Question

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

x y
2 3
3 28
4 53
5 78


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

Answer 1

To find the linear equation that fits the given data, we can first observe the change in \( y \) as \( x \) increases.

Let's calculate the differences in \( y \):

- From \( x = 2 \) to \( x = 3 \): \( 28 - 3 = 25 \)
- From \( x = 3 \) to \( x = 4 \): \( 53 - 28 = 25 \)
- From \( x = 4 \) to \( x = 5 \): \( 78 - 53 = 25 \)

The difference in \( y \) is constant at 25, suggesting that the relationship between \( x \) and \( y \) is linear.

We can use the slope-intercept formula \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

From the differences, we can identify the slope \( m = 25 \).

Next, we can use one of the points to find \( b \). Let's use the point \( (2, 3) \):

\[
3 = 25(2) + b
\]
\[
3 = 50 + b
\]
\[
b = 3 - 50 = -47
\]

Now we have \( m \) and \( b \):

\[
y = 25x - 47
\]

Thus, the equation is:

\[
y = 25x - 47
\]