Question

Fill in the missing number to complete the linear equation that gives the rule for this table.

x y
4 –32
5 –40
6 –48
7 –56


y =
x

Answer 1

To find the linear equation that relates \(x\) and \(y\), we first look for a pattern in the values.

Let's analyze the change in \(y\) as \(x\) increases:

- When \(x = 4\), \(y = -32\)
- When \(x = 5\), \(y = -40\) (change in \(y = -40 - (-32) = -8\))
- When \(x = 6\), \(y = -48\) (change in \(y = -48 - (-40) = -8\))
- When \(x = 7\), \(y = -56\) (change in \(y = -56 - (-48) = -8\))

The change in \(y\) is consistent at \(-8\) for each increase of \(1\) in \(x\). This suggests a linear relationship with a slope of \(-8\).

Now we can express this linear relationship in the form of the equation:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.

Using the slope \(m = -8\), we can write:
\[
y = -8x + b
\]
To find \(b\), we can use one of the points in the table. Let's use the point \( (4, -32) \).

Substituting \(x = 4\) and \(y = -32\):
\[
-32 = -8(4) + b
\]
\[
-32 = -32 + b
\]
\[
b = 0
\]

Therefore, the equation is:
\[
y = -8x
\]

Thus, the completed equation is:
\[
y = -8x
\]