Question

The sum of five times a number and twice a second number is 18, but the difference of three times the first number and seven times the second number is -22. Find the two numbers using the elimination method.


A. -2 and -4
B. -2 and 4
C. 2 and -4
D. 2 and 4

Answer 1

Do you understand how to translate the sentences into a system of equations? :)

Answer 2

no

Answer 3

The sum of 5 times `a number`
and
twice `a second number`
is
18

Translates to:
5x
+
2y
=
18

So I have 5 times something, I'm calling that something x.
and 2 times something else, calling that something else y.

Answer 4

5x+2y=18

Answer 5

The difference of three times `the first number`
and
seven times `the second number`
is
-22

How bout this second sentence, any ideas? :)

Answer 6

3x+7y=-22 that's probably wrong

Answer 7

Mmm very close! ^^
When they use the word `difference`, they want us to subtract the two terms, not add them.

Answer 8

So, something like that, yah?
3x-7y=-22

Answer 9

oh okay but im still confused on the question because they don't have 2 or 4 in their sentence

Answer 10

So we've established a system of equations:\[\Large\rm 5x+2y=18\]\[\Large\rm 3x-7y=-22\]Two equations, two unknowns, so we can solve this.
We want to solve this using either `substitution` or `elimination`.
It looks like `elimination` will work a lot better for this particular problem.
You are familiar with that method? :)

Answer 11

For this problem, let's try to match up our y's.
We want the same coefficient on both y's.

Answer 12

So we need the LCM (Least common multiple) of 2 and 7.

Answer 13

Which just ends up being 14 in this case. (2 x 7).

So we want to multiply the first equation (the whole thing) by 7,
while we'll multiply the second equation by 2,
to get our 14s

Answer 14

Multiplying everything by 7 in the first equation gives us:
\[\Large\rm 7(5x+2y=18)\]\[\Large\rm 35x+14y=126\]

And for our second equation:\[\Large\rm 2(3x-7y=-22)\]\[\Large\rm 6x-14y=-88\]I multiplied everything by 2.

Answer 15

okay so I tried to do that to the numbers in the question I did 18 - -22 and got -4

Answer 16

Well we had to do this work getting our y's matched up before we could add or subtract the equations.
\[\Large\rm 35x+\color{orangered}{14y}=126\]\[\Large\rm 6x-\color{orangered}{14y}=-88\]

Answer 17

If we ADD these equations now, the y's will cancel out.

Answer 18

35 + 6 gives us 41 x's.

Answer 19

oh ok