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Solving direct variation problems and equations

How to solve direct variation problems

In schools, students are taught that if every y value in a function is achieved by multiplying every x value by a constant number, then that function is a direct variation. The formula used for direct variation is y=kx, where k is the constant of the direct variation. A constant in direct variation is a number which relates to the two variables which are directly proportional to each other. The steps followed in order to solve direct variation problems are:

  • Step 1: Get the correct direct variation equation. The equation used to solve the problems is y=kx. When solving a direct variation word problem, however, you must consider using other variables apart from x and y. In other words, use variables that apply to the direct variation problem being solved. Also understand the problem very well in order to determine if there are other changes like squares, square roots and cubes in the equation of direct variation. 
  • Step 2: k is known as the constant of proportionality or the constant of variation in direct variation. Use the information provided in the problem to calculate the value of k.
  • Step 3: Rewrite the direct variation equation y=kx substituting in the value of k.
  • Step 4: Use the direct variation equation you got in step 3 and the other remaining information in the problem to answer the questions being asked. Remember to include units in your final answer when solving word problems.


The y intercept is that point where the graph of the line crosses the y axis. Solving direct variation problems involves formulas or simple relationships where one variable is equal to one term. The term may be linear which is anything with a “x”, it can be quadratic which is anything with x2,,it can be more than one value such as (“r2h”), it can be a square root such as [Description: sqrt(4 – r^2)] or it can be anything else that can form a direct variation. But most of the time it is one term in a formula that is multiplied by a constant k. To be sure that an equation has a direct variation, the equation has to be a division or a multiplication. A direct variation will never involve subtraction or addition. Whenever you are given a set of related x and y values, plug every pair into the equation. If the result you get is a constant value k every time, then that is a direct variation.


Examples of variation equations and applications

A good example of a direct variation equation is the formula for the area of a circle [Description: A = (pi) r^2]. A varies directly with the square of the radius r. The constant of direct variation is [Description: k = (pi)]. Another good example of direct variation is distance and speed. If your speed is high you will go further within a short time. So as one variable goes up the other one goes up too and vice versa. In algebra, direct variation is expressed using equations. If you are told that y varies directly as x it means that when x is increased, y is increased by the same factor. In other words x and y have a constant ratio which is represented as y/x=k where k is the constant of proportionality. The direct variation relationship between x and y can also be represented as y=kx where k again is the constant of proportionality. The constant k can be calculated when given any point by dividing the y coordinate by the x coordinate. For instance, if there is a direct variation between x and y, and y is equal to 8 when x is equal to 4. The constant of variation k =8/4 which is equal to 2. And the equation to describe this direct variation is y=2x.

Newton’s second law is another good direct variation example. The law is often expressed as F=ma where F is the quantity of force applied to an object with mass m and acceleration a. Acceleration is expressed in the units meters/second2 and force is given in newtons. If a 175 force causes a shopping cart to accelerate with an acceleration of 2.5 meters/second2. To calculate the mass of the shopping cart is simply to solve the constant of proportionality. Compare the direct variation formulas y=kx and F=ma (the direct variation equation and Newton’s second law equation), the equation have the same form. Y is the force and x is the acceleration. 175=m*2.5. To find m 175/2.5=m=70. The constant of direct variation is equal to 70 which is the mass. To calculate the force needed to accelerate the cart at 6 m/s2, substitute 6 for a and evaluate the equation. When a is 6, F is 70*6 which is equal to 420 Newtons.

Other examples of direct variation are: if y varies directly as x and x=10 when y=2, the equation that describes this direct variation is k=2/10=1/5. Y=1/5. If y varies directly as x and the constant of proportionality is k=5/3, and you are asked to find the value of y when x is equal to 9. To substitute for y, y=5/3x=5/3(9) =15.The constant of direct variation k is the same for every point. For example, the ratio between the y coordinate of any point and the x coordinate of that point is constant. This implies that if you are given any points (x1, y1) and (x2, y2) which satisfy the equation y1/x1=k and y2/x2=k, then y1/x1=y2/x2 for any two points which satisfy the direct variation equation. If y varies directly as x and y is equal to 15 when x is equal to 10 and you are asked to find y when x is equal to 6, then 15/10=y/6. 3/2=y/6, 6(3/2) =y, so y is equal to 9.

How to plot variation graphs in the coordinate plane

The direct variation of a relationship can be further demonstrated in a graph. If you have all points of a graph, you will have a straight 45 degrees line to demonstrate linearity. The graph lines have lines starting from the origin and extending to infinity. A direct variation can be tested and proven with various ratios. If you graph all the different ratios that you attempt and assuming the ratios is proportional, you will come up with similarly shaped lines. Graphs help to visualize direct variation. They make the information about direct variation more convincing.

To plot a direct variation graph, you first determine the origin. The coordinate plane has two axes: the vertical and horizontal axes. The two axes intersect each other at a point which is referred to as the origin. A coordinate plane is where you graph a function. Another term used in graphing direct variation is the slope. You must learn about a direct variation slope before learning about linear equations. A slope of a line is its steepness. There are many other ways of describing a slope in direct variation. It can be described as the rise over the run, as the change in y over the change in x or it can also be described as the gradient of the line. To determine the slope of a line, you first identify two points on the line. You will then use these two points to figure out the slope.

A horizontal line in direct variation is flat and has no steepness. Mathematically it is expressed by saying that the horizontal line has a slope of zero. The vertical line does not have a defined slope. In calculating the slope, a slope is the change in y over the change in x. This implies that if you have two points in a line (0, 0) and (2, 2) the slope is (y2-y1)/(x2-x1) = (2-0)/ (2-0) =1. To graph a line with a given slope, you must figure out how to graph that line. A line is expressed in a slope intercept form y=mx+b. If you are given the equation y=2x+0 then the line has a slope 2 and a y intercept of 0. This implies that a direct variation equation is an equation is slope which passes through (0, 0) and its slope is equal to the constant of variation. And since the y intercept is zero then one point is (0, 0). But since the slope is 2 you will have to go up 2 for every 1 you go to the right. The line will go up two points for every one point it goes right. This implies that the change in y is 2 when the change in x is equal to 1 which is a direct variation.

Another direct variation graphic example is in y=kx equation. The equation y=kx can be thought of as an equation of the form y=mx+b where b=0 and m=k. To therefore graph a direct variation equation, you begin at the origin point (0, 0) and proceed as you would when graphing any slope. Alternatively, if you have one point, just draw a straight line between (0, 0) and that particular point and then extend the line on both ends. If y varies directly as x and the constant of variation is ½, you can represent the variation on a graph and the equation that describes the direct variation is y=1/2x. If you are required to calculate the constant of variation with a graph of direct variation, you simply need to calculate the slope of that graph.

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