Now, the aim is to prove that, if these relations hold true, then the points must be collinear. This means that the area of the triangle PQR is 0 , proving that the points are collinear. We took 3 random points on the curve, and proved that those 3 points are collinear.
This can only be possible if the curve is a line. This might be a little too complex for you. We now explore more general equations of lines with equations. Example Find the x and y intercepts of the graph of the equations given below.
Solution to the Questions in the Above Example. Free Mathematics Tutorials. About the author Download E-mail. The properties of the graph such as slope and x and y intercepts are also explored. Click on the button above "click here to start" and maximize the window obtained.
Do you know how it works? By the end of this short lesson, you will understand how it works. Linear equation formula is upon case to case and based on the number of variables and the variables used themselves. Firstly, the variables should be independent of each other. Suppose you have x as a variable, then you can't keep x 2 as another variable.
Secondly, the highest and the only degree of all variables in the equation should be 1. Let us have a look at the standard form of a linear equation with variables x and y:. The graph of a linear equation in one variable x forms a vertical line parallel to the y-axis and vice-versa, whereas the graph of a linear equation in two variables x and y forms a straight line. The reason an equation of degree one is called a linear equation is that its geometrical representation is a straight line.
Given above are a few examples of how we plot linear equations on a graph. Let's have a look at how to solve any equation. An equation is like a weighing balance with equal weights on both sides.
If we add or subtract the same number from both sides of an equation, it still holds. Similarly, if we multiply or divide the same number on both sides of an equation, it still holds. Now let's add 2 on both sides to reduce the LHS to 3x. This will not disturb the balance. Now let's divide both sides by 3 to reduce the LHS to x. This is just one way of solving such equations. One another and more efficient way is to solve linear equations graphically.
Example 1: A one-day International cricket match was organized in Nagpur. India and Sri Lanka were the two teams. X is zero, Y is one, two, three, 4. And just with these two points, two points are enough to graph a line, we can now graph it. So let's do that. So let me, oops, though I was using the tool that would draw a straight line. Let me see if I can So the line will look something like that. There you have it. I've just graphed, I've just graphed, this is the line that represents all the X and Y pairs that satisfy the equation 9X plus 16Y is equal to Now, I mentioned standard form's good at certain things and the good thing that standard form is, where it's maybe somewhat unique relative to the other forms we looked at, is it's very easy to figure out the x-intercept.
It was very easy to figure out the x-intercept from standard form. And it wasn't too hard to figure out the y-intercept either. If we looked at slope-intercept form, the y-intercept just kinda jumps out at you.
At point-slope form, neither the x nor the y-intercept kind of jump out at you. The place where slope-intercept or point-slope form are frankly better is that it's pretty easy to pick out the slope here, while in standard form you would have to do a little bit of work. You could use these two points, you could use the x and y-intercepts as two points and figure out the slope from there.
So you can literally say, "Okay, if I'm going from "this point to this point, my change in X "to go from eight to zero is negative eight.
Let me. So when you go from eight to zero, your change in X is equal to negative eight. And to go from zero to 4. So your slope, once you've figured this out, you could say, "Okay, this is going to be "change in Y, 4.
You get negative nine over Now once again, we had to do a little bit of work here. We either use these two points, it didn't just jump immediately out of this, although you might see a little bit of a pattern of what's going on here. But you still have to think about is it negative? Is it positive? You have to do a little bit of algebraic manipulation.
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