solve with system of equation with 3 variables

Question

The sum of a student's three scores is 198. If the first is 16 points more than the second, and the sum of the first two is 18 more then twice the third, what was the first score?

Answer 1

When solving a word problem identify what your unknowns are.  From reading the problem we have three unknowns

 

let x represent the first score,  y represent the second score,  z represent the third score

 

Next write the sentences as mathematical statements using your chosen variables.

 

The sum of the three scores (means add them all up) is 198 (is mean =) so one equation is

 

x + y + z = 198

 

Some students prefer to replace the words with there chosen variable and mathematical symbols when writing the equations.  For example instead of reading "the first" they read x.

 

If the first x is = 16 more than the second y + 16.  A common mistake is that students write x = 16 + y which is not correct.  The sentence states 16 more than.  More than what? The second.  So we are adding 16 to y.  Our second equation is

 

x = 16 + y

 

The sum of the first two x + y is = 18 more than twice the third 2z + 18.  Our last equation is 

 

x + y = 2z + 18

 

The system of equations is

 

eq1:  x + y + z = 198

eq2:  x = 16 + y

eq3:  x + y = 2z + 18

 

To solve the system you can use elimination, substitution or matrices.  I am not sure what method you have covered. I will leave the rest for you to do.  If you need more help just comment me.  Hint start with equations 1 and 3.

Answer 2

Let the scores be represented by x, y, and z.  From the stated problem then, the sum of all scores is 198.

 

1. x + y+ z = 198 is the first equation

 

Then, the first score (x) is 16 points more than the second score (y).  So,

 

2. x = y + 16

 

Finally, the result of adding the first two scores is 18 points more that 2 times the third score (z).  

 

3. x + y = 2z + 18

 

Now, the key to doing the rest is to use substitution and then elimination to solve for the one of the 3 variables.

 

Step 1:

 

Eq 2 tells us what x is so we substitute that x (y +16) for the x in Eqs 1 and 3.

 

For Eq1: x + y + z = 198, replace x with y + 16 to get:

 

y + 16 + y + z = 198; simplify to get 2y + z = 198 - 16 => 2y + z = 182 (a)

 

For Eq3: x + y = 2z + 18, again, replace x with y +16 to get:

 

y + 16 + y = 2z + 18; simplify to get 2y + 16 = 2z + 18 => 2y - 2z = 2. (b)

 

Now note we have two equations and two variables so we can use elimination to remove one of the variables.  If we subtract equation b from a, we get:

 

  2y + z = 182

- (2y - 2z) = 2

 

3z = 180

 

Now, divide both sides by 3:

 

3z/3 = 180/3 => z = 60

 

To solve for y, substitute 60 for z in one of the two equations a or b.

 

If a, then:

 

2y + 60 = 182 => 2y = 122 => y = 61

 

Finally, substitute y into equation 2 to find x:

 

x = 61 + 16 => x = 76

 

 

Answer 3

x=first score    y=second score   z=third score

 

sum of scores is 198          

x + y + z = 198                                               

 

               first is 16 more than second 

                 x = 16 + y  

                          

                       sum of first two is 18 more than twice the third

                                     x + y = 18 + 2z

 

 

 

Substitute for x by using these two equations...

x + y + z = 198               x = (16 + y)

(16 + y) + y + z = 198

 

2y + z + 16 = 198           now solve for z

             -16    -16

2y + z = 182             

-2y          -2y

       z = 182 - 2y

 

Now we have  z = (182 - 2y)     and     x = (16 + y) and we can substitute for x and z in the third equation

 

     x       + y = 18 +  2z

(16 + y)  + y = 18 +  2(182 - 2y)  now solve for y

        16 + 2y = 18 + 364 - 4y

        16 + 2y =       382  - 4y

              +4y                 +4y

       16  + 6y =  382

      -16              -16

               6y =  366

                6        6

                y = 61              

 

now use y to solve for x in this equation: 

x = 16 + y

x = 16 + 61

x = 77

 

now use x and y to solve for z in this equation:

x  +   y +  z = 198

77 + 61 + z = 198

      138 + z = 198

     -138        -138

              z =  60