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reinhard10158

Step-by-step explanation:

it is a linear function, if the change in y is constant in relation to the changes of x.

in other words, if the slope y change/x change is the same for every point.

when we look at the table, we see x changes constantly by +2.

and y changes constantly by -6.

so, the change rate/slope is constantly -6/2 = -3.

and therefore it is a linear function.

the equation looks like (the slope is always the factor of x)

y = -3x + b

b we get by using one of the x,y pairs like 1,4

4 = -3×1 + b = -3 + b

b = 7

so, the full equation is

y = -3x + 7

djtwinx017

Step-by-step explanation:

Given the table of values, {(-3, 16), (-1, 10), (1, 4), (3, -2), (5, -8)}:

We must determine whether the given relation or data can be modeled by a linear equation.  It is necessary to analyze the pattern that exists between changes in y-values, along with the changes in x-values.

X-coordinates:

Subtracting the x-coordinates provides a constant difference of 2:

  1. Starting with the first two rows, given by the ordered pairs, (-3, 16) and (-1, 10):  Subtract the two x-coordinates: -1 - (-3) = -1 + 3 = 2.
  2. For the second and third rows, given by the ordered pairs, (-1, 10) and (1, 4): Subtract the two x-coordinates: 1 - (-1) = 1 + 1 = 2.

If you perform these same steps throught the remaining rows, then you'll get the same difference of 2.

Y-coordinates:

We can perform the same steps for the y-coordinates. Subtracting the y-coordinates provides a constant difference of -6.

  1. Starting with the first two rows, given by the ordered pairs, (-3, 16) and (-1, 10):  Subtract the two y-coordinates: 10 - 16 = -6.
  2. For the second and third rows, given by the ordered pairs, (-1, 10) and (1, 4): Subtract the two y-coordinates: 4 - 10 = -6.  

If you perform these same steps throught the remaining rows, then you'll get the same difference of -6.

Slope

Since the constant change in y = -6, and the constant change in x = 2, then we can calculate for the slope as: Δy/Δx = -6/2 = - 3.  Hence, the constant rate of change = -3.

As demonstrated throughout this post, we can infer that there is a linear pattern that exists between the x- and y-values.

Therefore, the data can be modeled by a linear equation because the rate of change is constant.

Linear Equation:

In order to establish the linear equation that represents y as a function of x, we can use the slope, m = -3, and one of the given ordered pairs, (1, 4), to solve for the value of the y-intercept, b. Substitute these values into the slope-intercept form, y = mx + b:

y = mx + b

4 = (-3)1 + b

4 = -3 + b

Add 3 to both sides to isolate b:

4 + 3 = -3 + 3 + b

7 = b

Therefore, the linear equation that represents the given data is: y = -3x + 7.

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