Question

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

x	y
3	-54
4	-58
5	-62

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

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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 \((2, -50)\) and \((3, -54)\).
\( m=\frac{-54 - (-50)}{3 - 2}=\frac{-54 + 50}{1}=\frac{-4}{1}=-4 \)

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 \((2, -50)\) and \( m=-4 \).
Substitute into the formula: \( y-(-50)=-4(x - 2) \)
Simplify: \( y + 50=-4x+8 \)
Subtract 50 from both sides: \( y=-4x+8 - 50 \)
\( y=-4x - 42 \)

We can check with other points. For \( x = 4 \), \( y=-4(4)-42=-16 - 42=-58 \), which matches the table. For \( x = 5 \), \( y=-4(5)-42=-20 - 42=-62 \), which also matches.

Answer:

\( y=-4x - 42 \)