Slope-intercept form of the equation for the line?
Answer:
[tex]\large\boxed{y=-\dfrac{3}{10}x+\dfrac{1}{2}}[/tex]
Step-by-step explanation:
The slope-intercept form of an equation of a line:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
We have the points from the graph (-5, 2) and (5, -1).
Substitute:
[tex]m=\dfrac{-1-2}{5-(-5)}=\dfrac{-3}{10}=-\dfrac{3}{10}[/tex]
We have the equation in form:
[tex]y=-\dfrac{3}{10}x+b[/tex]
Put the coordinates of the point (5, -1) to the equation:
[tex]-1=-\dfrac{3}{10}(5)+b[/tex]
[tex]-1=-\dfrac{3}{2}+b[/tex]
[tex]-\dfrac{2}{2}=-\dfrac{3}{2}+b[/tex] add 3/2 to both sides
[tex]\dfrac{1}{2}=b\to b=\dfrac{1}{2}[/tex]
What is the vertex of the graph of g(x) = |x – 8| + 6? A (6, 8) B (8, 6) C (6, –8) D (–8, 6)
its (8,6)
to get your x, you set what's in the absolute value to 0
so
x-8=0
then subtract 8 on both sides to get
x=8
then your y is just the number to the right of the absolute value so
y=6
Answer: Option B
(8, 6)
Step-by-step explanation:
By definition, for an absolute value function of the form
[tex]f (x) = | x-h | + k[/tex]
the vertex of f(x) will always be at the point
(h, k)
In this case we have the function of value ansoluto:
[tex]g(x) = |x - 8| + 6[/tex]
Therefore in this case
[tex]h=8\\k=6[/tex]
Finally the vertex of the function g(x) is: (8, 6)
The answer is the option B
Use substitution to write an equivalent quadratic equation.
(3x + 2)2 + 7 (3x + 2) – 8 = 0
-
u2 + 74 - 8 = 0, where u = (3x + 2)2
u2 + 7u - 8 = 0, where u = 3x + 2
u2 + 70 - 8 = 0, where u = 7(3x + 2)
lu² +4-8=0
What’s the answer?
For this case we can make a change of variable, to obtain a quadratic equation of the form:
[tex]ax ^ 2 + bx + c = 0[/tex]
Making the change:[tex]u = 3x + 2[/tex]
Substituting the change we have:
[tex]u ^ 2 + 7u-8 = 0[/tex]
Thus, the correct option is:[tex]u ^ 2 + 7u-8 = 0[/tex]where [tex]u = 3x + 2[/tex]
Answer:
Option B
Answer:
second option: [tex]u^{2}+7u-8=0[/tex]
Step-by-step explanation:
We have the equation given:
[tex](3x+2)^{2}+7(3x+2)-8=0[/tex]
We can replace the variable in the quardatic equation.
So,
[tex]Putting\\u=3x+2[/tex]
Putting u in place of 3x+2 will give us:
[tex](u)^{2}+7(u)-8=0[/tex]
So the answer is:
[tex]u^{2}+7u-8=0[/tex]
So, the second option is correct ..
Consider this equation: 2x + 2 = 11 - x
Now consider the equation written as: 3x + 2 = 11
Which of the following is the correct property of equality that justifies rewriting the equation?
Answer: Addition property of equality
Step-by-step explanation: You added the x to the other side, which is clearly using addition. Hope this help!
what is the value of x?
Answer:
A. 68°
Step-by-step explanation:
sum of angles inside a triangle = 180°
75 + 37 + x = 180
x = 180 - 75 - 37
= 68°
The sum of angles inside a triangle is 180°. so option is A. 68°.
What is the angle sum property?The angle sum property of a triangle states that the sum of the interior angles of a triangle is 180 degrees.
Given angles are; 75 and 37.
The sum of angles inside a triangle = 180°
75 + 37 + x = 180
x = 180 - 75 - 37
x = 68°
Learn more about the triangles;
https://brainly.com/question/2773823
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Billy is helping to make pizzas for a school function. He's made 25 pizzas so far. His principal asked him to make at least 30 pizzas but no more than 75. Solve the compound inequality and interpret the solution.
30 ≤ x + 25 ≤ 75
Answer: Number of pizzas would be less than 5 but not more than 50.
Step-by-step explanation:
Since we have given that
Number of pizzas so far = 25
His principal asked him to make at least 30 pizzas but no more than 75.
According to question, we have
30 ≤ x + 25 ≤ 75
First we subtract 25 from both the sides:
[tex]30-25\leq x \leq 75-25\\\\=5\leq x\leq 50[/tex]
Hence, number of pizzas would be less than 5 but not more than 50.
The solution to the compound inequality is 5 x 50.
The compound inequality given is 30 x 25 75, where x represents the number of additional pizzas Billy needs to make to satisfy the principal's request. To solve for x, we need to isolate x in the inequality.
First, we subtract 25 from all parts of the compound inequality to shift the 25 pizzas already made to the other side of the inequality. This gives us:
30 - 25 x + 25 - 25 75 - 25
Simplifying the inequality, we get:
5 x 50
This means that Billy needs to make at least 5 more pizzas to reach the minimum requirement of 30 pizzas (since 25 + 5 = 30) and no more than 50 additional pizzas to not exceed the maximum allowed number of 75 pizzas (since 25 + 50 = 75).
Therefore, the number of additional pizzas Billy should make is any integer value between 5 and 50, inclusive. This ensures that the total number of pizzas made will be within the principal's specified range of 30 to 75 pizzas."
PLEASE HELP PLEASE
Tricia has the same number of nickels and dimes in her pocket. The coins total $3.60. How many of each coin does she have?
A) 12 nickels and 12 dimes
B) 24 nickels and 24 dimes
C) 36 nickels and 36 dimes
D) 48 nickels and 48 dimes
Answer:
B
Step-by-step explanation:
Find out the total value of each option and see which one gives a total of $3.60
For A, 12 nickels + 12 dimes = 12 x $0.05 + 12 x $0.10 = $1.80 ≠ $3.60
For B, 24 nickels + 24 dimes = 24 x $0.05 + 24 x $0.10 = $3.60 (correct)
For C, 36 nickels + 36 dimes = 36 x $0.05 + 36 x $0.10 = $5.40 ≠ $3.60
For D, 48 nickels + 48 dimes = 48 x $0.05 + 48 x $0.10 = $7.20 ≠ $3.60
Answer:
For my opinion, the best option it B.
The given line passes through the points (-4,-3) and (4,1) what is the equation in point-slop
e form, of the line that is perpendicular to the given line and passes through the point (-4,3)?
let's bear in mind that perpendicular lines have negative reciprocal slopes...... hmmmm what's the slope of the given line anyway?
[tex]\bf (\stackrel{x_1}{-4}~,~\stackrel{y_1}{-3})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{1}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{1-(-3)}{4-(-4)}\implies \cfrac{1+3}{4+4}\implies \cfrac{4}{8}\implies \cfrac{1}{2} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{\cfrac{1}{2}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{2}{1}}\qquad \stackrel{negative~reciprocal}{-\cfrac{2}{1}\implies -2}}[/tex]
so, we're really looking for a line whose slope is -2 and runs through (-4 , 3)
[tex]\bf (\stackrel{x_1}{-4}~,~\stackrel{y_1}{-3})~\hspace{10em} slope = m\implies -2 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-3)=-2[x-(-4)] \implies y+3=-2(x+4) \\\\\\ y+3=-2x-8\implies y=-2x-11[/tex]
A license plate consists of three letters followed by
three digits. How many license plates are possible if
no letter may be repeated?
Answer: 15,600,000
Step-by-step explanation:
26 x 25 x 24 =15,600 combinations of letter with no repeats
10 x 10 x 10 = 1000 with repeating numbers
15,600 x 1000 = 15,600,000
The total number of possible license plates is 26*25*24*10*9*8.
The number of possible license plates with no repeated letters can be calculated by multiplying the number of choices for each position. Since no letter can be repeated, the choices for the first position would be 26 letters, then 25, and then 24 for the three positions. For the digits, there are 10 choices for each position.
So, the total number of possible license plates would be 26*25*24*10*9*8.
Which graph represents a function with direct variation?
Answer:
The graph that represent direct variation in the attached figure
Step-by-step explanation:
A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]
In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin
The graph that represent direct variation in the attached figure
Answer:
option d
Step-by-step explanation:
Please help me! This is is rational function and I don’t know how to/ don’t remember how do this! How would I find and write the equation for it?
An answer is
[tex]\displaystyle f\left(x\right)=\frac{\left(x+1\right)^3}{\left(x+2\right)^2\left(x-1\right)}[/tex]
Explanation:
Template:
[tex]\displaystyle f(x) = a \cdot \frac{(\cdots) \cdots (\cdots)}{( \cdots )\cdots( \cdots )}[/tex]
There is a nonzero horizontal asymptote which is the line y = 1. This means two things: (1) the numerator and degree of the rational function have the same degree, and (2) the ratio of the leading coefficients for the numerator and denominator is 1.
The only x-intercept is at x = -1, and around that x-intercept it looks like a cubic graph, a transformed graph of [tex]y = x^3[/tex]; that is, the zero looks like it has a multiplicty of 3. So we should probably put [tex](x+1)^3[/tex] in the numerator.
We want the constant to be a = 1 because the ratio of the leading coefficients for the numerator and denominator is 1. If a was different than 1, then the horizontal asymptote would not be y = 1.
So right now, the function should look something like
[tex]\displaystyle f(x) = \frac{(x+1)^3}{( \cdots )\cdots( \cdots )}.[/tex]
Observe that there are vertical asymptotes at x = -2 and x = 1. So we need the factors [tex](x+2)(x-1)[/tex] in the denominator. But clearly those two alone is just a degree-2 polynomial.
We want the numerator and denominator to have the same degree. Our numerator already has degree 3; we would therefore want to put an exponent of 2 on one of those factors so that the degree of the denominator is also 3.
A look at how the function behaves near the vertical asympotes gives us a clue.
Observe for x = -2,
as x approaches x = -2 from the left, the function rises up in the positive y-direction, andas x approaches x = -2 from the right, the function rises up.Observe for x = 1,
as x approaches x = 1 from the left, the function goes down into the negative y-direction, andas x approaches x = 1 from the right, the function rises up into the positive y-direction.We should probably put the exponent of 2 on the [tex](x+2)[/tex] factor. This should help preserve the function's sign to the left and right of x = -2 since squaring any real number always results in a positive result.
So now the function looks something like
[tex]\displaystyle f(x) = \frac{(x+1)^3}{(x+2 )^2(x-1)}.[/tex]
If you look at the graph, we see that [tex]f(-3) = 2[/tex]. Sure enough
[tex]\displaystyle f(-3) = \frac{(-3+1)^3}{(-3+2 )^2(-3-1)} = \frac{-8}{(1)(-4)} = 2.[/tex]
And checking the y-intercept, f(0),
[tex]\displaystyle f(0) = \frac{(0+1)^3}{(0+2 )^2(0-1)} = \frac{1}{4(-1)} = -1/4 = -0.25.[/tex]
and checking one more point, f(2),
[tex]\displaystyle f(2) = \frac{(2+1)^3}{(2+2 )^2(2-1)} = \frac{27}{(16)(1)} \approx 1.7[/tex]
So this function does seem to match up with the graph. You could try more test points to verify.
======
If you're extra paranoid, you can test the general sign of the graph. That is, evaluate f at one point inside each of the key intervals; it should match up with where the graph is. The intervals are divided up by the factors:
x < -2. Pick a point in here and see if the value is positive, because the graph shows f is positive for all x in this interval. We've already tested this: f(-3) = 2 is positive.-2 < x < -1. Pick a point in here and see if the value is positive, because the graph shows f is positive for all x in this interval.-1 < x < 1. Pick a point here and see if the value is negative, because the graph shows f is negative for all x in this interval. Already tested since f(0) = -0.25 is negative.x > 1. See if f is positive in this interval. Already tested since f(2) = 27/16 is positive.So we need to see if -2 < x < -1 matches up with the graph. We can pick -1.5 as the test point, then
[tex]\displaystyle f(-1.5) = \frac{\left(-1.5+1\right)^3}{\left(-1.5+2\right)^2\left(-1.5-1\right)} = \frac{(-0.5)^3}{(0.5)^2(-2.5)} \\= (-0.5)^3 \cdot \frac{1}{(0.5)^2} \cdot \frac{1}{-2.5}[/tex]
We don't care about the exact value, just the sign of the result.
Since [tex](-0.5)^3[/tex] is negative, [tex](0.5)^2[/tex] is positive, and [tex](-2.5)[/tex] is negative, we really have a negative times a positive times a negative. Doing the first two multiplications first, (-) * (+) = (-) so we are left with a negative times a negative, which is positive. Therefore, f(-1.5) is positive.
Given f(x) = 17-X^2what is the average rate of change in f(x) over the interval [1, 5]?
Answer:
-6
Step-by-step explanation:
The formula for finding the rate of change is:
[tex]Rate\ of\ change=\frac{f(b)-f(a)}{b-a}[/tex]
The interval is [1,5]
So, a = 1 and b=5
[tex]f(1) = 17 - (1)^2\\ =17-1\\=16\\f(5) = 17-(5)^2\\=17-25\\=-8[/tex]
Putting in the values
[tex]Rate\ of\ change = \frac{f(5)-f(1)}{5-1} \\=\frac{-8-16}{5-1}\\= \frac{-24}{4}\\ =-6[/tex]
The average rate of change is -6 ..
Answer:
-6
Step-by-step explanation:
Determine the range of the function graphed above.
A. [4,∞)
B. [-4,0]
C. [0,4]
D. (-∞,4]
Answer:
D. (-∞,4]
Step-by-step explanation:
The range is the y values
The lowest y values is negative infinity
The highest y values is 4
( - inf, 4]
We use the parentheses since we cannot get to negative infinity, the bracket since we reach 4
Which expresión is equivalente to (4g3 h2 k4)3
Answer: [tex]\bold{64g^9h^6k^{12}}[/tex]
Step-by-step explanation:
[tex]\text{Apply the product rule:} (a^b)^c=a^{b*c}\\\\(4g^3h^2k^4)^3\\\\=4^{(1*3)}g^{(3*3)}h^{(2*3)}k^{(4*3)}\\\\=4^3g^9h^6k^{12}\\\\=64g^9h^6k^{12}[/tex]
The lines shown below are parallel. If the green line has a slope of --, what is
the slope of the red line?
Answer: the answer would be -1/11
Step-by-step explanation: the reason being the slope of the green is a negative slope because it is going down ward not upwards therefore it is negative and because the the lines are parallel they are equal so the slope of both lines are -1/11
f(-5) if f(x)=|x+1|
Answer:
The answer is 6.
Step-by-step explanation:
Plug in: f(-5)=|-5+1|
Because this an absolute problem -5 is positive within |x|
so therefore f(-5)= |6|
If two angels are congruent, then the sides opposite those angles are congruent. True or false.
Answer:
The statement is True
Step-by-step explanation:
The Isosceles Triangle Theorem states that;
If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. The converse of this statement is;
if two angles are congruent, then sides opposite those angles are congruent.
Answer:
True.
Step-by-step explanation:
To start with , remember that congruent angles have the same degree of measurement. For example, in an isosceles triangle, the base angles are congruent angles because they both measure 45°
The base angles theorem states that the sides next to congruent angles are equal.The statement is therefore True.If sides of the triangle are congruent, the opposite angles in the triangle are congruent .
Factor completely. 81x^4 − 1
The expression 81x⁴ - 1 can be factored completely by identifying it as a difference of squares. First, factor it into (9x² + 1)(9x² - 1) and then further factor (9x² - 1) into (3x + 1)(3x - 1). The final factored form is (9x² + 1)(3x + 1)(3x - 1).
We are given the expression 81x⁴ - 1 and need to factor it completely. This is a difference of squares, which can be written as:
81x⁴ - 1 = (9x²)² - 1²The difference of squares formula is a²- b² = (a + b)(a - b). Applying this formula, we get:
(9x²)² - 1² = (9x² + 1)(9x² - 1)Next, notice that 9x²- 1 is also a difference of squares:
9x² - 1 = (3x)²- 12Again using the difference of squares formula, we get:
(3x)² - 1 = (3x + 1)(3x - 1)Putting everything together, the complete factorization of 81x⁴ - 1 is:
81x⁴ - 1 = (9x² + 1)(3x + 1)(3x - 1)Maria has three red dresses, 2 white dresses, and one blue dress . What is the probability she will wear a blue dress at her party?
Answer:
1/6
Step-by-step explanation:
there are 6 dresses altogether and 1 blue dress.
Answer:
[tex]\frac{1}{6}[/tex]
Step-by-step explanation:
We have been given that Maria has three red dresses, 2 white dresses, and one blue dress. We are asked to find the probability that Maria will wear a blue dress at her party.
[tex]\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}[/tex]
We know that number of blue dresses is 1, so number of favorable outcomes is 1.
Total dresses = 3 Red dresses + 2 White dresses + 1 Blue dress = 6 dresses.
[tex]\text{Probability that Maria will wear a blue dress at her party}=\frac{1}{6}[/tex]
Therefore, our required probability is [tex]\frac{1}{6}[/tex].
2 arcs of a circle are congruent if and only if their Associated chords are perpendicular
This is False
That is because the Arcs can only be congruent if the Chords are also Congruent
Answer:
FALSE
Step-by-step explanation:
there 240 students in the middle school band. The band director is dividing the students into groups of 10. Into how many groups will the band director divide the students?
Answer:
Step-by-step explanation:
1 group = 10 students
x group = 240 students.
1/x = 10/240 Cross multiply
10x = 240 Divide by 10
10x/10=240/10 Do the division
x = 24
f(5)=2, find f^-1(2)
Answer: 5
Step-by-step explanation:
f(5) = 2 means that when x = 5, y = 2 --> (5, 2)
f⁻¹(2) is the inverse (when the x and y are swapped) --> when x = 2, y = 5
Find the difference:
[tex] \sqrt[ {8ab}^{3} ]{{ac}^{2} } - \sqrt[ {14ab}^{3} ]{ {ac}^{2} } [/tex]
Answer: The difference cannot be found because the indices of the radicals are not the same.
Step-by-step explanation:
To find the difference you need to subtract the radicals. But it is important ot remember the following: To make the subtraction of radicals, the indices and the radicand must be the same.
In this case you have these radicals:
[tex]\sqrt[ {8ab}^{3} ]{{ac}^{2} }- \sqrt[ {14ab}^{3}]{{ac}^{2} }[/tex]
You can observe that the radicands are the same, but their indices are not the same.
Therefore, since the indices are different you cannot subtract these radicals.
6a - 5b when a = -3 and b = 4
Given.
6a - 5b
Plug in values.
6(-3) - 5(4) =
-18 - 20 =
-38
For this case we have the following expression:
[tex]6a-5b[/tex]
We must evaluate the expression when:
[tex]a = -3\\b = 4[/tex]
Then, replacing the values we have:[tex]6 (-3) -5 (4)[/tex]
Taking into account that according to the law of the signs of multiplication, it is fulfilled that:
[tex]+ * - = -[/tex]
So:
[tex]-18-20 =[/tex]
Equal signs are added and the same sign is placed:
-38
Answer:
-38
How do we express one-half of the difference of a number cubed and one?
Answer:
1/2(x^3 - 1) or we could write it as (x^3 - 1) / 2.
Step-by-step explanation:
A number cubed is x^3. Difference of this and 1 is x^3 - 1.
Finall we have 1/2 of this:
= 1/2(x^3 - 1) or we could write it as (x^3 - 1) / 2.
Answer:
[tex]\frac{x^3-1}{2}[/tex]
Step-by-step explanation:
We must follow what the statement asks us, starting from the last thing and going forward from there.
A number cubed: [tex]x^3[/tex]
The difference of a number cubed and one: [tex]x^3-1[/tex]
And finally:
one-half of the difference of a number cubed and one, this would be one half of the expression we already found:
[tex]\frac{x^3-1}{2}[/tex]
Which of the following is the point-slope form of the line?
Answer:
A
Step-by-step explanation:
We can see that the slope is positive, which means that the x-term must be positive.
If you expand and simply both equations into the form y = mx + b, you will find that m is positive for A and negative for B, hence A is the correct answer.
7s+4m+2l=24
5s+3m+6l=30
3s+7m+10l
Answer:
486 - 160l
Step-by-step explanation:
m = 90 - 32l
s = -48 + 18l
l = stationary
Combine like-terms, evaluate, then you will arrive at this crazy answer.
Which shows the correct substitution of the values a,b and c from the equation 1=-2x+3x^2+1
Answer:
a = 3 , b = -2 , c = 0
Step-by-step explanation:
The given equation is:
1 = -2x + 3x^2 + 1
To find the correct substitution values of a, b and c. We need to convert t into the standard form first.
Standard form of a Quadratic equation is written as:
ax^2 + bx + c = 0 (where a is not equal to zero)
Converting the given equation into its standard form:
1 = -2x + 3x^2 + 1
-2x + 3x^2 + 1 - 1 = 0
3x^2 - 2x + 0 = 0
OR 3x^2 - 2x = 0
According to the equation
a = 3 , b = -2 , c = 0
gerard’s baby brother spends 7/8 of his day sleeping. How many hours does his baby brother sleep? i
[tex] \frac{7}{8} \times 24 = 21 \\ [/tex]
You multiply 7/8 to the number of hours in a day which 24
Gerard's baby brother sleeps for 7/8 of the day, which equals 21 hours when calculated (7/8 * 24 hours).
Explanation:Gerard's baby brother spends 7/8 of his day sleeping.
To calculate how many hours this is, we need to know how many hours there are in a full day.
A full day has 24 hours. So, we multiply 7/8 by 24 to find out the total hours spent sleeping.
Here's the step-by-step calculation:
Multiply the fraction of the day spent sleeping (7/8) by the total number of hours in a day (24).7/8 * 24 = 21 hours.Therefore, Gerard's baby brother sleeps for 21 hours a day.
A grid shows the positions of a subway stop and your house. The subway stop is located at (-5,2) and your house is located at (-9,9). what is the distance, to the nearest unit, between your house and the subway stop?
Answer:
8
Step-by-step explanation:
Calculate the distance (d) using the distance formula
d = √ (x₂ - x₁ )² + (y₂ - y₁ )²
with (x₁, y₁ ) = (- 5, 2) and (x₂, y₂ ) = (- 9, 9)
d = [tex]\sqrt{(-9+5)^2+(9-2)^2}[/tex]
= [tex]\sqrt{(-4)^2+7^2}[/tex]
= [tex]\sqrt{16+49}[/tex] = [tex]\sqrt{65}[/tex] ≈ 8 ( nearest unit )
Answer: The required distance between my house and the subway stop is 8 units.
Step-by-step explanation: Given that a grid shows the positions of a subway stop and my house. The subway stop is located at (-5,2) and my house is located at (-9,9).
We are to find the distance, to the nearest unit, between my house and the subway stop.
We will be using the following formula :
Distance formula : The distance between the points (a, b) and (c, d) is given by
[tex]D=\sqrt{(c-a)^2+(d-b)^2}.[/tex]
Therefore, the distance between the points (-5, 2) and (-9, 9) is given by
[tex]D\\\\=\sqrt{(-9-(-5))^2+(9-2)^2}\\\\=\sqrt{(-9+5)^2+7^2}\\\\=\sqrt{4^2+49}\\\\=\sqrt{16+49}\\\\=\sqrt{65}\\\\=8.06.[/tex]
Rounding to the nearest units, we get
D = 8 units.
Thus, the required distance between my house and the subway stop is 8 units.