Hi, for the question, "the sum of two numbers is equal to a third number(x+y=z). The sum of twice the first, three times the second, and twice the third is 7(2x+3y+2x=7). The third number is 3 more than the first(3+z=x), have I set the equations right? I got different answers than my book that still satisfy all the equations.

Question

Hi, for the question, "the sum of two numbers is equal to a third number(x+y=z). The sum of twice the first, three times the second, and twice the third is 7(2x+3y+2x=7). The third number is 3 more than the first(3+z=x), have I set the equations right? I got different answers than my book that still satisfy all the equations.

Answer 1

looks like you did this ..

first no ---- x
second no -- y
third no ----x+y

now... " The sum of twice the first, three times the second, and twice the third is 7"
2x + 3y + 2(x+y) = 7
2x + 3y + 2x + 2y = 7
4x + 5y = 7 (#1)

also ... "The third number is 3 more than the first"
x+y = x+3
y = 3

back in #1
4x + 5(3) = 7
4x + 15 = 7
4x = -8
x = -2

first no -- -2
2nd no -- 3
3rd no -- 1

Check:
is the third the sum of the first two ? YEAH
is twice the first plus three times the second and twice the third is 7 ?
(-4 + 9 + 2 = 7 YEAH)
is the third 3 more than the first ?
1 is indeed 3 more than -2
YEAHHH

Answer 2

Dear Reiny! U r a life-saver! Is there a short cut to how can I set the question or change word problems into math one? It seems that this is my problem! Also, why did my way worked as well with different answers? My answers satisfied all equations as well. Many thanks.

Answer 3

To determine if you have set the equations correctly, let's analyze each equation:

1. "The sum of two numbers is equal to a third number: x + y = z"
This equation represents the relationship that the sum of the first number (x) and the second number (y) is equal to the third number (z).

2. "The sum of twice the first, three times the second, and twice the third is 7: 2x + 3y + 2z = 7"
This equation expresses that when we take twice the first number, three times the second number, and twice the third number, the total sum is 7.

3. "The third number is 3 more than the first: z = x + 3"
This equation states that the third number (z) is equal to the first number (x) plus 3.

Based on the information provided, it appears that you have correctly set up the equations. However, it's worth noting that these equations form a system of linear equations, and it's possible to have multiple solutions that satisfy all the equations. So, even if your answers differ from the ones in the book, they can still be correct as long as they satisfy all the equations.

To find the solution(s) to this system of equations, you can use various methods such as substitution, elimination, or matrix operations.