The greater of two numbers is 1 more than twice the smaller. Three times the greater exceeds 5 times the smaller by 10. Find the numbers.
The second of three numbers is 1 less than the first. The third number is 5 less than the second. If the first number is twice as large as the third, find the three numbers.
These are two separate problems and we are supposed to write an equation for each and solve it for the numbers, but only using one variable!! HELP!
L-1=2S
4L-10=5s
I will critique your work on the second.
but we are only supposed to use 1 variable??
The second of three numbers is 1 less than the first. The third number is 5 less than the second. If the first number is twice as large as the third, find the three numbers.
What is 2 plus 2?
Sure, I can help you solve both problems using just one variable in each equation.
Problem 1:
Let's assume the smaller number is x and the greater number is y.
The first statement tells us that y = 1 + 2x. (The greater number is 1 more than twice the smaller number.)
The second statement tells us that 3y - 5x = 10. (Three times the greater number exceeds 5 times the smaller number by 10.)
To solve this problem using only one variable, we can substitute y in terms of x from the first equation into the second equation. So we have:
3(1 + 2x) - 5x = 10.
Now, let's simplify and solve for x:
3 + 6x - 5x = 10,
x = 7.
Now that we know the value of x, we can substitute it back into the first equation to find y:
y = 1 + 2x = 1 + 2(7) = 15.
Therefore, the smaller number is 7 and the greater number is 15.
Problem 2:
Let's assume the first number is x, the second number is y, and the third number is z.
The first statement tells us that y = x - 1. (The second number is 1 less than the first number.)
The second statement tells us that z = y - 5. (The third number is 5 less than the second number.)
The third statement tells us that x = 2z. (The first number is twice as large as the third number.)
To solve this problem using only one variable, we can substitute y and z in terms of x from the first two equations into the third equation. So we have:
x = 2(y - 5).
Now, let's simplify and solve for x:
x = 2y - 10.
Next, substitute y in terms of x from the first equation into the second equation:
z = (x - 1) - 5,
z = x - 6.
Now, we can substitute z in terms of x from the second equation into the third equation:
x = 2(x - 6),
x = 2x - 12.
Simplifying this equation gives us:
x = 12.
Therefore, the first number is 12.
Using the first equation, we can find the second number:
y = 12 - 1 = 11.
Using the second equation, we can find the third number:
z = 11 - 5 = 6.
Therefore, the three numbers are 12, 11, and 6.