the larger of two numbers is 7 less than twice the smaller number. if the sum of the number is 47, find both numbers.
Let's break down the information given step-by-step:
Step 1: Assign variables
Let's assign variables to the two numbers. We can call the larger number L and the smaller number S.
Step 2: Translate the given information into equations
The problem states that the larger number (L) is 7 less than twice the smaller number (2S).
This can be expressed as: L = 2S - 7.
The problem also states that the sum of the two numbers (L + S) is 47.
This can be expressed as: L + S = 47.
Step 3: Solve the system of equations
We have the following system of equations:
L = 2S - 7
L + S = 47
We can solve this system of equations using substitution or elimination.
Substitution:
Substitute the value of L from the first equation into the second equation:
(2S - 7) + S = 47
3S - 7 = 47
Add 7 to both sides:
3S = 54
Divide both sides by 3:
S = 18
Now substitute the value of S into one of the equations to find L:
L = 2(18) - 7
L = 36 - 7
L = 29
So the two numbers are S = 18 and L = 29.
To solve this problem, let's assign variables to represent the two numbers:
Let's call the larger number "L" and the smaller number "S."
According to the problem, the larger number (L) is 7 less than twice the smaller number (S). This can be written as:
L = 2S - 7 ----(Equation 1)
Also, the sum of the two numbers is 47. This can be expressed as:
L + S = 47 ----(Equation 2)
We have two equations with two unknowns (L and S), which allows us to solve for both numbers. Let's use substitution or elimination to find the values of L and S.
Method 1: Substitution
We already have Equation 1 (L = 2S - 7). Substitute this expression for L into Equation 2:
(2S - 7) + S = 47
Combine like terms:
3S - 7 = 47
Add 7 to both sides of the equation:
3S = 54
Divide both sides by 3:
S = 18
Now that we have the value of S, we can substitute it back into Equation 1 to solve for L:
L = 2(18) - 7
L = 36 - 7
L = 29
So, the larger number (L) is 29, and the smaller number (S) is 18.
Method 2: Elimination
Rearrange Equation 2 (L + S = 47) to solve for L:
L = 47 - S
Now substitute this expression for L into Equation 1:
(47 - S) = 2S - 7
Simplify the equation:
47 + 7 = 2S + S
54 = 3S
Divide both sides by 3:
S = 18
Again, substitute this value of S back into Equation 1 to find L:
L = 2(18) - 7
L = 36 - 7
L = 29
Hence, the larger number (L) is 29, and the smaller number (S) is 18.
Both methods yield the same result. Therefore, the larger number is 29, and the smaller number is 18.
x + 2x - 7 = 47
3x + 54
x = ?