# Systems of linear equations: courses of ships, speeds needed for boats to collide

#### elledo

##### New member
How do I write an equation that tells me how many times faster Boat A is going compared to Boat B?

Learning more about relative speed: Relative speed is the speed that one object is moving in relation to another. (In this case, it should really be called relative velocity!) In order for the two boats to collide, they will have to be going at particular speeds in relation to each other in order to collide. If one boat is farther away, it will have to be going faster than the other boat in order for them to collide. Perhaps one boat might have to go twice as fast if it is twice the distance away from the collision point. You will use the d = rt equation to determine how fast boat A will have to go, compared to boat B. Follow the steps below in order to complete the final sentence below. Show the work for each step.

(a) Plot the exact collision point on the picture and label it with its coordinates.

(b) Measure from the bow (front) of each boat to the collision point and, using the given scale, determine the distance each boat is from the collision point.

(c) Create a d = rAt and a d = rBt equation for each boat. Fill in the distance from part (b) for each boat, and then rearrange each equation solving for t.

(d) Keeping in mind that the time that each boat is traveling must be the same in order to collide, set the two equations from part (c) equal to each other. Solve for rA.

(e) Finish this part by typing up and completing the sentence "Boat A must be going ____ times as fast as boat B in order for them to collide." (The number in the blank should be exact -- in other words, a fraction.)

Note: Boat A is 70 miles away. Boat B is 90 miles away.

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#### elledo

##### New member
If anyone could help, that would be greatly appreciated. Thanks

#### elledo

##### New member
On C I got 70=RaT and 90=RbT.
D I got 70-Ra=t and 90-Rb=t.
So if they equal each other, I have 70-Ra=90-Rb.
I solved for Ra and got Ra=-20+Rb
How would I connect these or write a different equation in order to get how many times as fast Ra is going Rb?

#### Otis

##### Senior Member
On C I got 70=RaT and 90=RbT.
D I got 70-Ra=t and 90-Rb=t …
When we use variable names that contain more than one letter (like Ra and Rb), it helps to show multiplications, explicitly, as I did above by adding the red asterisks (multiplication symbols).

You made a mistake, solving the equations in part (c). You cannot subtract Ra because its multiplied by T. (If that expression were Ra + T, then you could subtract Ra.)

In order to solve each equation in part (c), you need to use division, not subtraction.

#### Otis

##### Senior Member
sorry I didn't make this clear, but A and B are subscripts …
I understood your meaning for Ra and Rb; I offered only a suggestion about notation.

Where is your corrected work for part C?

#### elledo

##### New member
ohhhhhhhhhh I just realized thank you so much