We have two unknowns: x and y. Now, we have to formulate 2 equations. The first would come from the use of the given ratio:
We use the distance formula to find the distance between coordinates:
3/4 = √[(x-4)²+(y-1)²] / √[(4-12)²+(1-5)²]
√[(x-4)²+(y-1)²] = 3√5
(x-4)²+(y-1)² = 45
x² - 8x + 16 + y² - 2y + 1 = 45
x² - 8x + y² - 2y = 28 --> eqn 1
The second equation must come from the equation of a line:
y = mx +b
m = (5-1)/(12-4) = 1/2
Substitute y=5 and x=12 for point (12,5)
5 = (1/2)(12) + b
b = -1
So, the second equation is
y = 1/2x -1 or x = 2 + 2y --> eqn 2
Solving the equations simultaneously:
(2 + 2y)² - 8(2 + 2y) + y² - 2y = 28
Solving for y,
y = -2
x = 2+2(-2) = -2
Therefore, the coordinates of point A is (-2,-2).
The slope can sometimes be called the gradient, and the equation for the gradient is (y2 - y1)/(x2 - x1). So therefore, you'd do: (-8 - 7)/(4 - -5) which is (-15)/9) which is -1 2/3 or -1.6 (recurring), which is your answer. I hope this helps! Let me know if I've confused you :)
the average rate of change is just the slope from (0,f(0)) to (1,f(1))
find f(0) and f(1)
slope from (0,3) to (1,3) is rise/run=(3-3)/(0-1)=0/-1
the average rate of change from x=0 to x=1 for f(x) is 0
The answer is: " 91 " .
___________________________________________________
→ " B = 91 " .
__________________________________________________
Explanation:
__________________________________________________
Given:
__________________________________________________
" A + B = 180 " ;
"A = -2x + 115 " ; ↔ A = 115 − 2x ;
"B = - 6x + 169 " ; ↔ B = 169 − 6x ;
_____________________________________________________
METHOD 1)
_____________________________________________________
Solve for "x" ; and then plug the solved value for "x" into the expression given for "B" ; to solve for "B"
_____________________________________________________
(115 − 2x) + (169 − 6x) =
115 − 2x + 169 − 6x = ?
→ Combine the "like terms" ; as follows:
+ 115 + 169 = + 284 ;
− 2x − 6x = − 8x ;
_________________________________________________________
And rewrite as:
" − 8x + 284 " ;
_________________________________________________________
→ " - 8x + 284 = 180 " ;
Subtract: "284" from each side of the equation:
→ " - 8x + 284 − 284 = 180 − 284 " ;
to get:
→ " -8x = -104 ;
Divide EACH SIDE of the equation by "-8 " ;
to isolate "x" on one side of the equation; & to solve for "x" ;
→ -8x / -8 = -104/-8 ;
→ x = 13
__________________________________________________________
Now, to find the value of "B" :
__________________________________________________________
"B = - 6x + 169 " ; ↔ B = 169 − 6x ;
↔ B = 169 − 6x ;
= 169 − 6(13) ; ===========> Plug in our "solved value, "13", for "x" ;
= 169 − (78) ;
= 91 ;
B = " 91 " .
__________________________________________________
The answer is: " 91 " .
____________________________________________________
→ " B = 91 " .
____________________________________________________
Now; let us check our answer:
____________________________________________________
→ A + B = 180 ;
____________________________________________________
Plug in our "solved answer" ; which is "91", for "B" ; as follows:
________________________________________________________
→ A + 91 = ? 180? ;
↔ A = ? 180 − 91 ? ;
→ A = ? -89 ? Yes!
________________________________________________________
→ " A = -2x + 115 " ; ↔ A = 115 − 2x ;
Plug in our solved value for "x"; which is: "13" ;
" A = 115 − 2x " ;
→ A = ? 115 − 2(13) ? ;
→ A = ? 115 − (26) ? ;
→ A = ? 29 ? Yes!
_________________________________________________
METHOD 2)
_________________________________________________
Given:
__________________________________________________
" A + B = 180 " ;
"A = -2x + 115 " ; ↔ A = 115 − 2x ;
"B = - 6x + 169 " ; ↔ B = 169 − 6x ;
→ Solve for the value of "B" :
_______________________________________________________
A + B = 180 ;
→ B = 180 − A ;
→ B = 180 − (115 − 2x) ;
→ B = 180 − 1(115 − 2x) ; ==========> {Note the "implied value of "1" } ;
__________________________________________________________
Note the "distributive property" of multiplication:__________________________________________________ a(b + c) = ab + ac ; <u><em>AND</em></u>:
a(b − c) = ab − ac .________________________________________________________
Let us examine the following part of the problem:
________________________________________________________
→ " − 1(115 − 2x) " ;
________________________________________________________
→ " − 1(115 − 2x) " = (-1 * 115) − (-1 * 2x) ;
= -115 − (-2x) ;
= -115 + 2x ;
________________________________________________________
So we can bring down the: " {"B = 180 " ...}" portion ;
→and rewrite:
_____________________________________________________
→ B = 180 − 115 + 2x ;
→ B = 65 + 2x ;
_____________________________________________________
Now; given: "B = - 6x + 169 " ; ↔ B = 169 − 6x ;
→ " B = 169 − 6x = 65 + 2x " ;
______________________________________________________
→ " 169 − 6x = 65 + 2x "
Subtract "65" from each side of the equation; & Subtract "2x" from each side of the equation:
→ 169 − 6x − 65 − 2x = 65 + 2x − 65 − 2x ;
to get:
→ " - 8x + 104 = 0 " ;
Subtract "104" from each side of the equation:
→ " - 8x + 104 − 104 = 0 − 104 " ;
to get:
→ " - 8x = - 104 ;
Divide each side of the equation by "-8" ;
to isolate "x" on one side of the equation; & to solve for "x" ;
→ -8x / -8 = -104 / -8 ;
to get:
→ x = 13 ;
______________________________________________________
Now, let us solve for: " B " ; → {for which this very question/problem asks!} ;
→ B = 65 + 2x ;
Plug in our solved value, " 13 ", for "x" ;
→ B = 65 + 2(13) ;
= 65 + (26) ;
→ B = " 91 " .
_______________________________________________________
Also, check our answer:
_______________________________________________________
Given: "B = - 6x + 169 " ; ↔ B = 169 − 6x = 91 ;
When "x = 13 " ; does: " B = 91 " ?
→ Plug in our "solved value" of " 13 " for "x" ;
→ to see if: "B = 91" ; (when "x = 13") ;
→ B = 169 − 6x ;
= 169 − 6(13) ;
= 169 − (78)______________________________________________________
→ B = " 91 " .
______________________________________________________
Answer:
n^2 - n - 12 = (n+3)(n-4)
Step-by-step explanation:
3 times - 4 = -12 ( two numbers when multiplied = -12, when added=-1)
3 + -4 = -1
