The difference between two numbers is 3. Eight times the larger number is 10 times the smaller number. Write a system of equations describing the given conditions. Then solve the system by the substitution method and find the two numbers.

Question

The difference between two numbers is 3. Eight times the larger number is 10 times the smaller number. Write a system of equations describing the given conditions. Then solve the system by the substitution method and find the two numbers.

Answer 1

x - y = 3

8x = 10y

Answer 2

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

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.

Answer 3

To solve this problem, we can start by defining two variables to represent the two unknown numbers. Let's use the variables "x" and "y", where "x" represents the larger number, and "y" represents the smaller number.

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.