Nine times the smaller of two numbers plus six times the larger number is -15. Nine times the smaller number minus six times the larger number is -75 .

Answer 1

There are two unknown numbers here, so assign variables to them.

Let the smaller number be x and the larger number be y.

"Nine times the smaller number" translates to 9x.

"Six times the larger number" translates to 6y.

Therefore, "Nine times the smaller number plus six times the larger number" translates to 9x+6y.

But what does that equal? -15.

So our first equation is 9x+6y=-15.

Simplifying this equation by dividing both sides by the GCF=3 gives:

Equation 1: 3x+2y=-5

Similarly, "Nine times the smaller number minus six times the bigger number equals -15" translates to the equation

9x-6y=-75.

Dividing both sides by the GCF=3 gives

Equation 2: 3x-2y=-25

To summarize, translating from verbal description to equations and simplifying we get the system of equations:

Equation 1: 3x+2y=-5

Equation 2: 3x-2y=-25.

We can use the addition/elimination method to solve this system by adding Equation 1+Equation 2 to get Equation 3:6x=-30

Solving this gives x=-5.

Plugging x=-5 into Equation 1 gives:

3(-5)+2y=-5

-15+2y=-5

2y=10

y=5.

So our solution to the system is x=-5 and y=5, or sometimes written as an ordered pair (-5,5).

To check your answer plug the x and y into the original equations as a good habit.

9x+6y=-15

9(-5)+6(5)=-15

-45+30=-15

-15=-15. Check.

9x-6y=-75

9(-5)-6(5)=-75

-45-30=-75

-75=-75. Check.

To summarize the final answer, the smaller number is x=-5 and the larger number is y=5.