Which equation models the linear relationship in the table below?
x y
-4 10
1 -5
6 -20
change in y/change in x
= (-5-10) / (1+4) = -3
or
= (-20+5) / (6-1) = -3
or
= (-20 -10) / (6+4) = -3
remarkable, the slope is constant and is -3
so
y = -3 x + b
now put any of those points in
10 = -3(-4) + b
b = -2
so
y = -3 x - 2
NOW
check that with another point like
-20 = -3(6) - 2 ?????
sure enough
-20 = -20
To determine the equation that models the linear relationship in the given table, we need to find the equation of a line that passes through these points.
Let's calculate the slope of the line using two of the given points (x₁, y₁) and (x₂, y₂):
(x₁, y₁) = (-4, 10)
(x₂, y₂) = (1, -5)
The slope formula is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the values, we get:
m = (-5 - 10) / (1 - (-4))
m = (-5 - 10) / (1 + 4)
m = -15 / 5
m = -3
Now that we have the slope, we can determine the y-intercept (b). We can choose any of the given points to calculate b. Let's use the point (1, -5):
y = mx + b
-5 = -3(1) + b
-5 = -3 + b
b = -2
Therefore, the equation that models the linear relationship in the table is:
y = -3x - 2
To determine which equation models the linear relationship in the table, we can calculate the slope (m) and the y-intercept (b) using the formula for the equation of a line: y = mx + b.
Step 1: Calculate the slope (m)
The slope is calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line.
Let's choose (-4, 10) and (1, -5) from the table.
m = (-5 - 10) / (1 - (-4))
m = -15 / 5
m = -3
Step 2: Calculate the y-intercept (b)
We can choose any of the points from the table and substitute the values of x, y, and m into the equation y = mx + b to solve for b.
Using the point (1, -5):
-5 = -3(1) + b
-5 = -3 + b
b = -2
Step 3: Write the equation
Now that we have the slope (m = -3) and the y-intercept (b = -2), we can write the equation:
y = -3x - 2
Therefore, the equation that models the linear relationship in the table is y = -3x - 2.