Systems of Equations Word Problem

Question

When the smaller of two consecutive integers is subtracted from three times the larger, the difference is 63. What are the numbers? Write and solve a system of 2 equations

Answer 1

Let x and y be the two integers, x the smallest and y the largest.

the largest integer y has to be 1 more than x since its a consecutive integer: so 1st equation: y = x + 1

"When the smaller of two consecutive integers is subtracted from three times the larger, the difference is 63": so 2nd equation 3(y) - x = 63

We know y = x + 1, so we substitute it into the second equation:

3(y) - x = 63 becomes

3(x + 1) - x = 63

Using the distributive property to distribute the "3" into the "x" and the "positive 1"

3(x) + 3(1) = 63

3x + 3 = 63

Now to isolate (solve) for x:

3x + 3 = 63

- 3 = -3

3x = 60

Divide by 3 on both sides of the equal sign:

3x = 60

3 3

x = 30: This smallest integer in 30. Thus, the largest integer is 31. y = 31

Answer 2

Let X and Y be the 2 variables, and Y be the largest.

We are told they are consecutive, so: Y = X + 1

Next, we are told that the smaller is subtracted from 3 times the larger, to get 63:

So: 3Y - X = 63

Rearrange the first equation to get Y - X = 1, then subtract this from the 2nd equation to get:

2Y = 62

So Y = 31 and X = 30

Check this by plugging the numbers into the second equation.

Answer 3

We will assign two variables, a and b, where a is the smaller of the two consecutive integers, and b is the larger of the two integers.

We can then create a system of equations:

b = a + 1 because b is one greater than a, the smaller of the two consecutive integers

3b - a = 63 because three times the larger integer minus the smaller is 63

We can substitute b in the second equation for a + 1:

3(a+1) - a = 63

so ...

3a + 3 - a = 63

2a + 3 = 63

2a = 60

a = 30

Now that we know that a is 30, we can substitute a back into the first equation to solve for b. We know it should be 31 because the two integers are consecutive, but this is a good way to check your work.

b = a + 1

b = 30 + 1

b = 31

Now we can double check again using the instructions of the word problem.

3 * 31 - 30 = 63

93 - 30 = 63

63 = 63

The two integers are 30 and 31