Question

Ralph wants to find the solution to a system of equations using y = x - 4 and y = x2 + 2x - 46. Molly says that Ralph can solve x2 + x - 42 = 0 to find the x coordinates of the solutions to his system. Explain and demonstrate why Molly is correct.

Answer 1

use substitution to solve this system of equations.
y= x-4
use that to replace y in the 2nd equation
y= x^2 + 2x - 46 becoms
x- 4 = x^2 +2x -46

simplify that last equation. what do you get ?

Answer 2

x-4=x^2+2x-46

Answer 3

by simplify, I mean "move" x and -4 to the other side of the equation
in other words, add -x to both sides, and add +4 to both sides. what do you get ?

Answer 4

0=x^2+x-42

Answer 5

right?

Answer 6

yes. notice that it matches

" Molly says that Ralph can solve x2 + x - 42 = 0 to find the x coordinates of the solutions to his system."

Answer 7

so the answer would be that molly just used substitution to simplify the two equations

Answer 8

yes, that is ok for the explain part. But they say

Explain and demonstrate why Molly is correct.

I think they want you to "demonstrate" by actually solving it, and showing that the answers work in both equations.

Answer 9

thanks! can u help me with another word problem?

Answer 10

Did you solve this one ?

Answer 11

yes

Answer 12

what did you get ?

Answer 13

i just showed the substitution process and explained why she did it

Answer 14

I think they also want you to solve the system of equations.

Answer 15

i got x^3=42

Answer 16

you start with
x^2+x-42 =0

you can factor this. Look at the last number -42
the - means the factors will have different signs (one +, the other -)
next, list the pairs of numbers that when you multiply them , you get 42
1,42
2,24
3,14
6,7

now look at the number in from the +1x (remember +x is the same as +1x)
the + means the larger factor is +. (Because of the first step we know the other smaller factor is -)
the factors add up to +1

look at our list, with the bigger number + and the smaller number -
-1+24= 23
-2+24= 22
-3+14= 11
-6+7= 1

the last pair give +1... those are our guys. we write
(x-6)(x+7)=0

Answer 17

**now look at the number in front of the +1x

Answer 18

you should practice factoring, because that is how you solve these problems
the last step is to say
a*b=0 means either a =0 or b=0
we use that idea with
(x-6)(x+7)=0
to say x-6=0 or x+7=0
x-6=0 means x=6
x+7= 0 (add -7 to both sides) --> x= -7
we got two answers for x. to find y, use y= x-4 (one of your equations)
we could use the other equation but it is more complicated. Choose the easy one)

Answer 19

-2 and -11?

Answer 20

also why two answers for x?

Answer 21

@phi

Answer 22

yes, you get two answers. (6,2) and (-7,-11)

you can "demonstrate" these are the solutions by using them in both of the original equations to show that they work.

The reason you have two solutions is because the second equation is a parabola, it curves.
Here is a picture.

Answer 23

okay thank you that really helped me! i understand that now. i have one problem left that i don't understand can u help?

Answer 24

Betty and Andrea have been solving systems of equations with one polynomial function of degree two and one linear function. Betty says there must always be two solutions, and Andrea says there will no solution. Using complete sentences, explain how Betty can be correct, how Andrea can be correct, and how they both can be wrong.

Answer 25

Did you look at the picture I posted up above. can you imagine a line that misses the parabola? Or what about a line that just touches the parabola once (makes a tangent) ?

Answer 26

i think i understand this now thanks!