The points $(x, y)$ represented in this table lie on a straight line. The point $(28, t)$ lies on the same line. What is the value of $t?$ \begin{tabular}{c|c}
$x$ & $y$ \\ \hline
1 & 7 \\
3 & 13 \\
5 & 19 \\
\end{tabular}
Looking at the table, we see that increasing $x$ by two results in an increase in $y$ of $6.$ Thus for every increase in $x$ of $1,$ $y$ increases by $3.$ If $x = 1$ gives $y = 7,$ then $x = 28 = 1 + 27$ gives $y = 7 + 27\cdot 3 = \boxed{88}.$
88
Level 3
Algebra
Problem: The points $(x, y)$ represented in this table lie on a straight line. The point $(28, t)$ lies on the same line. What is the value of $t?$ \begin{tabular}{c|c}
$x$ & $y$ \\ \hline
1 & 7 \\
3 & 13 \\
5 & 19 \\
\end{tabular}
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the value of $t$ for the point $(28, t)$ on the same line as the given points, we first need to determine the equation of the line that passes through the points in the table. 

The points provided are $(1, 7)$, $(3, 13)$, and $(5, 19)$. We can calculate the slope of the line using any two points. Let's use the points $(1, 7)$ and $(3, 13)$.

The slope $m$ is calculated as follows:
m = (y2 - y1) / (x2 - x1) = (13 - 7) / (3 - 1) = 6 / 2 = 3.

Now that we have the slope, we can use the point-slope form of the line equation, which is $y - y_1 = m(x - x_1)$. Using the point $(1, 7)$, we have:
y - 7 = 3(x - 1).

Expanding this, we get:
y - 7 = 3x - 3,
y = 3x + 4.

Now we have the equation of the line: $y = 3x + 4$. 

Next, we substitute $x = 28$ into the equation to find the corresponding value of $y$, which is represented by $t$:
t = 3(28) + 4.

Calculating this gives:
t = 84 + 4 = 88.

Thus, the value of $t$ is $88$.<|END_OF_LATENT|> The final answer is $88$.