x - y = 3
8x = 10y
8x = 10y
From the given conditions, we can write the following system of equations:
Equation 1: x - y = 3 (the difference between the two numbers is 3)
Equation 2: 8x = 10y (eight times the larger number is 10 times the smaller number)
Now, we will solve the system of equations using the substitution method.
Step 1: Solve Equation 1 for x
x - y = 3
x = y + 3
Step 2: Substitute the value of x from Step 1 into Equation 2
8x = 10y
8(y + 3) = 10y
Step 3: Distribute the 8 on the left side of the equation
8y + 24 = 10y
Step 4: Subtract 8y from both sides
24 = 2y
Step 5: Divide both sides of the equation by 2
12 = y
Step 6: Substitute the value of y into Equation 1 to find x
x - y = 3
x - 12 = 3
Step 7: Add 12 to both sides
x = 15
The solution to the system of equations is x = 15 and y = 12.
Given that the difference between the two numbers is 3, we can write the equation:
x - y = 3 ... (Equation 1)
We are also told that eight times the larger number (x) is equal to ten times the smaller number (y). We can write this equation as:
8x = 10y ... (Equation 2)
Now we have a system of two equations:
Equation 1: x - y = 3
Equation 2: 8x = 10y
To solve this system of equations using the substitution method, we can solve Equation 1 for x, and then substitute that expression into Equation 2.
Let's solve Equation 1 for x:
x - y = 3
Adding y to both sides gives:
x = y + 3
Now we can substitute this expression for x in Equation 2:
8(y + 3) = 10y
Let's simplify Equation 2:
8y + 24 = 10y
Subtracting 8y from both sides:
24 = 2y
Dividing both sides by 2:
12 = y
Now, we can substitute the value of y (which is 12) into Equation 1 to find x:
x - 12 = 3
Adding 12 to both sides:
x = 15
So, the two numbers are x = 15 and y = 12.