Question 476566
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Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. 
Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?)
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        The solution in the post by @Theo is incorrect.

        His equations are not adequate to the problem, and the answer is wrong.

        I came to bring a correct solution.



<pre>
Let x be the number of the 2-points problems and 
let y be the number of the 4-points problems.


Write equations as you read the problem

    x +  y =  35,      (1)

   2x + 4y = 100.      (2)


It can be solved by different methods: by the Substitution method or by the Elimination method.
I will solve it here using the Elimination method.


Multiply equation (1) by 2 (both sides).  Keep equation (2) as is.
You will get

   2x + 2y =  70,      (1')

   2x + 4y = 100.      (2')


Now subtract equation (') from equation (2').
The terms with '2x' will annihilate each other, and you will get

    4y - 2y = 100 - 70,

       2y   =     30,

        y   =     30/2 = 15.


So, y = 15.  To find x, substitute this value y = 15 in equation (1).

You will get

    x + 15 = 35  --->  x = 35 - 15 = 20.


<U>ANSWER</U>.  There are  20  2-points problems and  15  4-points problems.


<U>CHECK</U>.  Checking is a necessary part of the solution.

        Substitute the found values x and y into equation (1) to make sure 
        that the left side value coincides with the right side value.

        Substitute the found values x and y into equation (2) to make sure 
        that the left side value coincides with the right side value.
</pre>

At this point, the solution is completed in full.



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There is no any need to recheck the @josgarithmetic' solution, 
since he simply re-wrote it from my solution, with variations.