Question

this table of values represents a linear function.
x | y
2 | -1
4 | -5
6 | -9
enter an equation in the form $y = mx + b$ that represents the function defined by this table of values

Explanation:

Step1: Find the slope \( m \)

The slope \( m \) of a line passing through 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 points \((2, -1)\) and \((4, -5)\). Then \( x_1 = 2, y_1=-1, x_2 = 4, y_2=-5 \). So \( m=\frac{-5 - (-1)}{4 - 2}=\frac{-5 + 1}{2}=\frac{-4}{2}=-2 \).

Step2: Find the y - intercept \( b \)

We know the equation of the line is \( y=mx + b \), and we have \( m=-2 \). Let's use the point \((2, -1)\) (we can use any point from the table). Substitute \( x = 2, y=-1 \) and \( m=-2 \) into the equation: \( -1=-2\times2 + b \). Simplify the right - hand side: \( -1=-4 + b \). Add 4 to both sides of the equation: \( b=-1 + 4 = 3 \).

Step3: Write the equation

Now that we have \( m=-2 \) and \( b = 3 \), the equation of the line in the form \( y=mx + b \) is \( y=-2x+3 \).

Answer:

\( y=-2x + 3 \)