One integer is 8 more than another. Their product is 153. Find the integers

Question

One integer is 8 more than another. The product is 153. Find the integers 

Answer 1

Note to preface this problem: it is important with the language of "more than" and "less than" to write the equation/expression in the correct order. For example: 8 more than y = y + 8 (even though y + 8 and 8 + y  yield the same answer). Another example is x less than 10 = 10 - x, which yields the opposite value of x -10. So, one must be careful about the order when translating from word to mathematical expressions. Here are examples to remember less, less than, and is less than (and more, more than, and is more than):

 

8 less 5 = 8 -5 

8 less than 5 = 5 - 8

8 is less than 5: 8 < 5 (which is not true)

 

8 more 5 = 8 + 5

8 more than 5 = 5 + 8

8 is more than 5: 8 > 5

 

Back to the question asked:

 

Let x be the first integer and y be the second integer.

One integer is 8 more than another: x = y +8 (equation 1)

Their product is 153: x * y = 153 (equation 2)

 

This problem involves two equations with two variables. It is traditionally referred to as a system of linear equations with two variables.

 

One possible way of solving this question is by substitution: we take the equivalent of x in equation 1 and replace it in the equation 2 to get:

 

(y + 8) * y  = 153

y² + 8y = 153

y² + 8y - 153 = 0. You can now solve by Quadratic Formula or by factoring. 

 

^{By factoring, we get: (y -  9)*(y+17) = 0}

 

By the Zero Product Property, we can say that either (y - 9) = 0 OR (y+17) = 0

 

y -9 = 0 gives y = 9, which gives x = y + 8 = 9 + 8 = 17

 

OR 

 

y + 17 = 0 gives y = -17, which gives x = y + 8 = -17 + 8 = -9

 

I hope this helps,

Eli

 

Answer 2

Lets call "One integer" "y" and the other one "x"

we can re-write the first sentence as:

y=8+x

 

We can re-write the second sentence as:

y+x=153

 

This is two systems of equations, so solve this, use the substitution method:

 

if y=8+x

then we can replace the "y" in the second equation with what "y" equals:

(8+x)+x=153

 

Solve for x:

8+2x=153

-8         -8

2x=145

divide both sides by 2

x=72.5

 

Now that you know x, plug x into one of the two original equations to find y:

72.5+y=153

-72.5     -72.5

y=80.5

 

Therefore your solution is the first integer is 80.5 and the other integer is 72.5.

Answer 3

Let X be the smaller

 

 X ( X+8) = 153

 

 X^2 + 8X - 153 = 0

 

(X       -      9  )(X     +       17    ) = 0

 

 

X = 9  or   X = -17

 

so there are two possible answers since negative numbers are allowed (not restricted)

 

9 x 17 = 153 and diifer by 8

 

-17 x -9 = 153 and also differ by 8