The math says that y varies with x. What does this mean?
Direct variation is a relationship between two variables, x and y. When two variables vary directly, their ratio (the fraction ) always equals the same number. That number is called the Constant of Variation. When we don’t know the value of the constant of variation, we use the letter k to hold its place.
The values of x and y change based on the value of k, but their values also depend heavily on each other. In direct variation, whenever x gets bigger, y gets bigger. Whenever x gets smaller, y gets smaller. The reverse is also true. Whenever y gets bigger, x gets bigger. Whenever y gets smaller, x gets smaller.
The Direct Variation Equation
The general equation for direct variation is written
y = kx
We can read this in two ways: “y varies directly as x,” or “y equals k times x.”
Whenever we need to find the value of k, we can divide both sides of the equation by x. The resulting equation looks like this:
How do I tell if an equation has direct variation?
A direct variation equation can only have multiplication or division in it. A direct variation equation will never involve addition or subtraction.
If you are given a set of related x and y values, plug each pair into the equation
If you get the same value for k every time, then you have direct variation.
If you are looking at a graph, direct variation has two defining characteristics. The graph will be a straight line, and that line will go through the origin, which is the point (0, 0).
How do I solve a problem with direct variation?
Direct variation problems will often involve two major steps. The math will usually give you either a related pair of values for x and y, or they will give you an ordered pair (our notation for a point in the Cartesian Coordinate Plane; this is actually still a related pair of values). The math will also give you a value for one of the variables, but not the other. Your task is to find the missing related value.
Once you have identified your related values for x and y, your first step is to find k (the constant of variation). Once you know the value of k, your second step is to solve for the missing value.
y varies directly with x, and y = 35 when x = 7. What is the value of y when x = 9?
· Prep Step: Identify your related values.
o The problem tells us, “y = 35 when x = 7.” These are our related values.
· Step 1: Find the value of k.
o We can plug our related values and k into the general equation for direct variation. When we isolate k, we find its value.
y = kx
35 = k(7)
o We can divide both sides of the equation by 7 to isolate k.
5 = k
· Step 2: Find the missing value.
o The problem asks, “What is the value of y when x = 9?” We need to plug 9 in for x and 5 in for k, then solve for y.
y = 5(9)
y = 45
Do these related sets of values for x and y represent direct variation? If so, write an equation for the direct variation.
Set 1: (-2, 3), (-6, 7), and (4, -5)
Set 2: (4, 5), (8, 10), and (10, 12.5)
· Step 1: Plug all of the given values into the second direct variation equation and solve for k.
· Step 2: Compare the values of k that you found.
o If the fractions do not reduce easily, you can use division to convert them to decimals. To do so, you divide the numerator (top number) by the denominator (bottom number).
-1.5 = k, -1.167 = k, -1.25 = k
o These values of k are all different. This means that we do not have direct variation.
1.25 = k, 1.25 = k, 1.25 = k
o These values of k are all the same. This means that we do have direct variation.