What is the equation of a line that passes through the point (4, 2) and is perpendicular to the line whose equation is y=x/3−1 ?
Answers
The slope here is 1/3 as this equation is in y = mx+b form (m = slope, b = y intercept)
Using the starting point of 1/3, we will...
Flip the fraction: 1/3 ----> 3/1 = 3
Then flip the sign from positive to negative: 3 ----> -3
The original slope is 1/3
The perpendicular slope is -3
The two slopes multiply to -1
We'll use m = -3 along with x = 4 and y = 2
We generate those x,y values from (x,y) = (4,2) as the given point
Plug m = -3, x = 4, y = 2 into the equation below. Solve for b
y = mx+b
2 = -3*4 + b
2 = -12 + b
2+12 = -12+b+12
14 = b
b = 14
So because m = -3 and b = 14, we go from y = mx+b to y = -3x+14
--------------------------------------------------------------------------------
Final Answer: y = -3x+14
Visual proof can be found using software like GeoGebra as shown in the attached image.
Related Questions
simplify the expression (-2 -c) (-1)
Answers
multiply -1 to both -2 & -c
-1 x -2 = 2
-1 x -c = c
2 + c is your answer
hope this helps
what is the approximate value of q ? q+log26=2q+2
Answers
Two sides of a triangle are 9 cm and 12 cm long. What is the range of possible lengths for the third side? Enter your answer in the boxes below.
Answers
Answer:
Possible length range of third side is between 3<x<21.
Step-by-step explanation:
Sum of two sides of triangle cannot be equal or less than third side. Therefore, one side length of a triangle must be between subtract and sum of other two sides. Than, range of third side is:
[tex](12-9)<x<(12+9)\\3<x<21[/tex]
Prove that a parallelogram is a square iff its diagonals are both congruent and perpendicular
Answers
The lengths of the sides are a and b.
The lengths of the diagonals are 2x and 2y.
Because the diagonals are both congruent and perpendicular, therefore there are two right triangles as shown.
Note that x = y.
Because x = y, each right triangle is isosceles and has the angles 90°, 45° and 45°.
From the Pythagorean theorem,
For one right triangle,
a² = x² + y² = x² + x² = 2x².
For the other right triangle,
b² = y² + x² = x² + x² = 2x²
Therefore
a² = b²
a = b
It follows that all sides of the parallelogram are equal and each angle is
45+45 = 90°
Therefore the parallelogram is a square.
Eric has a gift card to his favorite restaurant. Every Tuesday he has the same meal at the restaurant. This table shows the balance of money in dollars left on Eric’s gift card at different times. Time (weeks) 2, 4 ,6 ,8 Balance ($) 125, 100, 75, 50 What is the equation that represents Eric’s gift card balance, where x represents the time and y represents the balance?
Answers
Answer:
y − 100 = -12.5 (x − 4)
Step-by-step explanation:
Pick two points on the line to calculate the slope:
m = (y₂ − y₁) / (x₂ − x₁)
m = (100 − 125) / (4 − 2)
m = -12.5
Next, use point-slope form to write the line equation.
y − y₁ = m (x − x₁)
y − 100 = -12.5 (x − 4)
∠A
and
∠B
are vertical angles with
m∠A=x
and
m∠B=4x−30
.
What is
m∠A
?
Answers
Harry enlarged a picture to a height of 30 inches. The original picture was 7" wide by 5" tall. Write a proportion to find the width of the enlargement and then solve for the width of the enlargement.
Answers
original height was 5" and it was enlarged to 30"
find the ratio by dividing 30 by 5
30 / 5 = 6
so ratio would be 1/6 meaning for every 1 inch of the original the new one is 6 inches
now you need to multiply the original width by 6 to find the new width
7" x 6 = 42"
new width = 42"
The sum of two numbers is 67 . the smaller number is 9 less than the larger number. what are the numbers? larger number: smaller number:
Answers
Smaller Number is L -9
Larger # + Smaller # = 67
L + L - 9 = 67
2L - 9 = 67
2L = 67 + 9
2L = 76
L = 76/2
L = 38
Smaller Number = Larger Number - 9
38 - 9
Smaller # = 29
Check. .. 38+29=67 CORRECT!
Larger #: 38
Smaller #: 29
To solve the problem, two equations were set up based on the conditions given. After substitution and simple algebraic operations, the larger number was found to be 38 and the smaller number to be 29.
The question involves finding two numbers where the sum is 67, and the smaller number is 9 less than the larger number.
Let's define the larger number as x and the smaller number as y. According to the problem:
x + y = 67 (Equation 1)y = x - 9 (Equation 2)Substitute Equation 2 into Equation 1:
x + (x - 9) = 67
2x - 9 = 67
2x = 67 + 9
2x = 76
x = 38
Now, we can find y using Equation 2:
y = x - 9
y = 38 - 9
y = 29
Therefore, the larger number is 38 and the smaller number is 29.
Write an algebraic expression to answer the question below. if 10 calculators cost d dollars, how much, in dollars, does each calculator cost?
Answers
(1-sin squared times tan is equal to ?
Answers
i) [tex]\sin^2x+\cos^2x=1[/tex]
ii) [tex]\tanx = \frac{\sinx}{\cosx} [/tex].
We want to simplify [tex](1-\sin^2x)\cdot \tanx[/tex].
Since, [tex]\sin^2x+\cos^2x=1[/tex], then [tex]1-\sin^2x=\cos^2x[/tex]. Thus, the expression
[tex](1-\sin^2x)\cdot \tanx[/tex]
becomes
[tex]\displaystyle{ \cos^2x \cdot \frac{\sin x}{cosx} = \cos x \sin x.[/tex]
Answer: [tex] \ sinx \ cosx[/tex], or [tex]\displaystyle{ \frac{\sin2x}{2} [/tex].
How to simplify rational expressions
Answers
[tex]\bf \textit{difference of squares} \\ \quad \\ (a-b)(a+b) = a^2-b^2\qquad \qquad a^2-b^2 = (a-b)(a+b) \\\\\\ \textit{also recall that }i^2=-1\\\\ -------------------------------\\\\[/tex]
[tex]\bf \cfrac{-3-7i}{7+10i}\cdot \cfrac{7-10i}{7-10i}\implies \cfrac{(-3-7i)(7-10i)}{(7+10i)(7-10i)}\implies \cfrac{(-3-7i)(7-10i)}{7^2-(10i)^2} \\\\\\ \cfrac{-21+30i-49i+70i^2}{7^2-(10^2i^2)}\implies \cfrac{-21+30i-49i+70\left( \boxed{-1} \right)}{7^2-\left[10^2\left( \boxed{-1} \right)\right]} \\\\\\ \cfrac{-21+30i-49i-70}{49-(-100)}\implies \cfrac{-91-19i}{149}\implies -\cfrac{91}{149}-\cfrac{19i}{149}[/tex]
Please help! The very last question has 2 answers.
Answers
P2. A
3.D i think
4. C and D maybe
What number needs to be added to both sides of the equation in order to complete the square? x2+14x=−32
Answers
49 is the number to be added to both the sides of the equation to complete the square.
The equation given in the question is
[tex]x^{2} +14x=-32[/tex]Which can be written as [tex]x^{2} +2\times (7x)=-32[/tex]
We have to convert it into square form.
Since the standard form of the equation in square form is
(a + b)² = a² + 2ab + b²
By comparing this standard equation with the given equation,
a = x
&
b = 7
By substituting the values of 'a' and 'x' in the standard square form,
x² + 14x + 7² = -(32) + 7²
x² + 14x + 49 = -32 + 49
(x + 7)² = 17
Therefore, 49 should be added in both the sides of the equation in order to complete the square.
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Tina is the top seller of the store. She earns a salary plus 2%. Tina's salary is $30,000 a year. This month her sales were $48,500. How much was Tina paid this month?
Answers
If the above is true, her monthly pay is calculated as follows:
Commission: 48500 * 0.02 = $970
Monthly salary: 30000 ÷ 12 = $2500
So monthly pay is $2500 + $970 = $3470
Lauren would like to have an average of 88 in her math class. There are five assignments with the following grades: 85, 100, 88, 80, 95. What grade will lauren need to make on the last assignment to meet her goal
Answers
85+100+88+90+95+80=528
528/6=80
If f(x) = 2/x and g(x) = x^2 what is f of g
Answers
is that right?
Answer: The required value is [tex]f(g(x))=\dfrac{2}{x^2}.[/tex]
Step-by-step explanation: We are given two functions as follows :
[tex]f(x)=\dfrac{2}{x},~~~~~~~g(x)=x^2.[/tex]
We are to find f of g, i.e., f(g(x)).
To find the required value, first we have to find the f value of the function g(x).
We have
[tex]f(g(x))=f(x^2)=\dfrac{2}{x^2}.[/tex]
Thus, the required value is [tex]f(g(x))=\dfrac{2}{x^2}.[/tex]
use exponential form to evaluate log8(2)
Answers
Answer: [tex]\frac{1}{3}[/tex]
Explanation:
let y = [tex]log_{8} (2)[/tex]
whenever we transfer base from log from one side to other it will remain be base after shifting to the other side.
Change log to exponent form [tex]\log_xm=n\\m =x^{n}[/tex]
here we have n = y, m=2, x=8 by substituting the values
we will get
[tex]8^{y} = 2[/tex] (1)
we can write 8 in power of 2 that is [tex]2^{3}[/tex]
we will rewrite the above equation (1) as
[tex]2^{3y} = 2[/tex]
we can equate the power when base is same then we will get
3y = 1
then
[tex]y=\frac{1}{3}[/tex]
The result of the logarithmic expression [tex]log _82=x[/tex] is 1/3
Let the given logarithmic function be x to have:
[tex]log _82=x[/tex]
This can be transformed into indices to have:
[tex]8^x = 2[/tex]
Solve the resulting indices for "x" a shown:
[tex](2^3)^x = 2^1\\2^{3x}=2^1[/tex]
Cancel the base and equate the power to have:
[tex]3x = 1\\x = \frac{1}{3}[/tex]
Hence the result of the logarithmic expression [tex]log _82=x[/tex] is 1/3
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Which explains why the graphs of geometric sequences are a series of unconnected points rather than a smooth curve?
The range contains only natural numbers.
The domain contains only natural numbers.
Exponential bases must be whole numbers.
Initial values must be whole numbers.
Answers
The correct answer is:
The domain contains only natural numbers.
Step-by-step explanation:The graphs of geometric sequences are a series of unconnected points rather than a smooth curve because:
The domain contains only natural numbers.
As the geometric sequence us given as:
[tex]a_1=a,a_2=ar,a_3=ar^2,....[/tex]
where a is the first term of the sequence and r is the common ratio of the geometric sequence.
Also [tex]a_n[/tex] denotes the nth term of the sequence where n belongs to natural numbers that is the domain of the function is natural numbers.
The reason why the graphs of geometric sequences are a series of unconnected points rather than a smooth curve is because: B. The domain contains only natural numbers.
What is a geometric sequence?A geometric sequence is typically a series of real and natural numbers that are calculated by multiplying the next number by the same number each time.
Mathematically, a geometric sequence is given by the expression:
[tex]a, \;ar, \; ar^{2} ....a_n[/tex]
Where:
r is the common ratio.a is the first term of a geometric sequence.In a geometric sequence, the graphs of geometric sequences comprises a series of unconnected points rather than a smooth curve is because the domain is made up of only natural numbers.
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A square piece of tile with sides 12 inches is cut along a diagonal, What is the length of the diagonal rounded to the nearest inch?
Answers
To find the length of the diagonal of a square tile, you can use the Pythagorean theorem. Given that the side length of the tile is 12 inches, the length of the diagonal is approximately 17 inches when rounded to the nearest inch.
Explanation:To find the length of the diagonal of a square tile, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the two sides of the right triangle are the sides of the square tile, and the hypotenuse is the diagonal. Given that the side length of the tile is 12 inches, we can calculate the length of the diagonal as follows:
Square the length of one side: 12 x 12 = 144
Multiply the result by 2: 144 x 2 = 288
Take the square root of the result: √288 ≈ 16.97 inches
Rounded to the nearest inch, the length of the diagonal of the square tile is 17 inches.
Final answer:
The length of the diagonal of the square, rounded to the nearest inch, is 17 inches.
Explanation:
The length of the diagonal of a square can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the two sides of the square are both 12 inches long. So, we have:
hypotenuse² = side1² + side2²
hypotenuse² = 12² + 12²
hypotenuse² = 144 + 144
hypotenuse² = 288
To find the length of the hypotenuse (or the diagonal), we take the square root of both sides:
hypotenuse = √(288)
hypotenuse = 16.97 inches (rounded to the nearest inch)
Find the inverse of g(x)
Answers
1) Replace f(x) with y.
2) Switch x and y.
3) Solve for y.
4) Replace y with f^(-1)(x).
2X +105+ 8X -20+10 X +94+12 X -15=900 solve for x
Answers
32x + 164 = 900
. - 164 -164
32x = 736
÷32x. ÷32x
x = 23
How many gallons of rocket fuel containing 80% liquid oxygen must be mixed with 250 gallons containing 40% liquid oxygen to produce a mixture containing 70% oxygen?
Answers
Answer:
750
Step-by-step explanation:
for others
750 gallon of fuel at 80% to mix with the 250 gallon at 40% in order to get a mixture at 70% liquid oxygen.
What is Percentage?
percentage, a relative value indicating hundredth parts of any quantity
We need to find how many gallons of 80%
8x, since we don't know how many gallons
mixed with 250 gall of 40%: 0.40(250)
and we want a mix at 70%. The mix will include the 0.8x and the 0.4(250) Like this:
8x+0.4(250)=0.7x+0.7*250
Multiply on both sides of the equation
0.8x+100= 0.7x+175 Subtract 100 on both sides
0.8x= 0.7x+75
Subtract 0.7 on both sides
0.1x= 75
Divide both sides by 0.1
x= 750
You need 750 gallon of fuel at 80% to mix with the 250 gallon at 40% in order to get a mixture at 70% liquid oxygen.
Hence, 750 gallon of fuel at 80% to mix with the 250 gallon at 40% in order to get a mixture at 70% liquid oxygen.
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Let f(x)=2x-1 and g(x)=4-x^2. Find (f o g)(x).
Answers
[tex]fog(x)=f(g(x)) \\ \\ \\ =2.(4-x^2)-1 \\ \\ \\ = -2x^2+7 \\ \\ \\ GOOD \quad LUCK [/tex]
Can someone please help me?
Answers
Answer: -1.9
Step-by-step explanation:just add up the values for the weight change
(-1.8)+(0.9)+(-1.4)+(0.4)
-1.9
Which expression represents " 5 less than twice x " ?
1. 2x-5
2. 5-2x
3. 2(5-x)
4. 2( x-5)
Answers
The mathemtical expression for the word expression 5 less than twice x " is 2x - 5 option (A) 2x -5 is correct.
What is an expression?It is defined as the combination of constants and variables with mathematical operators.
It is given that:
The word expression is:
5 less than twice x
Twice the x = 2x
5 less than twice the x:
= 2x - 5
Thus, the mathematical expression for the word expression 5 less than twice x " is 2x - 5 option (A) 2x -5 is correct.
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How to solve for quadratic equations with inequalities?
Answers
q, and p
The sum of two numbers is
40
. One number is
4
times as large as the other.
Answers
Factoriza completamente las siguientes expresiones;
1. 3a²-a
2. 7ab³-14b
3. x³-x²+x
4. 12x³-xy²
5. 5x-6x²+7x³
6. 7a+8ab+9a²
7.3x²-3xy+6xy²
Answers
2. 7ab³-14b = 7b(ab^2 - 2)
3. x³-x²+x = x(x^2 - x + 1)
4. 12x³-xy² = x(12x^2 - y^2)
5. 5x-6x²+7x³ = x(5 - 6x + 7x^2)
6. 7a+8ab+9a² = a(7 + 8b + 9a)
7.3x²-3xy+6xy² = 3x(x - y + 2y^2)
What skills and practices were important for the growth and spread of early civilizations?
Answers
Line is parallel to line b and m<7=135. Find m<4