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Direct Variation - Problem 1
MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
Direct variation goes through the point (0, 0) and can be written in the form y=kx. In other words, the line passes through the origin (has a y-intercept of 0). Calculate k the same way you would calculate slope (m) -- (y2-y1)/(x2-x1). Use (0,0) and the given x and y values as x1, y1 and x2, y2. Once you have the slope, substitute it in for k.
This is a problem that where I'm already told that x and y vary directly. Before I read any further I'm going to right down what I know about direct variation. Direct variation mean it goes through the point (0,0) and the equation is going to look like y equals kx so that's kind of why I'm going to keep in mind as I'm continuing to read the problem.
The value of y is 22 when the value of x is 2, find y when x equals 5.Okay well I'm going to write this as a point my y value is 22 my x number is 2 and I'm going to see if I have any ideas that come to me about what's in move about how to move on.
Well I have two points and I need to find the equation that's something I know how to do. You can either use the Point-Slope formula like this by calculating the slope first and then plugging in one of your point values.
Or you could use the Slope-Intercepts form because we know this intercept number is going to be 0. That's why I'm going to choose to use y equals mx plus b strategies because I know half the problem already. I know half the answer already. Let me erase this.
Okay if I want to have an equation that looks like y equals kx I'm going to think in back of my head that y equals mx because I know more about m. The way you find m is by doing y2 take away y1 on top of x2 take away x1.
So let's do that with these values 22 take way 0 over 2 take away 0 is equal to 22 over 2 or 11 that's my slope number or in direct variations slope is the same thing as k which we call the constant variation. This is really tricky because there is so much vocabulary and you are moving back and fourth between what you already know in this new business.
Just keep in mind that m or slope is the same thing as the constant variation.So the equation that describes this will look like y equals 11 x, that's not my answer because they asked me to find the y value but this is the equation I'm going to be working with.
Let me erase it and we'll move on. The second half of this problem that I need to do if find y when x is equal to 5. This will be very easy now that I know my equation. Instead of x right there I'm going to write 5 using substitution y will be 11 times 5 or y equals 55. That's my final answer I found the y value when x is 5.
And again I just thought of this as a line where I already knew the y intercept and then I found the slope plugged it in there for k which is the constant of variation.
So when you see problems like this and you are looking at the letter k wondering what to do think of k in the same way you treat m, a slope, try using that y2 minus y1 over x2 minus x1 formula.
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Sample Problems (5)
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Problem 1 7,340 views
If y varies directly with x and the value of y is 22 when the value of x is 2, find y when x = 5.
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A biker's distance traveled on flat terrain varies directly with its traveling time. If the biker has gone 90 miles after 2½ hours, how long will it take him to go 306 miles?
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