Question

Write the linear equation that gives the rule for this table.

x y
4 88
5 92
6 96
7 100


Write your answer as an equation with y first, followed by an equals sign.

Answer 1

To find the linear equation that fits the data in the table, we first observe the relationship between \( x \) and \( y \).

The values for \( x \) and \( y \) are:
- When \( x = 4 \), \( y = 88 \)
- When \( x = 5 \), \( y = 92 \)
- When \( x = 6 \), \( y = 96 \)
- When \( x = 7 \), \( y = 100 \)

Now, we can determine the slope (\( m \)) of the linear equation using the changes in \( y \) and \( x \):

From \( x = 4 \) to \( x = 5 \):
\[
\Delta y = 92 - 88 = 4
\]
\[
\Delta x = 5 - 4 = 1
\]
So, the slope \( m = \frac{\Delta y}{\Delta x} = \frac{4}{1} = 4 \).

Next, we can use one of the points to write the equation in point-slope form, then convert it to slope-intercept form. Using the point \( (4, 88) \):

1. Start with the point-slope form:
\[
y - 88 = 4(x - 4)
\]

2. Distribute the slope:
\[
y - 88 = 4x - 16
\]

3. Add 88 to both sides to isolate \( y \):
\[
y = 4x - 16 + 88
\]
\[
y = 4x + 72
\]

Thus, the equation that gives the rule for this table is:
\[
y = 4x + 72
\]