The larger of two numbers in 8 times the smaller. The difference between them is 42. Find the numbers. Write the equations. Find the numbers

smaller number -- x

larger number --- 8x

8x - x = 42
7x = 42
x = 6

numbers are 6 and 48

the larger of two numbers is three times the smaller number, and their difference id 34. find the two numbers.

Larger=L, Small=s, L=3S , L-S=34, 3s-s=34, 2s=34, S=34/2, S=17, check L=3(17)=51, 51-17=34, it checks

To solve this problem, we can set up a system of equations.

Let's assume that the smaller number is x, and the larger number is y.

According to the problem, we know that the larger number (y) is 8 times the smaller number (x). Therefore, we can write the equation:

y = 8x ----> Equation 1

We also know that the difference between the two numbers is 42. This means that when we subtract the smaller number (x) from the larger number (y), we get 42:

y - x = 42 ----> Equation 2

Now, we have a system of two equations with two variables. To solve this system, we can use the method of substitution or elimination.

Using the substitution method:

Substitute the value of y from Equation 1 into Equation 2:

8x - x = 42

Simplifying, we get:

7x = 42

Divide both sides of the equation by 7:

x = 6

Now that we have found the value of x, we can substitute it back into Equation 1 to find the value of y:

y = 8(6)
y = 48

Therefore, the two numbers are x = 6 and y = 48.

To summarize:
The smaller number is 6, and the larger number is 48.