QuestionThe sum of three numbers is 16 . The sum of twice the first number, 4 times the second number, and 5 times the third number is 54 . The difference between 7 times the first number and the second number is 44 . Find the three numbers. first number: $\square$ second number: $\square$ third number: $\square$

Studdy Solution

STEP 1

1. Let the three numbers be $x$ , $y$ , and $z$ .
2. The sum of the three numbers is 16.
3. The sum of twice the first number, 4 times the second number, and 5 times the third number is 54.
4. The difference between 7 times the first number and the second number is 44.
5. We need to find the values of $x$ , $y$ , and $z$ .

STEP 2

1. Set up the equations based on the problem statements.
2. Solve the system of equations to find the values of $x$ , $y$ , and $z$ .

STEP 3

Set up the equations based on the problem statements.
From the first statement, we have: $x + y + z = 16$
From the second statement, we have: $2x + 4y + 5z = 54$
From the third statement, we have: $7x - y = 44$

STEP 4

Solve the system of equations.
First, solve the third equation for $y$ : $y = 7x - 44$
Substitute $y = 7x - 44$ into the first equation: $x + (7x - 44) + z = 16$ $8x - 44 + z = 16$ $8x + z = 60$ $z = 60 - 8x$
Substitute $y = 7x - 44$ and $z = 60 - 8x$ into the second equation: $2x + 4(7x - 44) + 5(60 - 8x) = 54$ $2x + 28x - 176 + 300 - 40x = 54$ $-10x + 124 = 54$ $-10x = 54 - 124$ $-10x = -70$ $x = 7$

STEP 5

Now substitute $x = 7$ back into the equations for $y$ and $z$ .
For $y$ : $y = 7(7) - 44$ $y = 49 - 44$ $y = 5$
For $z$ : $z = 60 - 8(7)$ $z = 60 - 56$ $z = 4$
The three numbers are: first number: $\boxed{7}$ second number: $\boxed{5}$ third number: $\boxed{4}$

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