Answer:
if the ratio allowance/chores is constant, it is proportional
Step-by-step explanation:
The values x and y are directly proportional if they satisfy the equation ...
y = kx
for some constant k, the constant of proportionality.
__
If the relation that produces ordered pairs (chores, allowance) is directly proportional, all of the ratios (allowance)/(chores) will have the same value: the constant of proportionality (k).
If the ratios are not constant, Carlos's allowance is not directly proportional to the number of chores he completes.
Answer the following
The set A satisfying the given inequality is A = (-[tex]\infty[/tex], -10].
What are some properties of an inequality relation?Following are some facts which are true for an inequality relation:
Equal numbers can be added or subtracted from both sides of an inequality without affecting the inequality sign.The Inequality sign is unchanged if both sides are multiplied or divided by a positive number, but when multiplied or divided by a negative number the inequality sign is reversed.[tex]\frac{5x-2}{8} - \frac{3x-5}{10} &\ge& x+y\\\\\Rightarrow\;\; \frac{13}{40}x + \frac{1}{4}&\ge& x+y\\\\\Rightarrow\;\;\;\;\;\; -\frac{27}{40}x &\ge & y - \frac{1}{4}\\\\\Rightarrow\;\;\;\;\;\;\;\;\;\; -x &\ge & \frac{40}{27}\left( y-\frac{1}{4} \right).\hspace{1cm}(1)[/tex]
Since y ∈ B, -2 ≤ y ≤ 7. So,
[tex]\;\;\;\;\;\;\;\,-2 - \frac{1}{4}\; \le\; y - \frac{1}{4} \;\le\; 7 - \frac{1}{4}\\\\\Rightarrow\;\;\;\;\;\;\;\;\; -\frac{9}{4}\; \le\; y - \frac{1}{4} \;\le\; \frac{27}{4}\\\\\Rightarrow\;\;\; -\frac{9}{4}\cdot \frac{40}{27} \;\le\; \frac{40}{27} \left( y-\frac{1}{4} \right) \;\le \;\frac{27}{4}\cdot \frac{40}{27}\\\\\Rightarrow\;\;\;\;\;\;\;\; -\frac{10}{3} \;\le\; \frac{40}{27}\left( y - \frac{1}{4} \right)\; \le\; 10.[/tex]
The set {-x | inequality (1) holds ∀ y ∈ B} is [10, [tex]\infty[/tex]) i.e.
10 ≤ -x ≤ [tex]\infty[/tex].
Multiplying -1 throughout gives
-10 ≥ x ≥ -[tex]\infty[/tex].
x, thus, lies in the range A = (-[tex]\mathbf{\infty}[/tex], -10}.
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Disclaimer: The question was incomplete. Please find the full content below.
QuestionFind the set A such that for x ∈ A
[tex]\frac{5x - 2}{8} - \frac{3x - 5}{10} \ge x + y[/tex]
∀y ∈ B = {y ∈ R | -2 ≤ y ≤ 7}.
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If an employer reduces the working week from 40 hours to 35 hours, with no
loss of weedy pay, calculate the percentage increase in the hourly rate of pay.
Answer:
14.285714285714 %
Step-by-step explanation:
suppose the weekly payment is of x dollars
Then
Previous payment : $(x/40) per hour
New payment : $(x/35) per hour
Then
The percentage increase in the hourly rate of pay is :
[tex]=\frac{\frac{x}{35} -\frac{x}{40} }{\frac{x}{40} } \times100[/tex]
[tex]=\frac{\frac{5x}{1400} }{\frac{x}{40} } \times100[/tex]
[tex]=\frac{5x}{1400} \times \frac{40}{x} \times100[/tex]
[tex]=\frac{200}{1400} \times100[/tex]
[tex]=\frac{1}{7} \times100[/tex]
[tex]=14.285714285714[/tex]
Which of the following points is NOT visible if the viewing window of a graphing utility is set at:
Xmin : 6
Xmaz: 6
Ymin : —6
Ymaz : 6
The points D(-7, 5), E(-8, -3), and H(10, -5) will not be visible the answers are D, E, and H.
What is graphing utility?A graphing calculator is a portable computer that can plot graphs, solve multiple equations at once, and carry out other variable-related activities.
We have points shown in the coordinate plane.
As the window of a graphing utility is set at:
X(min) : -6
X(max): 6
Y(min) : -6
Y(max) : 6
The points out of 6 will not be visible.
We can draw a square as shown in the attached picture.
The points D(-7, 5), E(-8, -3), and H(10, -5) will not be visible.
Thus, the points D(-7, 5), E(-8, -3), and H(10, -5) will not be visible the answers are D, E, and H.
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timmy makes a table to keep track of the number of visitors to his new websites as time progresses: is the data he collected best modeled by an exponential or linear function
Answer:
Show the process
[tex]\sqrt{6} /\sqrt{27}[/tex]
[tex]\mathrm{Apply\:the\:laws\:of\:exponents}:\quad \dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b }},\:\quad \:a\ge 0,\:b\ge 0[/tex]
[tex]\dfrac{\sqrt{6}}{\sqrt{27}}=\sqrt{\dfrac{6}{27}}[/tex]
[tex]=\sqrt{\dfrac{6}{27}}[/tex]
[tex]\mathrm{ Cancel \ \ \dfrac{6}{27} \ \ \ ; \ \ \dfrac{2}{9} }[/tex]
[tex]=\sqrt{\dfrac{2}{9}}[/tex]
[tex]\mathrm{Apply\:the\:laws\:of\:exponents}:\quad \sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b }},\:\quad \:a\ge 0,\:b\ge 0[/tex]
[tex]=\sqrt{\dfrac{2}{9}}[/tex]
[tex]\sqrt{9}=3[/tex]
[tex]=\dfrac{\sqrt{2}}{3} \ \ === > \ \ \ Answer[/tex]
[tex]\red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}[/tex]
The least common denominator of 5/6, 2/3, and 3/5 is:
Answer:
30
Step-by-step explanation:
¿Cuáles son las soluciones x1 y x2 de la ecuación x2 − 5x − 6 = 0?
Answer:
x1 = 4 y x2= 2
Step-by-step explanation:
escuchame
x2 - 5x -6 = 0
(4)^2 - 5(2) - 6 = 0
16 - 10 - 6 = 0
6 - 6 = 0
de nade :)
PLEASE HELP I WILL MARK BRAINLEST
#1
Error at sign shift due to -Correct version
[tex]\\ \rm\Rrightarrow (x^3+2x^2-x)-(3x^3-3x^2+2x-4)[/tex]
[tex]\\ \rm\Rrightarrow x^3+2x^2-x-3x^3+3x^2-2x+4[/tex]
[tex]\\ \rm\Rrightarrow x^3+5x^2-3x+4[/tex]
#2
Not multiplied 3x with x[tex]\\ \rm\Rrightarrow (x^2-3x+2)(x+1)[/tex]
[tex]\\ \rm\Rrightarrow x(x^2-3x+2)+x^2-3x+2[/tex]
[tex]\\ \rm\Rrightarrow x^3-3x^2+2x+x^2-3x+2[/tex]
[tex]\\ \rm\Rrightarrow x^3-2x^2-x+2[/tex]
Sandhu lights a candle at 6 pm. It is 30 cm high. At 7pm, the candle is 27 cm high. At 8pm, it is 24cm high. a) How many cm does the candle burn down every hour? b) How high is the candle at 11 pm?
A box has 1 red marble, 1 blue marble, and 4 green marbles of the same shape and size. dan draws a marble randomly from the box, replaces it, and then draws another marble randomly. what is the probability of drawing two green marbles in a row? explain your answer.
[tex]|\Omega|=6^2=36\\|A|=4\cdot3=12\\\\P(A)=\dfrac{12}{36}=\dfrac{1}{3}\approx33.3\%[/tex]
Answer: 1/5
Step-by-step explanation:
There are three kinds of Marbles. (M) marbles
(G) green: 4m
(R) red: 1m
(B) blue: 1m
Dan pulled a green marble out leaving
(G) green: 3m
So, the probability of picking a green marble is 1/5
sorry if I'm wrong/correct me if I'm wrong.
Write the equation of the line that passes through the points (8,-1) and (2,-5) in standard form, given that the point-slope form is y+1=3/3(x-8)
The equation of the line that passes through the points (8,-1) and (2,-5) is 2x - 3y = 19.
The given coordinate are (8, -1) and (2, -5).
What is the slope of an equation?In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.
The slope of the equation is [tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1}}[/tex].
The standard form of an equation is Ax + By = C.
The given the point-slope form is [tex]y+1=\frac{2}{3} (x-8)[/tex].
Multiply each term by 3 of the equation.
[tex](y+1) \times3=\frac{2}{3} (x-8) \times3[/tex]
⇒3y + 3 = 2x - 16
Subtract 2x from both sides of a equation.
That is, -2x + 3y + 3 = - 16
Subtract 3 from both sides of a equation.
That is, -2x + 3y = - 19
Multiply each term by - 1 from both sides of an equation.
That is, 2x - 3y = 19
Therefore, the equation of the line that passes through the points (8,-1) and (2,-5) is 2x - 3y = 19.
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A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6.
Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°
Answer:
m∠5 + m∠6 =180°m∠2 + m∠3 = m∠6m∠2 + m∠3 + m∠5 = 180°Step-by-step explanation:
m∠5 + m∠6 = 180° - This is true because the exterior angle and interior angle form a linear pair, and are thus supplementary.
m∠2 + m∠3 = m∠6 - This is true by the exterior angle theorem.
m∠2 + m∠3 + m∠5 = 180° - This is true because the interior angles of a triangle add to 180 degrees.
The three right options are:
m∠5 ₊ m∠6 = 180° which represents a linear pair.
∠2 ₊ ∠3 = ∠6 showing the exterior angle property of triangle.
m∠2 m∠3 m∠5 = 180° as the sum-triangle
sum-triangleproperty.
Given,
the interior angles of the triangle are 2,3,5
the exterior angle at ∠2 is ∠1.
the exterior angle at ∠3 is ∠4.
the exterior angle at ∠5 is ∠6.
Now, take into account a triangle with external angles as ∠1 , ∠4 and ∠6.
Apply the exterior angle property of a triangle.
The sum of the opposing internal angles determines the outer angle of a triangle.
For exterior ∠1 we get
∠1 = ∠5 ₊ ∠3
Similarly, for exterior ∠4 we get
∠4 = ∠5 ₊ ∠2
Then for exterior ∠6 we get
∠6 = ∠2 ₊ ∠3
∴ through the exterior angle property we get the correct option as
m∠6 = m∠2 ₊ m∠3
Now we know the sum-triangle property that sum of the angles in triangle is equal to 180°
∴ m∠2 ₊ m∠3 ₊ m∠5 = 180°
We know the linear pair property i.e, If the angles follow the point where the two lines cross, they are considered to be linear. A linear pair's angles add up to 180° in every case.,
∴ m∠5 ₊ m∠6 = 180°
Hence we get the statements which match the diagram are:
m∠5 ₊ m∠6 = 180°
m∠2 ₊ m∠3 ₊ m∠5 = 180°
m∠6 = m∠2 ₊ m∠3
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Pls help. !!!!!!!!!!!!!!!!!! (Give steps if you can)
Answer:
[tex]144 cm^{2}[/tex]
Step-by-step explanation:
Step 1: Find the true length of each side
We know that the perimeter is 56cm and the ratio of AD:AB:DC:BC is 5:12:6:5. It is safe to assume that this ratio is the simplified version, which means that if we multiply each of these numbers by a scale factor of [tex]x[/tex], then we get the true length of each side, so we would get [tex]5x, 12x, 6x, 5x[/tex] for the sides. We also know that when we add these numbers, we get the perimeter, which is 56. So, we can create the equation [tex]5x + 12x + 6x + 5x = 56[/tex]. Then, solve the equation for x, which would be 2, and go back and multiply each number with 2 to get the following:
AD = 5*2 = 10cm, AB = 12*2 = 24cm, DC = 6*2 = 12cm, BC = 5*2 = 10cm
Step 2: Calculate the Height
Now we know the lengths of each side, lets take a look at the formula for the area of a trapezium: [tex]\frac{base_{1} + base_2}{2}*height[/tex]. So, the next step would be to calculate the height. We can draw the height by drawing a line straight down from the vertex D and another straight down from the vertex C. These lines are both the height, and will be the exact same length. We can calculate that the height is 8 because AB is 24 cm, but the length from the height at vertex D to the height at vertex C is 12 cm, so that means that the base of each triangle is 12/2 = 6 cm. Then we can use Pythagorean theorem to figure out that the height is 8cm.
Step 3: Calculate the Area
Lastly, we can calculate the area. The formula is [tex]\frac{base_{1} + base_2}{2}*height[/tex]. In this case, the numbers would be [tex]\frac{12 + 24}{2} * 8 = 144cm^{2}[/tex]
I know this is a lot, so let me know if you have any questions!
A sequence of numbers begins with 12 and progresses geometrically. Each number is the previous number divided by 2.
Which value can be used as the common ratio in an explicit formula that represents the sequence?
One-half
2
6
12
The common ratio of the given Geometric progression is one-half.
Let a be first term and r common ratio of the Geometric progression.
The Geometric sequence is the form a,ar,ar^2, ...
Given that the sequence begins with 12, i.e a = 12
and also given that the sequence are in Geometric Progression.
By dividing the preceeding number by 2, we get the successive number
Then the Geometric sequence is 12, 12×1/2, 12×1/4, 12×1/8, 12×1/16 ...
= 12, 6, 3, 3/2, 3/4, ...
Common ratio = Second term / First term
= 6/12
=1/2
Therefore, the common ratio of the given geometric progression is one-half.
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Answer:
its a
Step-by-step explanation:
hi! The work I have is solving linear equations with fractions, i only have practice exercises so they should probably be a bit simple, shown work would be appreciated <3
Just 4 questions
1) y / 3 + 6 = -3 / 2
2) -2 / 3 r - 3 = 3
3) 2 / 3 - 2x = x
4) 7x / 2 + -5 / 2 = 9/2
I’ll add a photo of the problems in a simple format
The value of the variables are y = -45/2, r = -9, x = 2/9 and x = 2
How to solve the variables?(1) y/3 + 6 = -3/2
Multiply through by 3
y + 18 = -9/2
Subtract 18 from both sides
y = -18 - 9/2
Take LCM
y = (-18 * 2 - 9)/2
Evaluate
y = -45/2
(2) -2r/3 - 3 = 3
Multiply through by 3
-2r - 9 = 9
Add 9 to both sides
-2r = 18
Divide by 2
r = -9
(3) 2/3 - 2x = x
Multiply through by 3
2 - 6x = 3x
Add 6x to both sides
2 = 9x
Divide by 9
x = 2/9
(4) 7x/2 + -5/2 = 9/2
Multiply through by 2
7x - 5 = 9
Add 5 to both sides
7x = 14
Divide by 7
x = 2
Hence, the value of the variables are y = -45/2, r = -9, x = 2/9 and x = 2
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Casey picked 6 pounds of strawberries. Karen picked 48 ounces of blueberries. Jude picked 2 pounds of raspberries. They are going to make berry pies. The table shows the amount of berries needed for 1 pie.Select all of the true statements below.
A.
Together, Karen and Jude picked the same weight of fruit as Casey picked.
B.
Casey picked 2 times the weight of fruit that Karen picked.
C.
There are enough berries to make 6 berry pies.
D.
If they make as many pies as they can with the berries they picked, there will be 4 pounds of strawberries left over.
E.
To make 8 berry pies, they will need 1 more pound of blueberries and 2 more pounds of raspberries.
Answer:
Option B is correct
Answer:
B there is enough raspberries to make 4 pie they can 6 pies if they 2 more of raspberries
On a coordinate plane, a right triangle has points A (negative 5, negative 6), B (negative 5, negative 2), C (negative 2, negative 2). Examine the right triangle ABC. Which rise and run would create a similar right triangle on the same line? a rise of 6 and a run of 8 a rise of 8 and a run of 6 a rise of 6 and a run of 5 a rise of 5 and a run of 6
The rise and run that would give us a similar right triangle on the same given line is: B. a rise of 8 and a run of 6.
What is the Rise and Run of Points on the Same Line?The rise/run of any two points on the same line are have the same slope value.
Given the coordinates:
A(-5, -6)B(-5, -2)The rise of triangle ABC = change in y = 4 units
The run of triangle ABC = change in x = 3 units
Rise/run = 4/3
Therefore, a rise of 8 and a run of 6 would give us: 8/6 = 4/3.
This will give us a triangle similar to triangle ABC.
Therefore, the answer is: B. a rise of 8 and a run of 6.
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What are the solutions to the following system of equations?
y = x² + 7x-5
2x - y = -9
Answer:
Step-by-step explanation:
y = x² + 7x - 5 ............ (1)
2x - y = - 9 ................ (2)
(1) ------> (2)
2x - ( x² + 7x - 5 ) = - 9
- x² - 5x + 14 = 0 ........ (3)
- 1 × (3)
x² + 5x - 14 = 0
( x + 7 )( x - 2 ) = 0
[tex]x_{1}[/tex] = - 7 and [tex]x_{2}[/tex] = 2
2( - 7) - [tex]y_{1}[/tex] = - 9 ⇒ [tex]y_{1}[/tex] = - 5
2(2) - [tex]y_{2}[/tex] = - 9 ⇒ [tex]y_{2}[/tex] = 13
The solutions of system are
( - 7, - 5 ) and ( 2, 13 )
2 more questions and 100 points answer asap!
Answer:
-6
Step-by-step explanation:
Well you just plug in the value 3 for x. This will give you the following equation: [tex]\frac{4(3+3)(3+1)}{(3+5)(3-5)}[/tex] which simplifies to [tex]\frac{4(6)(4)}{(8)(-2)}[/tex] which further simplifies to [tex]-\frac{96}{16}[/tex] which can further be simplified by dividing the two numbers and that results in -6
Answer:
A
Step-by-step explanation:
it really only means to replace every x by 3 and then simply calculate.
4(3+3)(3+1) / ((3+5)(3-5)) =
= 4×6×4 / (8×-2) = 96/-16 = -6
Question 10 of 10
Which of the following are necessary when proving that the diagonals of a
rectangle are congruent? Check all that apply.
A. Corresponding parts of similar triangles are similar.
B. Corresponding parts of congruent triangles are congruent.
C. Opposite sides are congruent.
D. Opposite sides are perpendicular.
what is the median in this list
3 6 9 7 4 6 7 0 7
i need the median the range and the mode
ANYTHING HELPS
Median: 6
Range: 9
Mode: 7, appeared 3 times
What is the equivalent exponential expression for the radical expression below? √3+7
Answer:
09 l
Step-by-step explanation:
p0-9
The equivalent exponential expression for the given radical expression √(3 + 7) is √10.
To convert the given radical expression √(3 + 7) into its equivalent exponential expression. This process involves using exponentiation to represent the radical form.
The given radical expression is √(3 + 7). To find its equivalent exponential expression, we need to express the radical in terms of an exponent.
Recall that a radical expression of the form √(a) is equivalent to raising 'a' to the power of 1/2. In this case, 'a' is the quantity inside the radical, which is 3 + 7. So, we can write:
√(3 + 7) = (3 + 7)^(1/2).
Next, we simplify the expression inside the parentheses:
3 + 7 = 10.
Now, the expression becomes:
(10)^(1/2).
To further simplify, we raise 10 to the power of 1/2. Recall that raising a number to the power of 1/2 is the same as finding the square root of that number. So:
(10)^(1/2) = √10.
Therefore, the equivalent exponential expression for the given radical expression √(3 + 7) is √10.
To summarize, we transformed the radical expression √(3 + 7) into its equivalent exponential expression √10 by using the rule that √(a) is equal to a^(1/2). By raising the expression inside the radical to the power of 1/2, we simplified it and found the square root of 10 as the final result.
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Hi can I have help on this question with a simple explanation as I find things difficult to understand. Thank you :)
Step-by-step explanation:
Line L → 2x - 3y = 6
Where a = 2 and b = -3
• So, we can use this formula to find the gradient of Line L.
m = -a/b
m = -2/-3
m = 2/3 IS THE ANSWER
Answer:
gradient = 2/3
Step-by-step explanation:
Here is another way....manipulate the equation to y = mx+ b form
where m = slope ( gradient)
2x - 3y = 6 add 3 y to both sides
6 + 3y = 2x subtract 6 from both sides
3y = 2x - 6 divide by 3
y = 2/3 x - 2 gradient = 2/3
What is the value of x?
Enter your answer in the box.
...°
I need help answering these questions for geometry :)
1. What happens to the measure of the interior angles as the number of sides of the polygon increases? Explain your answer.
2.What happens to the measure of the exterior angles as the number of sides of the polygon increases? Explain your answer.
4. Explain what happens to the total sum of interior angles as the number of sides in the polygon changes.
5. Explain what happens to the total sum of exterior angles as the number of sides in the polygon changes.
The answers to the question have been written below
The changes to a polygon1. If the number of sides of the polygon should increase then the measure of the interior angle would increase by 180.
2. A regular polygon has the sum of its external angles as 360. This is solved as 360/n = 0
Hence the value of the exterior would move towards 0 as its sides increases.
4. As the total sum of the of the polygon changes, it would tend towards infinity.
5. The total sum of the exterior angles as the number of sides changes would tends towards 360 degrees.
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Solve the problem 4(a+3)=-32
[tex]4(a + 3) = - 32 \\ 4a + 12 = - 32 \\ 4a = - 32 - 12 \\ 4a = - 44 \\ a = - 11[/tex]
Hope it helps
Please give brainliest
Step-by-step explanation:
4(a+3)=-32
4(a+3)/4=-32/4
(a+3)=-8
a+3-3=-8-3
a=-11
Find the nth term of this number sequence
1, 7, 13, 19, ...
Answer:
now we have to follow up with the formula n² - n¹
which is 7-1= 6
n³-n²
which is 13-7=6
so therefore the nth term=6
A water sample shows 0.033 grams of some trace element for every cubic centimeter of water. Marques uses a container in the shape of a right cylinder with a diameter of 13.2 cm and a height of 19.7 cm to collect a second sample, filling the container all the way. Assuming the sample contains the same proportion of the trace element, approximately how much trace element has Marques collected? Round your answer to the nearest tenth.
Answer:
89.0 grams of trace element
Step-by-step explanation:
Step 1Find the volume of the sample container.
Given:
Container is a right cylinder with d = 13.2 cm and h = 19.7 cmVolume formula:
V = πr²h = πd²h/4Substitute to get:
V = π(13.2)²(19.7)/4 = 2695.9 cm³ (rounded)Step 2Find the volume of the trace element, using the given proportion:
2695.9*0.033 = 89.0 grams (rounded)The data set represents the number of minutes dan spent on his homework each night. 30, 35, 35, 45, 46, 46, 52, 55, 57 which box plot correctly represents the data?
The box plot that represents the data is the first option.
What box plot correctly represents the data?A box plot is used to study the distribution and level of a set of scores. The box plot consists of two lines and a box. The end of the first whisker represents the minimum number and the end of the second whisker represents the maximum number.
On the box, the first line to the left represents the lower (first) quartile. The next line on the box represents the median. The third line on the box represents the upper (third) quartile.
Minimum value = 30
Maximum value = 57
1st quartile = 1/4 (n + 1)
1/4 x 10 = 2.5th term = 35
Third quartile = 3/4 x (n + 1 )
3/4 x 10 = 7.5 term = 53
Median = 46
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PLEASE HURRY WILL GIVE POINTS AND BRAINLYEST TO FIRST RIGHT!!!
Answer:
(0, 4) and (-1, 0)
Step-by-step explanation:
Given system of equations:
[tex]\begin{cases}y=2x^2+6x+4\\y=-4x^2+4\end{cases}[/tex]
Solve by substitution
Substitute the first equation into the second:
[tex]\implies 2x^2+6x+4=-4x^2+4[/tex]
Add 4x² to both sides:
[tex]\implies 2x^2+6x+4+4x^2=-4x^2+4+4x^2[/tex]
[tex]\implies 6x^2+6x+4=4[/tex]
Subtract 4 from both sides:
[tex]\implies 6x^2+6x+4-4=4-4[/tex]
[tex]\implies 6x^2+6x=0[/tex]
Factor out 6x from the left side:
[tex]\implies 6x(x+1)=0[/tex]
Therefore:
[tex]\implies 6x=0 \implies x=0[/tex]
[tex]\implies x+1=0 \implies x=-1[/tex]
To find the y-coordinates of the found x-values, substitute the found values of x into one of the equations:
[tex]x=0 \implies -4(0)^2+4=4 \implies (0,4)[/tex]
[tex]x=-1\implies -4(-1)^2+4=0\implies (-1,0)[/tex]
Therefore, the solutions to the system of equations are:
(0, 4) and (-1, 0)
Answer:
Solutions:
a) x = 0, y = 4 ⇒ (0, 4)
b) x = -1, y = 0 ⇒ (-1, 0)
Step-by-step explanation:
Given system of equations:
a) y = 2x² + 6x + 4
b) y = -4x² + 4
1. Substitute the value of y in the second equation into the first equation:
⇒ -4x² + 4 = 2x² + 6x + 4
2. Solve for x:
⇒ -4x² + 4 = 2x² + 6x + 4 [subtract 4 from both sides]
⇒ -4x² + 4 - 4 = 2x² + 6x + 4 - 4
⇒ -4x² = 2x² + 6x [subtract 2x² from both sides]
⇒ -4x² - 2x² = 2x² - 2x² + 6x
⇒ -6x² = 6x [subtract 6x from both sides]
⇒ -6x² - 6x = 6x - 6x
⇒ -6x² - 6x = 0 [factor out -6x from the equation]
⇒ -6x(x + 1) = 0
Two cases:
a)
⇒ -6x = 0 [divide both sides by -6]
⇒ -6x ÷ -6 = 0 ÷ -6
⇒ x = 0
b)
⇒ x + 1 = 0 [subtract 1 from both sides]
⇒ x + 1 - 1 = 0 - 1
⇒ x = -1
3. Find the value of y by substituting the found x values into one of the given equations:
a) x = 0:
⇒ y = -4x² + 4
⇒ y = -4(0)² + 4
⇒ y = -4(0) + 4
⇒ y = = 0 + 4
⇒ y = 4
coordinate: (0, 4)
b) x = -1:
⇒ y = -4x² + 4
⇒ y = -4(-1)² + 4
⇒ y = -4(1) + 4
⇒ y = -4 + 4
⇒ y = 0
coordinate: (-1, 0)
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