the difference of two numbers is 7. the larger number is 9 less than 3 times the smaller. find the numbers
a = b + 7
a = 3b - 9
Substitute b + 7 for a in the second equation and solve for b. With that value in the first equation, find a. To check, put both values into the second equation.
To find the two numbers, we will set up a system of equations based on the given information.
Let's assume the smaller number is represented by 'x' and the larger number by 'y'.
From the first statement, "the difference of two numbers is 7," we can write the equation:
y - x = 7 ... Equation 1
From the second statement, "the larger number is 9 less than 3 times the smaller," we can write the equation:
y = 3x - 9 ... Equation 2
Now we can solve the system of equations by substitution or elimination method.
- Substitution method:
From Equation 2, we can see that y = 3x - 9. We can substitute this expression for 'y' into Equation 1:
(3x - 9) - x = 7
Simplifying this equation:
3x - 9 - x = 7
2x - 9 = 7
2x = 7 + 9
2x = 16
x = 16/2
x = 8
To find the value of 'y', we can substitute the value of 'x' back into Equation 2:
y = 3(8) - 9
y = 24 - 9
y = 15
Therefore, the smaller number is 8, and the larger number is 15.
- Check:
Let's verify if our solution satisfies both equations:
1) From Equation 1: y - x = 7
15 - 8 = 7 (Correct)
2) From Equation 2: y = 3x - 9
15 = 3(8) - 9
15 = 24 - 9
15 = 15 (Correct)
So, our solution is correct. The smaller number is 8, and the larger number is 15.