Writing and Solving Linear Equations 05:39 minutes

Video Transcript

Transcript Writing and Solving Linear Equations

While trying to crack a case, Irving the investigator gets an anonymous message...by fax? What’s this? It looks like a city map with a bunch of locations marked. The message says that Irving can figure out the right meeting spot by calculating each triangle's angles from the provided hints. The fax tells him that figuring out the angles of each triangle will show him the RIGHT meeting point. Let's do this by helping Irving the investigator solve linear equations in one variable. First, let's translate the hints into algebraic language. In each hint, there are three unknown angles, which means it's a triangle. The first hint says that the angles that make up the first triangle are all the same. That's an equilateral triangle. Since they're unknown values and all the same we can just use the variable 'x' to represent each angle. The sum of the interior angles of a triangle add up to be 180 degrees. Therefore, when we sum up three of the same unknown values we can write it as a linear equation, 3x equals 180. Using the opposite operation of division, we can isolate our variable and see that 'x' equals 60. So each of the angles in the triangle is 60 degrees. Since all the angles in this triangle are the same measure, and Irving can't be in three places at once, this must not tell him the right meeting place. Irving can disregard this triangle. Let's take a look at the hint for the next triangle. For this triangle, the first measure is unknown. The second angle is three times the first plus 3 and 1/3. And the third angle is twice as large as the second angle. Though this problem contains mixed numbers, we can still solve it just like before by writing a linear expression. All we need to do is deal with it piece by piece. First, sum the variable expressions that represent the measurements of the three unknown angles. Then, set the expression equal to 180, the total interior angle measure of a triangle. Now we have a linear equation. To solve for 'x', first, use the distributive property by multiplying 2 by 3x and 2 by 3 and one third. Combine the like terms and then use opposite operations to isolate 'x'. Use the opposite operation of subtraction to undo the addition by 10, and then the opposite operation of division by ten on both sides of the equation to finish isolating the 'x', giving us 17. Next, we can substitute this back into our equations for each angle. Our first angle is, of course, 17 degrees. Plugging 17 into our equation for angle 2, we get 3 times 17 plus 3 and one third, 3 times 17 is 51. Then, adding our remaining numbers gives us 54 and one third degrees. Although the expression for angle 3 looks messy, remember, we already solved the part inside the parentheses and got 54 and one third! Just multiply that result by 2 and we have our answer! 108 and two-thirds degrees. This can't be it. This whole triangle's in the water. So unless Irving needs to buy scuba gear, this isn't it, either. Hmmm... we have one more triangle left. The second angle is twice as large as the first angle, so we can represent this by calling angle 1 'x' and expressing angle 2 in terms of 'x', giving us 2x. The third angle is three times the size of the first angle, so 3x. Like the first two triangles, this information can be written as a linear equation by summing the three, unknown angles. To solve for 'x', we first combine the like terms, 'x' plus 2x and 3x, giving us 6x. Next, we should divide both sides by 6 to isolate the 'x'. Therefore, 'x' equals 30. Now, let's evaluate each of the angles, using 30 for 'x'. This means the measure of angle 1 is 30 degrees, the measure of angle 2 is 60 degrees, and the measure of angle 3 is equal to 90 degrees. 90 degrees!? That's a right triangle. The hints have shown Irving that the triangle must be a 30-60-90 triangle. That must be it! The RIGHT meeting spot! Irving gets in his car and drives to the vertex of the right angle. What’s this? It was all just a scheme to have a surprise party!? Happy Birthday Irving!