Need i have attatched a screenshot! distance, rate, and time problem using linear equations

Step-by-step explanation:

speed = distance / time

The two trains are traveling toward each other at a combined speed of ...

... (590 km)/(2 h) = 295 km/h

The speed of the faster train will be half this value plus half their difference in speed:

... faster rate = (295 km/h + 21 km/h)/2 = 158 km/h

The slower rate is 21 km/h less:

... slower rate = 158 km/h - 21 km/h = 137 km/h

Comment on Sum and Difference Problems

After computing the combined speed, you have two numbers: the total speed of the two trains (their sum) and their speed difference. Then, the problem is to determine the two numbers that have that sum and difference.

One way to think about this is to consider the sum of the two numbers plotted on a number line, with a mark at half that value (the average of the two numbers). When the mark is centered at the halfway point, the distance above the mark is equal to the distance below the mark.

Moving the mark 1 unit upward takes 1 from the smaller number and adds it to the larger number, making the larger number 2 units more than the smaller number. You can see that if you want the larger number 21 units more than the smaller, you need to make the larger number have a value that is half of 21 more than the average. (Then the smaller number will be half of 21 less than the average, and their difference will be 21.)

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If you want to use equations, you can write them as ...

Adding these two equations together cancels the b term and gives

... 2a = (sum + difference)

And dividing by 2 gives

... a = (sum + difference)/2

This solution is generic and applies to all "sum and difference" problems.