Question

write the linear equation that gives the rule for this table
x | y
1 | 45
2 | 33
3 | 21
4 | 9
write your answer as an equation with y first, followed by...

Explanation:

Step1: Find the slope (m)

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the first two points \((1, 45)\) and \((2, 33)\).
\( m=\frac{33 - 45}{2 - 1}=\frac{- 12}{1}=- 12 \)

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We can use the point \((1, 45)\) and \( m=-12 \).
Substitute into the formula: \( y - 45=-12(x - 1) \)
Expand the right - hand side: \( y - 45=-12x+12 \)
Add 45 to both sides: \( y=-12x + 12 + 45 \)
Simplify: \( y=-12x+57 \)

We can verify this equation with other points. For \( x = 3 \), \( y=-12\times3 + 57=-36 + 57 = 21 \) (which matches the table). For \( x = 4 \), \( y=-12\times4+57=-48 + 57 = 9 \) (which also matches the table).

Answer:

\( y=-12x + 57 \)