In the coordinate plane, a line has a slope of 1. If points (w, 5) and (2, v) are on this line, then w + v =

A. 1 B. 2 C. 5 D. 7 E. 10

(D) Given the slope and two points on the line, you can use the slope formula to put the information together. Recall that the slope is the change in y divided by the change in x.

1 = (v – 5) / (2 – w)

2 – w = v – 5

7 = v + w

Or, put another way: w + v = 7, choice (D).

Another way to solve this question is to write an equation of a line. Since we know that the slope is 1. The equation is following: y = x + l, where l is some unknown constant. Points (w, 5) and (2, v) are on this line so we can plug them in:

5 = w + l v = 2 + l

In order to get rid of unknown l we deduct one from the other:

5 – v = w – 2 + l – l 5 – v = w – 2 7 = w + v

The answer is (D).

Could you, please, explain once again how we get those equations?

Post subject: Re: math: x,y-plane, complicated question.

Posted: Mon May 10, 2010 11:27 am

Joined: Fri Apr 09, 2010 2:11 pm Posts: 453

Let us go over this question once again, step-by-step. The key for solving this question is analyzing relations in between given facts on one side and understanding how it appears on the x,y-plane on the other.

First of all, let us see what we have:

1. Point (2, v). At first, we must think of it as v is some fixed yet unknown number. In any case of v, (2, v) is some point that lies on the line x = 2.

or

2. Point (w, 5). At first, we must think of it as w is some fixed yet unknown number. In any case of w, (w, 5) is some point that lies on the line y = 5.

or

3. A line has a slope of 1. So the equation of this line is y = x + l, where l is some fixed yet unknown number.

or

or

We see that solution statement doesn't exactly "fix" points (w, 5) and (2, v). But they sum w + v must be the same for any w, and v that satisfy the question statement. Therefore we can choose points (w, 5) = (2, 5) and (2, v) = (2, 5).

We can see that it matches the question statement. So in this case w = 2 and v = 5, so w + v = 7. Since we assume that w + v must be the same for any w, and v that satisfy the question statement, the answer is 7, choice (D).

This problem can also be solved analytically. First, let us think of what we have:

A formula for a slope of the line that contains points (w, 5) and (2, v) is

If you don't remember the formula for a slope you can use basic form for a line instead. y = bx + l

The slope b is 1. So in our case y = x + l

We know that line contains both points so we can plug those in:

5 = w + l v = 2 + l

Subtract one from another

5 – v = w + l – 2 – l 5 – v = w – 2 7 = w + v

The answer is choice (D).

We solved the problem in 3 different ways, each one of these is sufficient on its own. The key for solving this question is analyzing relations in between given facts on one side and understanding how it appears on the x,y-plane on the other.

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