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Writing Equations of Lines
Transcript
Writing equations of lines. Sometimes, the test will give you information and
ask you to come up with the equation of a line. Another question, solving for
the equation of the line will be very helpful in finding what the question asks. The test could expect you to find the equation of a line
或者从数字信息，这是斜率的值，这是一个点，这是y截距，诸如此类。
或者他们可以给你一张照片。如果给你一张这条线的图片，那么这两条线都应该相对容易阅读 the slope and the y-intercept from the picture, and these will allow you to create the equation. Here's a practice problem. Pause the video, and then we will talk about this.
好的，这条线穿过点A，30。A的价值是什么？ 很容易读懂这个图，y截距的值。y截距是1。 斜率，好吧，因为它穿过-2，我们有一个2的行程和一个1的上升，所以上升超过行程是1/2。
So the slope is one-half and the y-intercept is 1. So that means this has to be the equation of the line y equals one-half x+1, well, now we have an equation. So now we can plug in. We plug in A for x and 30 for y. Subtract 1, and then multiply by 2 to cancel the one-half.
我们得到A=58，这就是答案。这个测试可以给你一个斜率，然后是直线上的某个点。 坡度和单个点足以确定唯一的直线。您可以将坡度直接插入坡度截距窗体，y=mx+b。 So we would know m, initially not know the value of b but, because every point on the line must satisfy the equation of the line, 你可以在方程中插入x=y的给定点的坐标，这会给你一个方程，你可以解出b。
And once you solve for b, you have full slope intercept form. So here's a practice problem. 暂停视频，然后我们再谈这个。好的，我们这里给出的是一个斜率。 我们有一个负的三分之五的斜率，在这条线上有一个点，负的七分之二，我们想知道这条线的x截距是多少。
Well, we have two very different ways of going about this. And I'm gonna show them both. 第一个是我所说的代数解。在这里，我们要写y=mx+b。 我们知道m，所以，我们要插入负五个半的斜率 我们还要插入点的坐标，x等于负2，y等于7，我们要插入。
This will give us an equation for b. Of course, negative five-thirds times negative 2 is positive ten-thirds, 然后b等于7减去三分之十。我们找到一个公分母，然后我们减去得到y截距。 So now we have the full slope intercept form, the full y=mx+b form. Now what we need to find is the x-intercept.
当然，求直线的x截距的方法是把y设为0，然后求x。 所以乘以3，我们得到的x截距是第十一个五分之一，或者2.2，是一个小数。 So that is a completely algebraic way to solve the problem, and that's a perfectly correct way to solve the problem.
That will get the answer. I also wanna show a completely different way of thinking about this problem, what I would call a graphical solution. So we know that it goes through the point -2,7. 好吧，让我们想想这个坡度，那个坡度是什么意思？如果我们想靠近y轴截距，因为它有一个负数， x截距，我们需要向右移动，因为它有一个负斜率。
所以我们向右移动，它就会向下移动。所以我们可以以3的速度移动，然后把高度降低负5。 That would be a slope of negative five-thirds. And so, that would mean we move the x goes up by 3. 所以它从负2变成正1。y值下降5。
So it goes from 7 down to 2. So now we're a lot closer to the x-axis. 现在让我们从视觉上考虑一下。我们有点1，2，在这条线上。 Then there's the x-intercept. And notice that little triangle that we make there, 那一定是一个斜面三角形。
So in other words, if we just look at the absolute values, 2 over b, that's a rise over run, 2 over b. 等于绝对值5除以3。好吧，这样我们就可以解决那个小b了。 We get b equals six-fifths. Now, let's think about this.
We want the x-intercept. Well, we know the distance from the origin to 1,0, that's a distance of one. And then the distance that I'm pulling b there is a distance of six-fifths. So we'll just add to the six-fifths. 为了简单起见，我将它写成1.2。1.2加1，就是2.2的x截距。
所以这是一个图形化的方法。现在这可能是一种陌生的方式。 That's typically not the way they teach you to approach things in school, but I would point out, if you can solve the problem algebraically and also solve it 以图形的方式使用比例，你就能真正理解它。它将极大地增强你对纵坐标几何的理解 to think about both approaches.
The test could give you two points and expect you to find the equation of the line. 两点唯一地决定了一条直线。如果你得到这两点，你会怎么做？ 当然，从这两点，我们总能找到一个斜率。一旦我们又有了斜坡我们该怎么办， we'll plug either one of those points, it doesn't matter, either one of those points for the x and the y in the equation, plug the slope in, we can solve for b, and then we have full slope intercept form.
一旦你有了坡度，你就可以形象地思考这个问题。 有个问题。暂停视频，然后我们再谈这个。 Okay, line J passes through the points -3,-2, 1,1 and 7,Q.
Find the value of Q. Well again, we have a couple of ways of thinking about this. 我要展示的第一个解是代数解。所以我们需要找到主要的坡度。 We find the slope from those two known points between -3,-2 and 1,1. We have a rise of 3 and a run of 4.
所以上升超过了跑步的四分之三。这就是坡度。 Now, we'll plug that into y=mx+b. We can plug either one of the points in. I'm gonna say that plugging 1,1 is gonna be much easier than plugging in -3, -2. I get to choose, I'm gonna choose the easier one.
我将插入1，1，然后解出b，得到b等于四分之一。 So then I get three-quarters x plus one-quarter that is the slope intercept form. So now I have the equation of the line itself. I have the equation of line J.
Well now, I'm just gonna plug in to find Q. I'm just gonna plug in 7 for x, Q for y, multiply everything out, and what I get is 22 over 4, which I can simplify that, that becomes eleven-halves. So that's one way to solve the problem. Again, that's a perfectly valid way to solve the problem.
顺便说一句，我也可以写十一个半5.5。代数解总是能给你一个答案。 但这是另一种思考的方式。我们要用图解法。 同样，你必须找到斜坡。坡度是有关直线的主要信息。
如果可以找到斜坡，总是找到斜坡，它总是帮助你。 All right, so now let's think about this. We go through the point -3, -2, we go through the point 1,1. 我们想离第7点更近一点，所以我要超过4，再上升3，等等 that will put me at 5,4 and then I'm reasonably close.
所以现在从5，4，当然，从5到7的距离是2。 So that little slip triangle there has a run of 2, we'll call the height of it h and it must be true that rise over run, h over 2 equals the slope of the line. H over 2 has to equal three-quarters.
So we set up this proportion. We solve for h. H等于三个半。三个半，我可以写成1.5。 So now we can figure out the height of Q. So the bottom of the slope triangle this diagram is at a height of 4.
And then to get up to Q, it's just 4+h or 4+1.5, which gives us 5.5 so 这是一种解决完全相同问题的图解方法。再说一遍，当你练习的时候，如果你能用代数和 也可以使用比例来解决它的图形你会有一个更深入的理解坐标几何。
If we are given a well-labeled graph, we may be able to read the slope and the y-intercept from the graph itself. 好主意。一旦给我们两点，我们就能找到一个斜坡。 一旦我们有了一个点和一个斜率，我们就可以把它们插入y=mx+b来解b。记住，你经常可以选择用代数或 从图形上看，考虑与坡度相关的比例。
再说一次，如果你能用代数解和图解解两种方法来解决这个问题，你就能非常好地理解这个主题。
Read full transcript
或者他们可以给你一张照片。如果给你一张这条线的图片，那么这两条线都应该相对容易阅读 the slope and the y-intercept from the picture, and these will allow you to create the equation. Here's a practice problem. Pause the video, and then we will talk about this.
好的，这条线穿过点A，30。A的价值是什么？ 很容易读懂这个图，y截距的值。y截距是1。 斜率，好吧，因为它穿过-2，我们有一个2的行程和一个1的上升，所以上升超过行程是1/2。
So the slope is one-half and the y-intercept is 1. So that means this has to be the equation of the line y equals one-half x+1, well, now we have an equation. So now we can plug in. We plug in A for x and 30 for y. Subtract 1, and then multiply by 2 to cancel the one-half.
我们得到A=58，这就是答案。这个测试可以给你一个斜率，然后是直线上的某个点。 坡度和单个点足以确定唯一的直线。您可以将坡度直接插入坡度截距窗体，y=mx+b。 So we would know m, initially not know the value of b but, because every point on the line must satisfy the equation of the line, 你可以在方程中插入x=y的给定点的坐标，这会给你一个方程，你可以解出b。
And once you solve for b, you have full slope intercept form. So here's a practice problem. 暂停视频，然后我们再谈这个。好的，我们这里给出的是一个斜率。 我们有一个负的三分之五的斜率，在这条线上有一个点，负的七分之二，我们想知道这条线的x截距是多少。
Well, we have two very different ways of going about this. And I'm gonna show them both. 第一个是我所说的代数解。在这里，我们要写y=mx+b。 我们知道m，所以，我们要插入负五个半的斜率 我们还要插入点的坐标，x等于负2，y等于7，我们要插入。
This will give us an equation for b. Of course, negative five-thirds times negative 2 is positive ten-thirds, 然后b等于7减去三分之十。我们找到一个公分母，然后我们减去得到y截距。 So now we have the full slope intercept form, the full y=mx+b form. Now what we need to find is the x-intercept.
当然，求直线的x截距的方法是把y设为0，然后求x。 所以乘以3，我们得到的x截距是第十一个五分之一，或者2.2，是一个小数。 So that is a completely algebraic way to solve the problem, and that's a perfectly correct way to solve the problem.
That will get the answer. I also wanna show a completely different way of thinking about this problem, what I would call a graphical solution. So we know that it goes through the point -2,7. 好吧，让我们想想这个坡度，那个坡度是什么意思？如果我们想靠近y轴截距，因为它有一个负数， x截距，我们需要向右移动，因为它有一个负斜率。
所以我们向右移动，它就会向下移动。所以我们可以以3的速度移动，然后把高度降低负5。 That would be a slope of negative five-thirds. And so, that would mean we move the x goes up by 3. 所以它从负2变成正1。y值下降5。
So it goes from 7 down to 2. So now we're a lot closer to the x-axis. 现在让我们从视觉上考虑一下。我们有点1，2，在这条线上。 Then there's the x-intercept. And notice that little triangle that we make there, 那一定是一个斜面三角形。
So in other words, if we just look at the absolute values, 2 over b, that's a rise over run, 2 over b. 等于绝对值5除以3。好吧，这样我们就可以解决那个小b了。 We get b equals six-fifths. Now, let's think about this.
We want the x-intercept. Well, we know the distance from the origin to 1,0, that's a distance of one. And then the distance that I'm pulling b there is a distance of six-fifths. So we'll just add to the six-fifths. 为了简单起见，我将它写成1.2。1.2加1，就是2.2的x截距。
所以这是一个图形化的方法。现在这可能是一种陌生的方式。 That's typically not the way they teach you to approach things in school, but I would point out, if you can solve the problem algebraically and also solve it 以图形的方式使用比例，你就能真正理解它。它将极大地增强你对纵坐标几何的理解 to think about both approaches.
The test could give you two points and expect you to find the equation of the line. 两点唯一地决定了一条直线。如果你得到这两点，你会怎么做？ 当然，从这两点，我们总能找到一个斜率。一旦我们又有了斜坡我们该怎么办， we'll plug either one of those points, it doesn't matter, either one of those points for the x and the y in the equation, plug the slope in, we can solve for b, and then we have full slope intercept form.
一旦你有了坡度，你就可以形象地思考这个问题。 有个问题。暂停视频，然后我们再谈这个。 Okay, line J passes through the points -3,-2, 1,1 and 7,Q.
Find the value of Q. Well again, we have a couple of ways of thinking about this. 我要展示的第一个解是代数解。所以我们需要找到主要的坡度。 We find the slope from those two known points between -3,-2 and 1,1. We have a rise of 3 and a run of 4.
所以上升超过了跑步的四分之三。这就是坡度。 Now, we'll plug that into y=mx+b. We can plug either one of the points in. I'm gonna say that plugging 1,1 is gonna be much easier than plugging in -3, -2. I get to choose, I'm gonna choose the easier one.
我将插入1，1，然后解出b，得到b等于四分之一。 So then I get three-quarters x plus one-quarter that is the slope intercept form. So now I have the equation of the line itself. I have the equation of line J.
Well now, I'm just gonna plug in to find Q. I'm just gonna plug in 7 for x, Q for y, multiply everything out, and what I get is 22 over 4, which I can simplify that, that becomes eleven-halves. So that's one way to solve the problem. Again, that's a perfectly valid way to solve the problem.
顺便说一句，我也可以写十一个半5.5。代数解总是能给你一个答案。 但这是另一种思考的方式。我们要用图解法。 同样，你必须找到斜坡。坡度是有关直线的主要信息。
如果可以找到斜坡，总是找到斜坡，它总是帮助你。 All right, so now let's think about this. We go through the point -3, -2, we go through the point 1,1. 我们想离第7点更近一点，所以我要超过4，再上升3，等等 that will put me at 5,4 and then I'm reasonably close.
所以现在从5，4，当然，从5到7的距离是2。 So that little slip triangle there has a run of 2, we'll call the height of it h and it must be true that rise over run, h over 2 equals the slope of the line. H over 2 has to equal three-quarters.
So we set up this proportion. We solve for h. H等于三个半。三个半，我可以写成1.5。 So now we can figure out the height of Q. So the bottom of the slope triangle this diagram is at a height of 4.
And then to get up to Q, it's just 4+h or 4+1.5, which gives us 5.5 so 这是一种解决完全相同问题的图解方法。再说一遍，当你练习的时候，如果你能用代数和 也可以使用比例来解决它的图形你会有一个更深入的理解坐标几何。
If we are given a well-labeled graph, we may be able to read the slope and the y-intercept from the graph itself. 好主意。一旦给我们两点，我们就能找到一个斜坡。 一旦我们有了一个点和一个斜率，我们就可以把它们插入y=mx+b来解b。记住，你经常可以选择用代数或 从图形上看，考虑与坡度相关的比例。
再说一次，如果你能用代数解和图解解两种方法来解决这个问题，你就能非常好地理解这个主题。
