Slope-Intercept Equation of the Line
Given a line, we can express it in the form:
y = mx + b
Where m is the slope and b is the y-intercept.
We are given the equation of a line:
2y = 2x + 4
Dividing by 2:
y = x + 2
Comparing with the generic equation in slope-intercept form, we can see that m = 1 and b=2.
Now we have to find the equation of a line that is parallel to that line and passing through the given point.
Parallel lines have the same value of the slope. Thus, the slope of our required line is m'=1
The equation of the line is so far:
y = x + b'
We need to find the value of b'. Here comes handy the use of the point (-5,-1). Substituting in the equation:
-1 = -5 + b'
Solving for b':
b' = -1 + 5 = 4
Now we have the complete equation of the line as required:
y = x + 4
Show that if the diagonals of a quadrilateral bisect each other at right angles then it is a rhombus.
To Show that if the diagonals of a quadrilateral bisect each other at right angles then it is a rhombus.
Proof:
Let ABCD be a quadrilateral such that the diagonals bisect each other,
Therefore,
[tex]\begin{gathered} OA=OC\ldots(1) \\ OB=OD\ldots(2) \end{gathered}[/tex]the diagonal bisect at right angle.
Hence,
[tex]\begin{gathered} \angle AOB=90^{\circ} \\ \angle BOC=90^{\circ} \\ \angle COD=90^{\circ} \\ \angle AOD=90^{\circ}\ldots(3) \end{gathered}[/tex]to prove: ABCD is rombus,
Rombus: its is a parallelogram, with all the sides equal.
so, to prove ABCD a parallelogram.
consider the triangle,
[tex]\begin{gathered} triangleAOD\text{ and triangle }COB, \\ OA=OC \\ \angle AOD=\angle COB \\ OD=OB \\ \end{gathered}[/tex]thus, traingle
[tex]\text{AOD}\cong COB[/tex]consider the sides, AD and BC
with the transversal ac,
The angles,
[tex]\angle OAD\text{ AND }\angle OCB[/tex]are alterntaive angles. they are equal.
this implies, AD is parallel BC.
similarly, AB is parallel to DC.
Hence, AD II BC and AB II DC.
In ABCD the opposite sides are parallel,
This implies, ABCD is parallelogram.
Now, to prove that ABCD is a rombus.
for that all the sides of ABCD should be equal.
now, consider the triangle AOD and COD.
[tex]\begin{gathered} OA=OC \\ \angle AOD=\angle COD \\ OD=OD\text{ common side} \end{gathered}[/tex]By SAS congruent rule,
Traingles,
[tex]AOD\cong COD[/tex]Thus, by CPCT Corresponding parts of congruent triangles ,
AD= CD
we know that,
AD=CB and CD=AB
Thus, AD=CD=CB=AB.
hence, all the sides are eqaul and ABCD is parallelogram.
So, ABCD is a rhombus.
Trapezoid A'B'C'D was formed after a translation and reflection.YA61AB41D2The amount of unitstrapezoid ABCD wastranslated is the nextnumber of yourcombinationReturn-621D4Α'В"to the-6Dungeon
The trapezoid on the left side of the y-axis, is first of all moved 6 units down (negative 6 units on the y axis).
Next its moved 6 units to the right (positive 6 units on the x-axis).
Then it rotates 180 degrees clockwise, and the transformation from trapezoid ABCD to A'B'C'D' is complete.
2 and the probability that event A occurs given 2 In an experiment, the probability that event B occurs is 3 6 that event B occurs is 7 What is the probability that events A and B both occur? Simplify any fractions.
In order to find the probability that events A and B both occurs, we can use the following formula:
[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]So we have that:
[tex]\begin{gathered} \frac{6}{7}=\frac{P(A\cap B)}{\frac{2}{3}} \\ 7\cdot P(A\cap B)=6\cdot\frac{2}{3} \\ 7\cdot P(A\cap B)=4 \\ P(A\cap B)=\frac{4}{7} \end{gathered}[/tex]Simplify using the distributive property.8(y + 12)8 y + 9620 + y8 y + 1220 y
Solution:
Concept:
The distributive property of multiplication states that when a number is multiplied by the sum of two numbers, the first number can be distributed to both of those numbers and multiplied by each of them separately, then adding the two products together for the same result as multiplying the first number by the sum.
The expression is given below as
[tex]\begin{gathered} 8(y+12) \\ =8\times y+8\times12 \\ =8y+96 \end{gathered}[/tex]Hence,
The final answer is
[tex]\Rightarrow8y+96[/tex]A right rectangular prism's edge lengths are 10.5 inches, 5 inches, and 2 inches. How many unit cubes with edge lengths of 0.5 inch can fit inside the prism?A)105 unit cubesB)210 unit cubes460 unit cubesD)840 unit cubes5)
Given
A right rectangular prism's edge lengths are 10.5 inches, 5 inches, and 2 inches.
To find how many unit cubes of edge length 0.5 inches can fit inside the prism.
Explanation:
It is given that, the volume of the rectangular prism is,
[tex]\begin{gathered} Volume=l\times b\times h \\ =10.5\times5\times2 \\ =105in^3 \end{gathered}[/tex]Since the edge length of 0.5inch.
Then,
[tex]\begin{gathered} Volume\text{ of rectangular prism}=n\times Volume\text{ of a cube} \\ 105=n\times(0.5)^3 \\ n=\frac{105}{0.125} \\ n=840 \end{gathered}[/tex]Hence, the number of cubes is 840 unit cubes.
Jan Sara and maya ran a total of 64 miles last week. Jan and maya ran the same amount and Sarah 8 miles less then maya. how many miles did Sarah run
Let:
x = Distance traveled by Jan
y = Distance traveled by Sara
z = Distance traveled by maya
Jan Sara and maya ran a total of 64 miles last week, so:
[tex]x+y+z=64_{\text{ }}(1)[/tex]Jan and maya ran the same amount and Sarah 8 miles less then maya. therefore:
[tex]\begin{gathered} x=z_{\text{ }}(2) \\ y=z-8_{\text{ }}(3) \end{gathered}[/tex]Replace (2) and (3) into (1):
[tex]\begin{gathered} z+z-8+z=64 \\ 3z=64+8 \\ 3z=72 \\ z=24mi \end{gathered}[/tex]Replace z into (2):
[tex]\begin{gathered} y=24-8 \\ y=16mi \end{gathered}[/tex]Sara ran 16 mi
9 x+3=9 3x=9 3+x=9 x=9-3 x=9=3 < those are the answers
Based on the diagram of the figure, the equation is:
x + 3 = 9
a proton is a positively charged particle found in the nuclei of atoms a proton has a diameter of 1.5 x 10^-15 meters How is this written in standard form. A) 0.000000000000015 meters B) 0.0000000000000015 meters C) 0.00000000000000015 meters D)0.000000000000000015 meter
Graph the line.y-1= 1/5 (x+4)
We are given the following equation:
[tex]y-1=\frac{1}{5}(x+4)[/tex]Using the distributive property:
[tex]y-1=\frac{1}{5}x+\frac{4}{5}[/tex]Adding 1 to both sides
[tex]y=\frac{1}{5}x+\frac{4}{5}+1[/tex]Solving the operations:
[tex]y=\frac{1}{5}x+\frac{9}{5}[/tex]To graph this line we need two points through which the line passes. The first point can be obtained by making x = 0:
[tex]\begin{gathered} y=\frac{1}{5}(0)+\frac{9}{5} \\ y=\frac{9}{5} \end{gathered}[/tex]Therefore, the first point is (0,9/5).
The second point can be obtained by making x = 1, we get:
[tex]\begin{gathered} y=\frac{1}{5}(1)+\frac{9}{5} \\ y=\frac{10}{5}=2 \end{gathered}[/tex]Therefore, the point is (1,2). Now we plot both points and join them with a line. The graph is:
If you apply the changes below to the absolute value parent function, f(x) = [X,which of these is the equation of the new function?• Shift 2 units to the left.• Shift 3 units down.O A. g(x) = 5x + 21 - 3O B. g(x) = (x - 21 - 3O c. g(x) = 5x + 31 - 2O D. g(x) = \x - 31 - 2
h is the translation left or right
h > 0 the function shifts to the right h units
h < 0 the function shifts to the left h units
k is the translation up or down
k > 0 the function shifts up k units
k < 0 the function shifts down k units
For the given transformations:
Shift 2 units to the left. h: -3
Shift 3 units down. k: -3
Then, the function g(x) is:
[tex]\begin{gathered} g(x)=\lvert x-(-2)\rvert-3 \\ \\ g(x)=\lvert x+2\rvert-3 \end{gathered}[/tex]I need help to simplify 3x (x² - x - 2) + 2x (3 - x) - 7x. I've tried to solve the problem three times and have gotten 2x² - 2x - 6, then, x² - 1x - 6, then, 3x³ - 5x² - 13x, I can't figure out what I'm doing wrong.
Given the initial expression,
[tex]3x(x^2-x-2)+2x(3-x)-7x[/tex]Simplify it as shown below
[tex]\begin{gathered} =3x*x^2-3x*x-3x*2+2x*3-2x*x-7x \\ =3x^3-3x^2-6x+6x-2x^2-7x \end{gathered}[/tex][tex]\begin{gathered} =3x^3-3x^2-2x^2-7x \\ =3x^3-5x^2-7x \end{gathered}[/tex]Thus, the answer is 3x^3-5x^2-7xSelect ALL the expressions that have the same value as 9s
1. 9+s
2. 9*s
3. 3s + 6s
4. 3(3s)
5. s(5+4)
6. 3s * 3s
7. s+s+s+s+s+s+s+s+s2. 3s+2t=-4-6s-10t=-7What’s the value of s ?
ANSWER
2, 3, 4, 5, and 7 are the correct expressions that have the same values as 9s
STEP-BY-STEP EXPLANATION
Using Simultaneous equation
3s + 2t = -4 .............................................(1)
-6s - 10t = -7 .............................................(2)
multiply equation 1 by 2 and multiply equation 2 by 1:
equ 1 x 2: 6s + 4t = -8 ...............................(3)
equ 2 x 1: -6s -10t = -7 ................................(4)
Add equation 3 and 4 together
0s -6t = - 15
Divide through by -6:
[tex]\begin{gathered} t\text{ = }\frac{-15}{-6} \\ t\text{ = }\frac{5}{2} \end{gathered}[/tex]substitute the value of t into equation 1:
[tex]\begin{gathered} 3s\text{ + 2t = -4} \\ 3s\text{ + 2(}\frac{5}{2})\text{ = -4} \\ 3s\text{ + 5 = -4} \\ 3s\text{ = -4 -5} \\ 3s\text{ = -9 } \\ s\text{ = }\frac{-9}{3} \\ s\text{ = -3} \\ \end{gathered}[/tex]Now solving for the expression that has the same value as 9s:
Note: 9s = 9(-3) = -27
1. 9 + s = 9 - 3 = 6
2. 9 * s = 9 * -3 = -27
3. 3s + 6s = 3(-3) + 6(-3) = -9 -18 = -27
4. 3(3s) = 9s = 9(-3) = -27
5. s(5 + 4) = s(9) = -3(9) = -27
6. 3s * 3s = 3(-3) * 3(-3) = -9 * -9 = 81
7. s+s+s+s+s+s+s+s+s = -3-3-3-3-3-3-3-3-3 = -27
Hence, 2, 3, 4, 5, and 7 are the correct expressions that have the same values as 9s.
If: x+y+z=2-x+3y+2z=84x+y=4Find the value of x, y and z
We have
[tex]\begin{gathered} x+y+z=2 \\ -x+3y+2z=8 \\ 4x+y=4 \end{gathered}[/tex]We have with the third equation
[tex]y=4-4x[/tex]We substitute in the first and second equation
[tex]\begin{gathered} x+4-4x+z=2 \\ -3x+z=-2 \end{gathered}[/tex][tex]\begin{gathered} -x+3(4-4x)+2z=8 \\ -x+12-12x+2z=8 \\ -13x+2z=8-12 \\ -13x+2z=-4 \end{gathered}[/tex]Then we have
[tex]z=-2+3x[/tex]We substitute
[tex]\begin{gathered} -13x+2(-2+3x)=-4 \\ -13x-4-6x=-4 \\ -19x=0 \\ x=0 \end{gathered}[/tex]if x=0
[tex]z=-2[/tex]and if x=0
[tex]y=4[/tex]ANSWER
x=0
y=4
z=-2
I need help with my math
we have
8.13x10^5
convert to standard form
10^5=100,000
substitute
8.13x10^5=8.13*(100,000)=813,000
therefore
answer is
813,000Tim bought a new car for 25,000 one year later the value of the car decrease to 20,000 what is the percentage of the decrease of the car
Answer:
The percentage decrease in the value of the car is;
[tex]20\text{\%}[/tex]Explanation:
Given that the initial price of the car is;
[tex]25,000[/tex]And after one year the price decreased to;
[tex]20,000[/tex]The percentage change in the price will be;
[tex]\begin{gathered} \text{ \%P }=\frac{25000-20000}{25000}\times100\text{\%} \\ \text{ \%P }=\frac{5000}{25000}\times100\text{\%} \\ \text{ \%P }=0.2\times100\text{\%} \\ \text{ \%P }=20\text{\%} \end{gathered}[/tex]Therefore, the percentage decrease in the value of the car is;
[tex]20\text{\%}[/tex]Brady wants to purchase a skateboard that costs $245. So far, he has saved $98 and plans to savean additional $25 per week.Part A:If w represents the number of weeks, write the inequality that represents how many weeks it willtake Brady to save at least $245.
number of weeks: w
cost of the skateboard: 245
Amount saved: 98
Additional per week: 25
98+25w ≥ 245
Since he have to save at least 245, the expression must be equal or greater to 245
solve[tex]3 - \frac{x}{2} \geqslant 15[/tex]the equation
GIVEN:
We are given the following inequality;
[tex]3-\frac{x}{2}\ge15[/tex]Required;
To solve the inequality for x.
Step-by-step solution;
We begin by collecting like terms. Subtract 3 from both sides;
[tex]\begin{gathered} 3-3-\frac{x}{2}\ge15-3 \\ \\ -\frac{x}{2}\ge12 \end{gathered}[/tex]Now we cross multiply;
[tex]-x\ge24[/tex]We now multiply both sides of the inequality by -1.
Note that when an inequality is multiplied or divided by a negative value, then the inequality sign flips over.
Therefore;
[tex]\begin{gathered} -x\ge24 \\ \\ -x(-1)\ge24(-1) \\ \\ x\leq-24 \end{gathered}[/tex]ANSWER:
[tex]x\leq-24[/tex]find the slope of the line. 5x-2y=7
Thus 5/2 is the slo
Write an equation and solve.The supplement of an angle is 63º more than twicethe measure of its complement. Find the measure ofthe angle.
The measure of the angle is 63 degrees
Explanation:Let the angle in question be x degrees.
The supplement is (180 - x) degrees, and the complement is (90 - x) degrees.
Given that the supplement is 63 degrees more than twice the measure of its complement, we have the equation:
180 - x = 2(90 - x) + 63
Solving for x in the above:
180 - x = 180 - 2x + 63
2x - x = 180 - 180 + 63
x = 63
Ahmed takes out a loan charging 6.7% simple interest for 10 years.
At the end of 10 years Ahmed pays back $1278 in just interest. Round your answer
to the nearest penny. The original amount of the loan (principal) was
A/
Round your answer to the nearest penny..
Answer:
$1907.46
Step-by-step explanation:
You want the principal amount of a 10-year loan that earns $1278 in simple interest at the annual rate of 6.7%.
Simple InterestThe interest is given by the formula ...
I = Prt
Solving for P gives ...
P = I/(rt)
P = $1278/(0.067·10) ≈ $1907.46
The amount of the loan was $1907.46.
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Which graph shows the transformation of the function f(x)=e^x where the function is translated four units to the right, vertically compressed by a factor of 1/3, and translated down five units then translated five units down?
The graph that shows the transformation of the function f(x) = e^x is option D.
Step - by - Step Explanation
What to find? The transformation of the function f(x)=e^x.
Given:
• f(x) = 4^x
,• Vertially compresses by a factor 1/3
,• Translated four units to the right.
,• Translated down five units.
Note that:
• If f(x) shifts up m- units, then we have f(x) + m.
,• If f(x) shifts down n-units then we have f(x) - n.
,• If f(x) shifts right p - units, then we have f(x - p).
,• If f(x) shifts left q - units, then we have f(x+q).
From the given question, f(x) is translated four units to the right., hence e^x becomes eˣ⁻⁴
f(x) is further compressed by a factor of 1/3, the function becomes 1/3 eˣ⁻⁴.
Finally, the function is translated down five units, hence, the function becomes:
[tex]f(x)=\frac{1}{3}e^{x-4}-5[/tex]The graph of the function after the translation is
Solve for X using cosine law along with with written explanation.
Given:
The two sides and angle of triangle are
[tex]\begin{gathered} a=19m \\ b=25m \\ \angle C=65\degree \end{gathered}[/tex]Required:
To find the value of X.
Explanation:
By cosine rule
[tex]\begin{gathered} X=\sqrt{a^2+b^2-2ab\cos C} \\ \\ =\sqrt{19^2+25^2-2\times19\times25\cos65} \\ \\ =24.17m \end{gathered}[/tex]Final Answer:
The value of X is
[tex]X=24.17m[/tex]Last week, Shelly rode her bike a total of 30 miles over a three-day period. On the second day, she rode LaTeX: \frac{4}{5}45 the distance she rode on the first day. On the third day, she rode LaTeX: \frac{3}{2}32 the distance she rode on the second day
We make expressions for each afirmation
Where X is the first day, Y second day and Z the third
1. the sum of the 3 days gives us 30
[tex]X+Y+Z=30[/tex]2. Second day is 4/5 of the first day
[tex]Y=\frac{4}{5}X[/tex]3.Third day is 3/2 of the second day
[tex]Z=\frac{3}{2}Y[/tex]Whit the expressions I try to represent everything as a function of X
I must represent Z in function of X, for this I can replace Y of the second expression in the third expression
[tex]\begin{gathered} Z=\frac{3}{2}(\frac{4}{5}X) \\ Z=\frac{12}{10}X \\ Z=\frac{6}{5}X \end{gathered}[/tex]So I have:
[tex]\begin{gathered} Y=\frac{4}{5}X \\ Z=\frac{6}{5}X \\ \end{gathered}[/tex]And I can replace on the first expression
[tex]\begin{gathered} X+Y+Z=30 \\ X+(\frac{4}{5}X)+(\frac{6}{5}X)=30 \end{gathered}[/tex]I must find X
[tex]\begin{gathered} (1+\frac{4}{5}+\frac{6}{5})X=30 \\ 3X=30 \\ X=\frac{30}{3} \\ X=10 \end{gathered}[/tex]So, if I have X I can replace on this expressions to find de value:
[tex]\begin{gathered} Y=\frac{4}{5}X \\ Z=\frac{6}{5}X \end{gathered}[/tex]Where X is 10
[tex]\begin{gathered} Y=\frac{4}{5}\times10 \\ Y=\frac{40}{5}=8 \\ \\ Z=\frac{6}{5}\times10 \\ Z=\frac{60}{5}=12 \end{gathered}[/tex]To check:
[tex]\begin{gathered} X+Y+Z=30 \\ (10)+(8)+(12)=30 \\ 30=30 \\ \end{gathered}[/tex]The result is correct, therefore:
[tex]\begin{gathered} X=10 \\ Y=8 \\ Z=12 \end{gathered}[/tex]someone help me
please and thank you
Answer: Answer C
Step-by-step explanation:
Because it is a reflection of the Y-axis, the X-coordinates would remain the same but the Y-coordinates would change.
8. The square of a number decreased by 3 times the number is 28. Find allpossible values for the number.
We need to find a number x
We know its square (x²) decreased by 3 times it (3x) is 28. Then
x² - 3x = 28
x² - 3x - 28 = 0 [Simplifying the equation]
Since - 7 + 4 = - 3 and (-7) (4) = - 28, we factor the polynomial:
(x - 7)(x + 4) = 0 [Factoring the polynomial]
When a multiplication, like (x - 7)(x + 4), equals cero?
it equals cero if and only if
(x - 7) = 0 or (x + 4) = 0
Then, simplifying both equations
x = 7 or x = - 4
Answer: x = 7 or x = - 4
Given parallelogram ROCK and m R =159, find m 20.
Answer:
m∠C = 159
Explanation:
In parallelograms, opposite angles have the measure. Since angle C is opposite to angle R, we can write the following expression:
m∠C = m∠R
m∠R is equal to 159, so m∠C is equal to:
m∠C = 159
So, the answer is m∠C = 159
Debra has ridden 6 miles of a bike course. The course is 15miles long. What percentage of the course has Debra ridden so far
From the question
Miles covered for bike course = 6 miles
Total miles of course = 15 miles
In percentage, this becomes
Let z = percentage of miles covered
Hence
[tex]z=\frac{\text{miles covered}}{Total\text{ miles}}\times100\text{\%}[/tex]Substitute in the values to get
[tex]z=\frac{6}{15}\times100\text{\%}[/tex]Simplify:
[tex]\begin{gathered} z=\frac{2}{5}\times100\text{\%} \\ z=2\times20\text{\%} \\ z=40\text{\%} \end{gathered}[/tex]Therefore, the percentage of the course Debra has ridden so far is 40%
What is the probability of a person who plays equation IQ being in grade 6? Enter the answer as a percentage, round to the nearest tenth place.
ANSWER
28.3%
EXPLANATION
The total number of people that play Equation IQ is 300 - we can know this by adding either the more right column or the lowest row of the table.
From those people, 85 are in grade 6. The probability of a person that plays Equation IQ being in grade 6 is:
[tex]P(\text{grade 6})=\frac{\#\text{ people in grade 6}}{\#\text{ total number of players}}=\frac{85}{300}\approx0.2833\ldots[/tex]To write it as a percentage we just have to multiply it by 100:
[tex]0.283\times100=28.3\text{ \%}[/tex]how do I solve them to know what is the correct answer
For:
[tex]8x^3+16x^2[/tex]Factor 8x² out of the expression:
[tex]8x^2(x+2)[/tex]-------------------------------------------------------------------
For:
[tex]\begin{gathered} 2x^2-x+8x-4 \\ \end{gathered}[/tex]Add like terms:
[tex]2x^2+7x-4[/tex]The coefficient of x² is 2 and the constant term is -4. The product of 2 and -4 is -8. The factors of -8 which sum to 7 are -1 and 8, thus:
[tex]\begin{gathered} 2x^2+7x-4=4(2x-1)+x(2x-1) \\ so\colon \\ 4(2x-1)+x(2x-1)=(2x-1)(x+4) \end{gathered}[/tex]The graph below shows an office worker's annual salary:What are the domain and range of the function? Why are the x-values nonnegative?
Okay, here we have this:
Considering that the domain refers to the possible values of x that can be substituted in the correspondence rule of a function. We can see in the function that the domain is [0, +∞) (because there is an arrow that indicates that it continues to increase).
And as the range is the set of numbers that depend on the substitution (tabulation) of the values that "x" can take. We can see that the range is [5000, +∞).
And the number of years is non-negative because the years that elapse are always counted forward, that is, they begin to count from zero.