Step-by-step explanation:
he concept of direct variation is summarized by the equation below.
We say that yy varies directly with xx if yy is expressed as the product of some constant number kk and xx.
Cases of Direct Variation
However, the value of kk can’t equal zero, i.e. k \ne 0k
=0.
Case 1: k > 0k>0 (kk is positive)
If xx increases then the value of yy also increases, or if xx decreases then the value of yy also decreases.
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Case 2: k < 0k<0 (kk is negative)
If xx increases then the value of yy decreases, or if xx decreases then the value of yy increases.
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If we isolate kk on one side, it reveals that kk is the constant ratio between yy and xx. In other words, dividing yy by xx always yields a constant output.
k=y/x
kk is also known as the constant of variation, or constant of proportionality.
Examples of Direct Variation
Example 1: Tell whether yy varies directly with xx in the table below. If yes, write an equation to represent the direct variation.
Solution:
To show that yy varies directly with xx, we need to verify if dividing yy by xx always gives us the same value.
Since we always arrived at the same value of 22 when dividing yy by xx, we can claim that yy varies directly with xx. This constant number is, in fact, our k = 2k=2.
To write the equation of direct variation, we replace the letter kk by the number 22 in the equation y = kxy=kx.
When an equation that represents direct variation is graphed in the Cartesian Plane, it is always a straight line passing through the origin.
Think of it as the Slope-Intercept Form of a line written as
y = mx + by=mx+b where b = 0b=0
Here is the graph of the equation we found above.
Example 2: Tell whether yy varies directly with xx in the table below. If yes, write an equation to represent direct variation.
Solution:
Divide each value of yy by the corresponding value of xx.
The quotient of yy and xx is always k = - \,0.25k=−0.25. That means yy varies directly with xx. Here is the equation that represents its direct variation.
Here is the graph. By having a negative value of kk implies that the line has a negative slope. As you can see, the line is decreasing from left to right.
In addition, since kk is negative we see that when xx increases the value of yy decreases.
Example 3: Tell whether if yy directly varies with xx in the table. If yes, write the equation that shows direct variation.
Solution:
Find the ratio of yy and xx, and see if we can get a common answer which we will call constant kk.
It looks like the kk-value on the third row is different from the rest. In order for it to be a direct variation, they should all have the same kk-value.
The table does not represent direct variation, therefore, we can’t write the equation for direct variation.
Example 4: Given that yy varies directly with xx. If x = 12x=12 then y = 8y=8.
Write the equation of direct variation that relates xx and yy.
What is the value of yy when x = - \,9x=−9?
a) Write the equation of direct variation that relates xx and yy.
Since yy directly varies with xx, I would immediately write down the formula so I can see what’s going on.
We are given the information that when x = 12x=12 then y = 8y=8. Substitute the values of xx and yy in the formula and solve kk.
We will use the first point to find the constant of proportionality kk and to set up the equation y = kxy=kx.
Substitute the values of xx and yy to solve for kk.
The equation of direct proportionality that relates xx and yy is…
We can now solve for xx in (xx, - \,18−18) by plugging in y = - \,18y=−18.
Example 6: The circumference of a circle (CC) varies directly with its diameter. If a circle with the diameter of 31.431.4 inches has a radius of 55 inches,
Write the equation of direct variation that relates the circumference and diameter of a circle.
What is the diameter of the circle with a radius of 77 inches?
a) Write the equation of direct variation that relates the circumference and diameter of a circle.
We don’t have to use the formula y = k\,xy=kx all the time. But we can use it to come up with a similar set-up depending on what the problem is asking.
The problem tells us that the circumference of a circle varies directly with its diameter, we can write the following equation of direct proportionality instead.
The diameter is not provided but the radius is. Since the radius is given as 55 inches, that means, we can find the diameter because it is equal to twice the length of the radius. This gives us 1010 inches for the diameter.
The equation of direct proportionality that relates circumference and diameter is shown below. Notice, kk is replaced by the numerical value 3.143.14.
