Answers
The change in internal energy of the system is 0 J because the energy input (50 J) precisely equals the energy output (50 J), assuming no work is done by the system, according to the first law of thermodynamics.
Explanation:The question is asking about the change in internal energy of a system given that there is some energy transfer into and out of the system. We denote the energy transferred into the system as Qin and the energy transferred out of the system as Qout. For this question, Qin is 50 J and Qout is 50 J.
The first law of thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Qin) minus the work done by the system (W) and the heat leaving the system (Qout). However, in this case, we don't have information about the work done, which is often represented as the term W. If we assume no work is done, the equation simplifies to ΔU = Qin - Qout.
Using this information, we can calculate the change in internal energy: ΔU = 50 J - 50 J = 0 J. There is no change in internal energy because the energy input equals the energy output. Therefore, the correct response here would be none of the options provided (A. 20 J, B. 30 J), but instead, it would be 0 J.
Related Questions
A woman has 14 different shirts: 10 white shirts and 4 red shirts. If she randomly chooses 2 shirts to take with her on vacation, then what is the probability that she will choose two white shirts? Show your answer in fraction and percent, round to the nearest whole percent.
Answers
Answer:
1/5; 20%
Step-by-step explanation:
If she is only choosing 2 shirts, and they both have to be white, the probability of choosing those 2 white shirts out of 10 white shirts is 2/10 or, equivalently, 1/5. In percent form, this is 20%
The probability of a woman randomly choosing two white shirts from a collection of 14 shirts (10 white and 4 red) is approximately 49%, derived through the process of combinations calculation in the field of probability.
Explanation:The question relates to the field of probability in mathematics. To calculate her probability of selecting two white shirts, we first need to understand the number of total possible outcomes when she is selecting 2 out of the 14 shirts. This is represented by the combination formula C(n, r) = n! / r!(n-r)!, where n is the total number of items to choose from and r is the number of items to choose. Using this combination formula, we find that there are C(14, 2) = 91 total possible combinations of shirts she could end up with.
Next, we need to find the number of combinations that involve her picking 2 white shirts. As there are 10 white shirts, this is represented by C(10, 2) = 45. So, the probability of her taking two white shirts is then the number of desired outcomes divided by the total number of outcomes, or 45 / 91 which simplifies to approximately 0.4945 or 49% when rounded to the nearest whole percent.
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Solve the 3 × 3 system shown below. Enter the values of x, y, and z. X + 2y – z = –3 (1) 2x – y + z = 5 (2) x – y + z = 4 (3)
Answers
[tex]\boxed{x=1, \y=-1, \ z=2}[/tex]
Step-by-step explanation:We will use the Gaussian elimination method to solve this problem. To do so, let's follow the following steps:
Step 1: Let's multiply first equation by −2. Next, add the result to the second equation. So:
[tex]\begin{array}{ cccc }~~ x&+~~2~ y&-~~~~~ z&~=~-3\\&-~~~5~ y&+~~3~ z&~=~11\\~~ x&-~~~~~ y&+~~~~ z&~=~4\end{array}[/tex]
Step 2: Let's multiply first equation by −1. Next, add the result to the third equation. Thus:
[tex]\begin{array}{ cccc }~~ x&+~~2~ y&-~~~~~ z&~=~-3\\&-~~~5~ y&+~~3~ z&~=~11\\&-~~~3~ y&+~~2~ z&~=~7\end{array}[/tex]
Step 3: Let's multiply second equation by −35, Next, add the result to the third equation. Therefore:
[tex]\begin{array}{ cccc }~~ x&+~~2~ y&-~~~~~ z&~=~-3\\&-~~~5~ y&+~~3~ z&~=~11\\&&+~~\frac{ 1 }{ 5 }~ z&~=~\frac{ 2 }{ 5 }\end{array}[/tex]
Step 4: solve for z, then for y, then for x:
[tex] \frac{ 1 }{ 5 } ~ z & = \frac{ 2 }{ 5 } \\ \\ \boxed{z & = 2}[/tex]
[tex]-5y+3z &= 11\\-5y+3\cdot 2 &= 11\\ \\ \boxed{y &= -1}[/tex]
By substituting [tex]y=-1 \ and \ z=2[/tex] into the first equation, we get the [tex]x[/tex]. So:
[tex]x+2(-1)-2=-3 \\ \\ x-2-2=-3 \\ \\ \boxed{x=1}[/tex]
Answer:
X= 1, y= -1, z= 2
Step-by-step explanation:
Analyze the zeros of f(x) =x^4 - 3x^3 - 2x^2 + 3x + 5.
Determine the number of possible positive real zeros and the number of possible negative real zeros
a. positive 1; negative 3 or 1
b. positive 1; negative 2 or 1
c. positive 3 or 1; negative 1
d. positive 2 or 1; negative 1
Answers
Answer:
c
Step-by-step explanation:
c. positive 3 or 1; negative 1
The sum of all but one interior angle of a heptagon is 776°. The final angle must have a measure of degrees.
Answers
Answer:
The final angle is 124°
Step-by-step explanation:
1. Getting the total sum of all angles:
The general rule to get the sum of all interior angles is as follows:
sum of interior angles = (n-2)*180
where n is the number of sides
We know that a heptagon has 7 sides, therefore:
sum of interior angles of a heptagon = (7-2)*180 = 5 * 180 = 900°
2. Getting the missing angle:
We know that the total sum of all 7 angles is 900°
We also know that the sum of 6 of those angles = 776°
This means that:
measure of the last angle = 900 - 776
measure of last angle = 124°
The final angle is 124°
Hope this helps :)
Answer:
124 degrees
Step-by-step explanation:
Rosana is tracking how many customers she can serve in a morning. She listed the number of customers served per hour in the following table:
Hour (x) Number of Customers f(x)
1 3
2 5
3 7
Determine if these data represent a linear function or an exponential function, and give the common difference or ratio.
A. This is a linear function because there is a common difference of 2.
B. This is an exponential function because there is a common ratio of 2.
C. This is a linear function because there is a common difference of 3.
D. This is an exponential function because there is a common ratio of 3.
Answers
Answer:
A. This is a linear function because there is a common difference of 2
Step-by-step explanation:
Differences are ...
5 - 3 = 27 - 5 = 2The differences of 2 are common, so this is an arithmetic function.
Option: A is the correct answer.
A. This is a linear function because there is a common difference of 2.
Step-by-step explanation:Linear function--
A function is said to be linear if the rate of change is constant.
i.e. the function is increasing by a fixed constant.
Here the table is given by:
Hour (x) Number of Customers f(x)
1 3
2 5
3 7
From the table of values we observe that as the value of increasing by 1 the value of y is increasing by 2.
i.e. the rate of change is constant i.e. 2.
Also, the common difference is 2.
since,
5-3=2
and 7-5=2
Hence, the table represents a linear function.
The soccer team collected $800 at a car wash fundraiser. They charged $5.00 for small vehicles and $10.00 for larger vehicles. The amount collected can be modeled by the equation. What is the equation?
Answers
Answer:
The equation is 5x + 10y = 800
Step-by-step explanation:
* Lets explain what is the equation and how can we make one
- The equation is a mathematical statement that two things are equal.
- It consists of two expressions, one on each side of an equal sign (=).
* Lets solve the problem
- There are two types of vehicles, large one and small one
- The cost of washing the small vehicle is $5
- The cost of washing the larger vehicle is $10
- They collected from washing both is $800
- To make an equation for the amount collected let the number of the
small vehicles is x and the number of the large vehicles is y
∵ x is the number of the small vehicles were washed
∵ y is the number of the large vehicles were washed
∵ The cost of washing the small vehicle is $5
∵ The cost of washing the larger vehicle is $10
∴ The money collected from the small vehicles = 5 × x = 5x
∴ The money collected from the large vehicles = 10 × y = 10y
- find the total money collected from both
∴ The amount of money collected from both = 5x + 10y
∵ The amount of money collected is $800
- Equate the two expressions above
∴ 5x + 10y = 800
* The equation is 5x + 10y = 800
Solve for x.
5(2x - 1) = 6
x = 1/10
x = 11/10
x = 1/2
Answers
Answer:
Distributive Law : x= [tex]\frac{11}{10}[/tex]
Step-by-step explanation:
5(2x - 1) = 6
Using Distributive law on right hand side we get
5*2x - 5 * 1 = 6
10x - 5 = 6
Now we add 5 on both hand sides
10x-5+5=6+5
10x= 11
Now we divide both sides by 10
[tex]\frac{10x}{10} = \frac{11}{10}[/tex]
Hence we get
x= [tex]\frac{11}{10}[/tex]
what is the exact value of cos 90degrees as found on the unit circle?
Answers
Answer:
it is 0 because cosine is the x and sine is the y. Sin is 1 and cos is 0
I
f point A is located at (–15, –7) and JKL has vertices at J(–22.5, –10.5), K(19.5, –3.0), and L(–4.5, 15.0), find the scale factor ABC to JKL of the dilation from
Answers
Answer:
1.5
Step-by-step explanation:
The coordinate values of point J are 1.5 times those of point A, so if dilation is about the origin, the scale factor is 1.5.
If the dilation is about some other point, this problem statement does not contain enough information to tell what the scale factor might be.
This week, 300 tickets were sold to the school play. This is 120 percent of the number of tickets sold last week. How many tickets were sold last week for the school play?
Answers
Answer:
250 tickets
Step-by-step explanation:
300/1.2 or 120%
= 250 tickets
Answer:
250 tickets
Step-by-step explanation:
A rectangle has a length that is 5 inches greater than its width, and its area is 104 square inches. The equation (x + 5)x = 104 represents the situation, where x represents the width of the rectangle.
The first step in solving by factoring is to write the equation in standard form, setting one side equal to zero. What is the equation for the situation, written in standard form?
Choose one of the best answers.
A. x² – 99 = 0
B. x² – 99x = 0
C. x² + 5x + 104 = 0
D. x² + 5x – 104 = 0
Please explain your answer?
If your answers is wrong, it's going mark your answer report and it's called "improper answer."
No need to spam answers, if your answer is spam is going to be report.
Don't copied or paste answers with someone else answers from other sites. If you copied and paste answers from other websites and it mark your answer report and it's called "plagiarism."
Thank you!
-Charlie
Answers
Answer:
D
Step-by-step explanation:
(x + 5)x = 104
Distribute:
x² + 5x = 104
Subtract 104 from both sides:
x² + 5x − 104 = 0
Taylor fires a toy rocket from ground level. The height of the rocket with respect to time can be represented by quadratic function. If the toy rocket reaches a maximum height of 34 feet, 3 seconds after it was fired, which of the following functions could represent the height, h, of the rocket t seconds after it was fired?
A) h(t)=-16(t-3)²+34
B) h(t)=-16(t+3)²+34
C) h(t)=16(t-3)²+34
D) h(t)=16(t+3)²+34
Answers
Answer:
The function in choice
A) [tex]h(t) = -16(t - 3)^{2} + 34[/tex]
could possibly represent this relationship.
Step-by-step explanation:
Consider the vertex form of parabolas with a local extrema at [tex](x_{0}, y_{0})[/tex].
[tex]y = a(x - x_{0})^{2} + y_0[/tex].
Note the minus sign in front of [tex]x_0[/tex] in this expression.
The coefficient [tex]a[/tex] cannot be zero. The value of [tex]a[/tex] depends on the direction and width of the parabola's opening:
[tex]a > 0[/tex] if the parabola opens upwards, and[tex]a < 0[/tex] if the parabola opens downwards.The width of the opening decreases as the value of [tex]a[/tex] increases.For this parabola,
[tex]a < 0[/tex] since the parabola opens downwards: the height of the rocket will eventually decrease as the rocket falls back to the ground;[tex]t_0 = 3[/tex], and[tex]h_0 = 34[/tex].Among the four functions, only the function in A) meets the requirements.
Twenty-five blue and twenty-five yellow marbles are placed in a jar. Thirty-four marbles are removed from the jar and put in a second jar. What is the positive difference between the number of blue marbles in the second jar and the number of yellow marbles remaining in the first jar?
Answers
Answer:
a) Let b represent the number of blue marbles placed in the second jar. Then 34-b is the number of yellow marbles in the second jar, and 25 - (34-b) is the number of yellow marbles remaining in the first jar. That value simplifies to b-9. The positive difference between b and (b-9) is 9.
b) Let n, d, p represent the numbers of nickels, dimes, and pennies, respectively.
.. (5n + 10d + p)/(n + d + p) = 7 ... the average value of the collection is 7 cents per coin
.. (5(n-1) + 10d + (p+5))/((n-1) + d + (p+5)) = 6 ... replacing one nickel with pennies changes the average to 6 cents per coin
Multiplying by the denominator and subtracting the right side of each resulting equation gives
.. 10d + 5n + p = 7d + 7n + 7p
.. 10d + 5n + p = 6d + 6n + 6p + 24
Subtracting the second equation from the first gives
.. 0 = d + n + p - 24 ... tells us the total number of coins is 24. This tells us the total value is $1.68 = 24*$0.07.
This sum will consist of at least 3 pennies. If that is all the pennies, then the remaining 21 nickels and dimes must add to $1.65. If all were nickels, the value of the 21 coins would be $1.05. We have $0.60 more than that. For each nickel traded for a dime, we gain $.05, so 12 of the 21 nickels must be traded for dimes to raise the total coin value to $1.65.
Alternatively, you can write two equations in the two unknowns:
.. d + n = 21
.. 10d + 5n = 165
Subtracting 5 times the first equation from the second gives
.. 5d = 60 ... which gives the same answer as reasoned above.
There are 3 pennies, 9 nickels, 12 dimes. (If there are 8 pennies to start, then the number of nickels is zero, so it is impossible to trade one for 5 pennies. The solution shown is the only one.)
If carol travels 30 miles and her car averages 48 miles per hour, how long will it take carol to complete the distance?
Answers
[tex]\bf \begin{array}{ccll} miles&hours\\ \cline{1-2} 48&1\\ 30&x \end{array}\implies \cfrac{48}{30}=\cfrac{1}{x}\implies \cfrac{8}{5}=\cfrac{1}{x}\implies 8x=5 \\\\\\ x=\cfrac{5}{8}\qquad \textit{37 minutes and 30 seconds}[/tex]
jackson bought 555 ounces of raisins for \$4$4dollar sign, 4. How much do raisins cost per ounce? \$$dollar sign How many ounces of raisins can be bought for \$1$1dollar sign, 1? ounces
Answers
Answer:
$.80 and 1.25
Step-by-step explanation:
hope this helps
Find the area of a regular hexagon with an apothem 11.4 yards long and a side 13 yards long. Round your answer to the nearest tenth.
Answers
Answer:
Step-by-step explanation:
area=6×11.4×13÷2=444.6 yd²
Final answer:
The area of a regular hexagon with an apothem of 11.4 yards and a side of 13 yards is approximately 444.6 square yards when rounded to the nearest tenth.
Explanation:
To find the area of a regular hexagon with a given apothem and side length, you can use the formula for the area of a regular polygon:
Area = (1/2) × Perimeter × Apothem
In this case, the perimeter of the hexagon can be calculated by multiplying the length of one side by 6 (since a hexagon has 6 sides). Therefore, the Perimeter = 13 yards × 6 = 78 yards.
Now, plug the values into the area formula:
Area = (1/2) × 78 yards × 11.4 yards
Area = (1/2) × 889.2 square yards
Area = 444.6 square yards
So, the area of the regular hexagon is approximately 444.6 square yards, when rounded to the nearest tenth.
Pia printed two maps of a walking trail. The length of the trail on the first map is 8 cm. The length of the trail on the second map is 6 cm.
(a) 1 cm on the first map represents 2 km on the actual trail. What is the scale factor from the map to the actual trail? What is the length of the actual trail?
(b) A landmark on the first map is a triangle with side lengths of 3 mm, 4 mm, and 5 mm. What is the scale factor from the first map to the second map? What are the side lengths of the landmark on the second map? Show your work.
Answers
Answer:
Two trail maps:
Trail on the first map is 8 cm
Trail on the second map is 6 cm
Scale on first map is 1 cm : 2 km
A. What is the scale factor from the map to the actual trail?
For the first map, the scale factor is 1 cm: 2km. Therefore the actual trail is 8 centimeters * 2 kilometers = 16 km.
The scale factor of the second map is 16 km / 6 cm = 2.67 km : 1 cm
B. The length of the actual trail is 16 kilometers.
Answer:
Given:
Two trail maps:
Trail on the first map = 8 cm
Trail on the second map = 6 cm
Scale on first map = 1 cm : 2 km
A) What is the scale factor from the map to the actual trail?
For the first map, the scale factor is 1 cm: 2km. Therefore the actual trail is 8 centimeters * 2 kilometers = 16 km.
The scale factor of the second map is 16 km / 6 cm = 2.67 km : 1 cm
B) The length of the actual trail is 16 kilometers.
Step-by-step explanation:
two lengths of a triangle are show
Answers
I can't give you the correct answer but you can guess for it, cause I can give you the the scop of the length of KL.
So, as we know this;
The sum of the two sides of a triangle is greater than the third and the difference between the two sides is less than the third.
10+5 >= KL >= 10-5
15 >= KL >= 5
based on your option, the answer should be:
9 in
I have a model motorcycle that is all metal parts and is 18" long. Since the size all the models online are given in a scale, What scale to an actual motorcycle do I have and where do I look to find a value?
Answers
Answer:
See below.
Step-by-step explanation:
The model may have the scale written somewhere. Try looking for it.
If you can't find it, then you can calculate this way.
Search online for the real motorcycle's length.
Then divide the scale model's length, 18 inches, by the real length. Express that as a ratio. That is the scale.
Let's say the length of a real motorcycle of this type is 90 inches.
Divide 18 inches by 90 inches.
18/90 = 1/5
The scale, then, is 1:5.
Write the equation of the graph shown in factored form.
Answers
let's take a peek at the graph.
the graph of the equation touches the x-axis at 1, 2 and 3, at 1 it crosses it, at 2 it crosses it, at 3 is doesn't cross it, it simply bounces off of it.
on the roots/solutions the graph crosses the x-axis, they have ODD multiplicity, on the solutions it bounces off of it, they have EVEN multiplicity, so then.
x = 1 => x - 1 = 0 <------ one factor
x = 2 => x - 2 = 0 <------ another factor
x = 3 => x - 3 = 0 <------ another factor, BUT with even multiplicity
(x-1)¹(x-2)¹(x-3)² = f(x)
(x-1)(x-2)(x-3)² = f(x).
A farmer wants to plant peas and carrots on no more than 200 acres of his farm. If x represents the number of acres of peas and y represents the number of acres of carrots for solution (x, y), then which is a viable solution?
A.(−50, 160)
B.(80, 160)
C.(75, −200)
D.(60, 135)
Answers
Answer:
Option D.(60, 135)
Step-by-step explanation:
Let
x -----> the number of acres of peas
y ----> the number of acres of carrots
we know that
The inequality that represent the problem is equal to
[tex]x+y\leq 200[/tex]
so
Verify each case
case A) (-50,160)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]-50+160\leq 200[/tex]
[tex]110\leq 200[/tex] ----> is true
The point is a solution for the inequality, but is not a viable solution because the number of acres can not be a negative number
case B) (80,160)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]80+160\leq 200[/tex]
[tex]240\leq 200[/tex] ----> is not true
therefore
The point is not a solution
case C) (75,-200)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]75-200\leq 200[/tex]
[tex]-125\leq 200[/tex] ----> is true
The point is a solution for the inequality, but is not a viable solution because the number of acres can not be a negative number
case D) (60,135)
substitute the value of x and the value of y in the inequality and then compare the result
[tex]60+135\leq 200[/tex]
[tex]195\leq 200[/tex] ----> is true
therefore
The point is a viable solution
What is the solution to this system of equations?
x + 2y = 4
2x + 4z = 8
3y − z= 10
A. (-6, 5, 5)
B. (2, -4, 1)
C. no solutions
D. (1, 8, -2)
E. infinite solutions
Answers
Answer:
A. (-6, 5, 5)
Step-by-step explanation:
A suitable calculator can reduce the augmented matrix for the system.
Matthew earned $325 giving haircuts. He charged $25 for each haircut. How many haircuts did he give? PLEASE ANSWER AS SOON AS POSSIBLE (nvm i got it)
Answers
Answer: 13
Step-by-step explanation:
Divide 325 by 25
According to this partial W-2 form, How much money was paid in FICA taxes?
Answers
FICA taxes on a W-2 form include Social Security and Medicare taxes, summed up at a standard rate of 7.65% of gross wages. To know the total FICA taxes paid, one must multiply the gross wages by 7.65% based on the provided rates.
Explanation:The question is asking for the total FICA taxes paid, which includes both Social Security and Medicare taxes. The information provided indicates that the employee pays 6.2% for Social Security and 1.45% for Medicare, which are standard rates for FICA tax deductions. To calculate the total amount of FICA taxes, add these percentages together to get a combined rate of 7.65%. If you know the gross wages, you multiply this amount by 7.65% to find the total FICA tax contribution on your W-2 form. If the actual gross wages earned are unknown, the calculation cannot be completed without this figure.
It is important to note that your employer also matches the FICA tax contributions, and some economists argue that this employer contribution effectively reduces employees' wages, meaning that employees might bear the total cost of these taxes indirectly.
Please answer this multiple choice question for 27 points and brainliest!!
Answers
Answer:
A
Step-by-step explanation:
The equation will be in the form
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (0, 5) and (x₂, y₂ ) = (2.5, 0) ← 2 points on the line
m = [tex]\frac{0-5}{2.5-0}[/tex] = - 2
The line crosses the y- axis at (0, 5) ⇒ c = 5
y = - 2x + 5 or
y = 5 - 2x → A
How many terms are in the expression shown below?
2x3 - 10x2 +Z
2
3. 4. 5
Answers
Answer:
3 terms. All of the numbers are considered terms.
Step-by-step explanation:
If it asked for variables, it would be 2.
In order to be accepted into a top university, applicants must score within the top 5% on the SAT exam. Given that the test has a mean of 1000 and a standard deviation of 200, what is the lowest possible score a student needs to qualify for acceptance into the university?
Answers
Answer:
1329
Step-by-step explanation:
A suitable calculator can tell you the minimum score is 1329.
Using the concept of z-scores and normal distributions, the lowest possible SAT score a student needs to qualify for acceptance into a top university, that requires top 5% of SAT scores, is approximately 1328.
Explanation:The subject of your question falls under Mathematics, particularly statistics. The problem involves understanding SAT scores and how they apply to scholarships. The problem also asks for the understanding of z-scores and how they apply to normal distributions.
In this case, the SAT score requirement is stated to be within the top 5% of all test-takers. In terms of z-scores, this requirement matches with a z-score of approximately 1.64, since z-scores of this value represent the cut-off for the top 5% in a normal distribution.
To calculate the corresponding SAT score, we use the formula for z-score which is Z = (X - μ) / σ, where X is the SAT score, μ is the mean and σ is the standard deviation. We rearrange the formula to find X, where X = Z * σ + μ. Since we know the mean (μ = 1000), standard deviation (σ = 200), and Z (Z = 1.64), we can substitute these values to get X = 1.64 * 200 + 1000 = 1328.
Therefore, the lowest possible score a student needs to qualify for acceptance into the university is approximately 1328.
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Given the equation 6/x=3/k+2, and the constraints x ≠ 0 and k ≠-2, what is x in terms of k?
A) x=2k+4
B) x=2k+12
C) x=2k-1/4
D) x=1/4+12
Answers
Answer:
A) 2k+4
Step-by-step explanation:
To solve for terms of x, we need to. she sure that x is on its own side while every other term is on the other side.
To start, we will cross multiply. If we have a/b=c/d, then ad=bc
We can perform the same calculation here,
6/x=3/(k+2)
6(k+2)=3x
Divide both sides by 3
2(k+2)=x
x=2k+4
The number –1,000 can be used to indicate a(n) A. increase in inventory of 1,000 units. B. harvest of 1,000 bushels of wheat. C. withdrawal of $1,000 from a checking account. D. receipt of $1,000.
Answers
Answer:
C. withdrawl of $1000 from a checking account
Step-by-step explanation:
When you withdraw money from a checking account, you lose the amount of money which you withdraw. In this situation, you are withdrawing $1000, which means you have -1000 dollars in your checking account
Please mark for Brainliest!! :D Thanks!!
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What does the graph of an equation reveal about the solutions to the equation? How do you solve a system of equations approximately using graphs and tables?
Answers
Final answer:
The graph of an equation reveals all possible solutions, and a system of equations can be solved graphically by finding the intersection points of their lines. Changing the slope or intercept allows manipulation of these lines, and growth rates show the rate of increase as perce
Explanation:
The graph of an equation illustrates the relationship between two variables and reveals the set of all possible solutions that satisfy the equation. When graphing a line, the slope dictates the direction and steepness of the line, while the intercept indicates where the line crosses the y-axis. To manipulate a line on a graph, you can change the slope or the intercept, altering its position and angle.
To solve a system of equations using graphs, plot each equation on the same set of axes. The point where the lines intersect represents the solution to the system, which is the set of values that satisfies all equations simultaneously. If using tables, input different values for one variable and solve for the other to find ordered pairs. When the pairs from different equations match, you've found an approximate solution. A growth rate can be determined by calculating the percentage change and can be interpreted as the rate at which a quantity increases over time.
Learning how to read and manipulate a graph is critical to interpret and express equations visually. Graphs not only convey solutions but also the behavior of equations, such as trends and growth rates, which are essential in understanding and communicating mathematical concepts effectively.