex]\[ -3 + k = 2 \implies k = 2 + 3 \implies k = 5 \][/tex]

So, the value of [tex]\( k \)[/tex] is:
[tex]\[ k = 5 \][/tex]