Writing Linear Equations 06:13 minutes

Video Transcript

Transcript Writing Linear Equations

Back in the day when the west was still a frontier, W.J. Palmer was in his office planning to build a railroad track between two cities. He wanted to make the track a straight line so that it'd be the shortest route possible. To figure out how to do so, he uses linear equations. Let's take a look at the map. We can turn it into a coordinate system.

We know the coordinates of the two ciities that will be connected by the train tracks. Palm Valley is at point P (2, 3) and Wildwood Crest is at point W (12, 8). Because the track will be a straight line between the two points, we can write a linear equation to represent this line. In order to write the equation in slope-intercept form, y = mx +b, we need to find the slope, m, and the y-interecept, b.

Calculate slope

To calculate slope, we'll use the formula m= y2 minus y1 over x2 minus x1. So let's plug in our points. We'll call point (2, 3) (x1, y1) and point (12, 8) (x2, y2). When plugged into the formula, we get 8 minus 3 over 12 minus 2... , ...which is 5 over 10. This reduces to 1/2, so the slope is 1/2. Now that we have m, we can put that into our formula.

We also need to choose a point to plug in for x and y. Let's use the coordinates of Palm Valley, (2, 3). Now we have 3 equals 1/2 times 2 plus b. 1/2 times 2 is 1. To find b, you need to subtract 1 from both sides of the equals sign. B equals 2. Now that you know m is 1/2 and b is 2, you know the equation of this line is y= 1/2 x + 2 W. J. Palmer's plans are looking great.

New track

But suddenly he realizes that the railroad will go through a forest that's home to the famous gold rush bugs. Since W.J. Palmer is a big bug enthusiast, he wants to save the bugs and decides to make a Plan B. Palmer decides to start from a different point in Palm Valley. The point is (2, 4). Palmer also wants this track to be parallel to his original track. Let's write the equation for the line of the new track.

Parallel lines have the same slope, so we can keep m equal to 1/2 again. Let's use our slope-intercept formula again and plug in (2, 4) to find our y-intercept, b, for the new route. So we have 4 equals 1/2 times 2 plus b. 1/2 times 2 is 1. So 4 equals 1 plus b. Subtract 1 from both sides. B equals 3.

If we plug in m and b, the new equation is y = 1/2 x + 3. The new railroad track is built and everything looks great! But W. J. Palmer found out that his little bug friends have to cross the tracks to get to their favorite field of flowers. Many bugs have died on their way across the tracks. So Palmer decides to build a bridge perpendicular to the tracks so that the bugs can cross the tracks safely.

Find the equation

The bugs are located in the forest at point B (6, 5). In order to build the bridge, we need to find the equation of the line that is perpendicular to the track and goes through (6, 5). There are two ways to find the slope of a perpendicular line. The first way is to make sure that the product of the slopes of two perpendicular lines equals -1.

The perpendicular line

We know the first slope is 1/2. So if we plug in 1/2 for m1, we can solve for m2 by multiplying both sides by 2. -1 multiplied by 1/2 is -2 -1 divided by 1/2 is -2. The second way to find the slope of a perpendicular line is to remember that perpendicular slopes are negative reciprocals of each other. Negative means that we will need to multiply by -1. Reciprocal means that we will flip the numerator and denominator. -2 over 1 simplifies to -2.

So we got the slope of the line that's perpendicular to the track, but we still need to find b. If we use slope-intercept form again, y = mx +b, we'll plug in -2 for m and use the point (6, 5) for x and y. So the equation will be 5 = -2 times 6 plus b -2 times 6 is -12. Since we have a negative 12, we will need to add 12 to both sides.

5 plus 12 is 17, so b is 17. If we plug in the values for m and b, the equation for the bridge is y = -2 x + 17. And this, my children, was the story of W. J. Palmer--a true friend to all beings on 6 legs.

1 comment
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    Funny and informative! It's great! Keep up the good work!

    Posted by Plano Kellmeyer, 8 months ago