Two buses leave new york at the same time traveling in opposite directions. one travels at an average speed of 79mph and the other at 72mph. in how many hours will they be 302 miles apart? let x=x= time of travel for the first bus and y=y= time of travel for the second bus. then the system that models the problem is {x=y79x+72y=302 solve the system by using the method of addition.

know idea how to do this

Step-by-step explanation:

It usually works well to use the method the problem statement tells you to use. Here, the system of equations is given as ...

For purposes of solution by addition, it works well to have all the variables on the same side of the equation. That means, we want to rewrite the first equation as ...

  x - y = 0

Since the signs of the y term are opposites in the two equations, we can multiply this rewritten equation by 72 to get an equation with the y-coefficient being the opposite of that of the original second equation:

  72x -72y = 0

Now, we add this equation with the second equation. The left side of the equal sign is the sum of all the left-side terms, and the right side of the equal sign is the sum of the right-side terms:

  (72x -72y) + (79x +72y) = (0) + (302)

  151x = 302 . . . . . simplify. The y-terms cancel, which is the point of using this method.

  302/151 = x = 2 . . . . divide both sides by the coefficient of x

Since the first equation tells us x=y, we know y=2 also. Both buses travel 2 hours to get to be 302 miles apart.

About "method of addition"

This is a method of solving a system of linear equations wherein equations are added for the purpose of eliminating one of the variables. (It is also called the method of elimination for that reason.) The basic idea is that one or both of the equations are multiplied by values that result in the coefficients of one of the variables being opposites. Then, when those (multiplied) equations are added, the opposite coefficients add to give 0 and that variable is eliminated.

Here, we chose to eliminate the y-variable by multiplying one of the equations by 72. We rearranged the equation first, but that is not strictly necessary. If you don't, then the addition would look like ...

  (72×(equation 1)) + (equation 2)

  (72x) + (79x +72y) = (72y) + (302)

  151x + 72y = 302 + 72y . . . . . simplify

Then 72y can be subtracted from both sides, eliminating the variable y from the equation.

About solving equations

All of algebra is based on the notion that an equation remains true and the values of the variables remain unchanged if you do the same thing to both sides of the equation. So, when we say "multiply by 72", we mean "multiply both sides of the equation by 72." When we say "divide by 151", we mean "divide both sides of the equation by 151."

In the example immediately above, we added 72x to one side of the equation 79x+72y=302, and we added 72y to the other side. We can do this because the equation 72x=72y means that 72x and 72y are the same thing. That is, we added the same thing to both sides of the equation 79x+72y=302.

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