To solve the given problem, let us go through the necessary steps to complete the table and determine if each input [tex]\( x \)[/tex] has exactly one output [tex]\( y \)[/tex] for the equation [tex]\( y = 15x - 2 \)[/tex].
1. Equation Analysis:
The equation provided is a linear function: [tex]\( y = 15x - 2 \)[/tex]. In this form, [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. A linear equation has a direct one-to-one relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. For each [tex]\( x \)[/tex] value, there is exactly one [tex]\( y \)[/tex] value.
2. Complete the table:
We will calculate the missing [tex]\( y \)[/tex] values corresponding to the given [tex]\( x \)[/tex] values (3 and 6):
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 15(3) - 2 = 45 - 2 = 43
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = 15(6) - 2 = 90 - 2 = 88
\][/tex]
3. Fill the table:
With these calculations, the table should look like this:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & -47 \\
\hline
0 & -2 \\
\hline
3 & 43 \\
\hline
6 & 88 \\
\hline
\end{array}
\][/tex]
4. Determine if each input has exactly one output:
The given linear equation [tex]\( y = 15x - 2 \)[/tex] implies a one-to-one relationship. Therefore, each [tex]\( x \)[/tex] value has exactly one corresponding [tex]\( y \)[/tex] value. As a result, each input does indeed have exactly one output.
Hence, the completed table would show that each [tex]\( x \)[/tex] value corresponds to one unique [tex]\( y \)[/tex] value, and therefore, the answer to the question is:
Yes.
