eight times a number plus five times another number is -13. The sum of the two numbers is 1

Answer 1

Let's have x be the first number and y be the second number. So in this equation, we have 8x + 5y = -13. We also know that x + y = 1. In this case, we need to have the second equation be equal to either x or y and substitute it back into the original equation. For this, I will solve for y.

x + y = 1

y = 1 - x

Now that we've solved for y, we will put 1 - x back into the first equation, and then simplify and solve for x, as follows:

8x + 5(1 - x) = -13 Initial equation

8x + 5 - 5x = -13 Factor 5 into (1 - x)

3x + 5 = -13 Subtract 5x from 8x

3x = -18 Subtract 5 from both sides

x = -6 Divide by 3 from both sides

Now that we've solved for x, we can now choose to solve the original equation, 8x + 5y = -13 or x + y = 1 to solve for y. To make it easier for ourselves, let's choose the second equation. Since we already know what x is, we can plug that into the equation as follows:

x + y = 1 Initial equation

-6 + y = 1 Plugged 6 into x since we found out that x = -6

y = 7 Added 6 to both sides

Now, we found that x = -6 and y = 7. The sum of these two numbers do, in fact, equal to 1, because if you put -6 and 7 back into the second equation, you do find that they equal 1. Also, for an added benefit, we can put the two numbers back into the original equation and check to make sure that it does equal -13. For this, however, we will replace -13 with z.

8x + 5y = -13 Initial equation

8x + 5y = z Replaced -13 with z, as stated above

8(-6) + 5(7) = z Since x = -6 and y = 7, we replace them in the equation

-48 + 35 = z Simplify

-13 = z Add both numbers

Now, we have fully confirmed that x = -6 and y = 7.