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Answer:

#P_C->(x_c,y_c)=(-6,+16)#

Full details shown. With practice you will be able to do this calculation type with very few lines.

Explanation:

#color(blue)("The meaning of 'collinear'")#

Lets split it into two parts

#color(brown)("co"->"together".# Think about the word cooperate
#color(white)("ddddddddddddd")#So this is 'together and operate.'
#color(white)("ddddddddddddd")#So you are doing some operation (activity)
#color(white)("ddddddddddddd")#together

#color(brown)("liniear".->color(white)("d")# In a strait line.

#color(brown)("collinear")-># co =together, linear =on a strait line.

#color(brown)("So all the points are on a strait line")#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Answering the question")#

#color(purple)("Determine the gradient (slope)")#

The gradient for part is the same as the gradient for all of it

Gradient (slope) #->("change in y")/("change in x")#

Set point #P_A->(x_a,y_a)=(2,8)#
Set point #P_B->(x_b,y_b)=(6,4)#
Set point #P_C->(x_c,y_c)=(-6,y_c)#

The gradient ALWAYS reads left to right on the x-axis (for standard form)

So we read from #P_A " to " P_B# thus the we have:

Set gradient# -> m="last "-" first" #

#color(white)("d")"gradient " -> m=color(white)("d")P_Bcolor(white)("d")-color(white)("d")P_A #

#color(white)("dddddddddddd")m=color(white)("d,")(y_b-y_a)/(x_b-x_a) #

#color(white)(dddddddddddddddddddd") (4-8)/(6-2) = -4/4=-1#

Negative 1 means that the slope (gradient) is downward as you read left to right. For 1 across there is 1 down.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(purple)("Determine the value of "y)#

Determined that #m=-1# so by direct comparison

#P_C-P_A =m = (y_c-y_a)/ (x_c-x_a) = -1#

#color(white)("dddddddddddd.d") (y_c-8)/ (-6-2) = -1#

#color(white)("dddddddddddddd.") (y_c-8)/ (-8) = -1#

Multiply both sides by (-8)

#color(white)("ddddddddddddddd.") y_c-8 = +8#

Add 8 to both sides

#color(white)("ddddddddddddddddd.")y_c color(white)("d")=+16#

Tony B

Answer:

#x=-1/2, -2/3#

Explanation:

We can solve this quadratic with the strategy factoring by grouping. Here, we will rewrite the #x# term as the sum of two terms, so we can split them up and factor. Here's what I mean:

#6x^2+color(blue)(7x)+2=0#

This is equivalent to the following:

#6x^2+color(blue)(3x+4x)+2=0#

Notice, I only rewrote #7x# as the sum of #3x# and #4x# so we can factor. You'll see why this is useful:

#color(red)(6x^2+3x)+color(orange)(4x+2)=0#

We can factor a #3x# out of the red expression, and a #2# out of the orange expression. We get:

#color(red)(3x(2x+1))+color(orange)(2(2x+1))=0#

Since #3x# and #2# are being multiplied by the same term (#2x+1#), we can rewrite this equation as:

#(3x+2)(2x+1)=0#

We now set both factors equal to zero to get:

#3x+2=0#

#=>3x=-2#

#color(blue)(=>x=-2/3)#

#2x+1=0#

#=>2x=-1#

#color(blue)(=>x=-1/2)#

Our factors are in blue. Hope this helps!

Answer:

The distance is #8#

Explanation:

The easiest way is to use the distance formula, which is kinda tricky:
#d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2#

That looks really complex, but if you take it slowly, I'll try and help you through it.

So let's call #(-6,7)# Point 1. Since points are given in the form #(x,y)# we can deduct that

#-6 = x_1# and #7 = y_1#

Let's call #(-1,1)# Point 2. So:

#-1 = x_2# and #1=y_2#

Let's plug these numbers into the distance formula:
#d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2#
#d = sqrt((-1 - -6)^2 + (1 - 7)^2#
#d = sqrt((5)^2 + (-6)^2#
#d = sqrt(25 + 36#
#d = sqrt61#
#d ~~ 7.8# rounded to the nearest whole unit is #8#

This is quite the hard subject, and is best taught by someone who knows how to explain well! This is a really good video about the distance formula:
Khan Academy distance formula video

Answer:

Answer:-#" "color(red)(1/n)# and #color(red)(1/m#

Explanation:

  • If the roots of an equation #color(red)(ax^2+bx+c=0# is #color(blue)(alpha,beta#, then we can write as per rule that

    • #color(red)(alpha+beta)=-b/a#
    • #color(red)(alpha cdot beta)=c/a#
  • As per given condition, we can write that

    • #color(red)(m+n)=-b/a#
    • #color(red)(m cdot n)=c/a#
  • We will determine some values now for further use.

    • #color(red)(-b/c)=(-b/a)/(c/a)=(m+n)/(m cdot n)#
    • #color(red)(a/c)=1/(m cdot n#
  • If the roots of the equation #color(red)(cx^2+bx+a=0# is #color(blue)(alpha,beta#, then,

    • #color(red)(alpha+beta)=-b/c=(m+n)/(m cdot n)" "...(1)#
    • #color(red)(alpha cdot beta)=a/c=1/(m cdot n#
  • #color(red)(alpha-beta)#

#=sqrt((alpha+beta)^2-4 cdot alpha cdot beta)#

#=sqrt(((m+n)/(m cdot n))^2-4 /(m cdot n))#

#=(m-n)/(m cdot n)" "...(2)#

  • From #(1)" & " (2)#,
    • #color(red)(alpha)=((m+n)/(m cdot n)+(m-n)/(m cdot n))/2=1/n#
    • #color(red)(beta)=((m+n)/(m cdot n)-(m-n)/(m cdot n))/2=1/m#

Hope this helps....
Thank you...

Answer:

Kate has 50 pens.

Explanation:

If Jimmy gives 7 pens to Kate, that means Kate got 7 pens from Jimmy. Let the number of pens Jimmy and Kate have be #P_j# and #P_k# respectively.

From the question, we can deduce that

#P_j=P_k*128%rarr#equation 1

#P_j-7=P_k+7rarr#equation 2

From equation 1,

#P_j=1.28P_k#

From equation 2,

#P_j=P_k+14#

Since #P_j=P_j#,

#1.28P_k=P_k+14#

Hence,

#0.28P_k=14#

Solve,

#P_k=50#

Answer:

#-102#

Explanation:

#-50-4((-10-4(-3+1)^2)/(-2))#

To simplify this, we will use PEMDAS, shown here:
www.coolmath.com

This is a common method to simplify expressions, and you can remember it using:
Please Excuse My Dear Aunt Sandy

We also start from inside to outside.

First thing we do is simplify the parenthesis:
#-50-4((-10-4(-2)^2)/(-2))#

Exponent:
#-50-4((-10-4(4))/(-2))#

Multiplication:
#-50-4((-10-16)/(-2))#

Subtract on numerator:
#-50-4((-26)/(-2))#

Division:
#-50-4(13)#

Multiplication:
#-50-52#

And lastly, subtraction:
#-102#

Hope this helps!

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