The solution is [tex](1,1)[/tex]
Step-by-step explanation:
The system of linear equation is
[tex]y=-3x+4[/tex] and [tex]y+\frac{1}{3} y=\frac{4}{3}[/tex]
Using substitution method, let us substitute [tex]y=-3x+4[/tex] in the equation
[tex]y+\frac{1}{3} y=\frac{4}{3}[/tex]
[tex]-3 x+4+\frac{1}{3}(-3 x+4)=\frac{4}{3}[/tex]
Multiplying the term [tex]-3x+4[/tex] by [tex]\frac{1}{3}[/tex], we get,
[tex]-3 x+4-x+\frac{4}{3}=\frac{4}{3}[/tex]
Subtracting both sides by [tex]\frac{4}{3}[/tex],
[tex]-3x+4-x=0[/tex]
Simplifying, we get,
[tex]\begin{aligned}-4 x+4 &=0 \\-4 x &=-4 \\x &=1\end{aligned}[/tex]
Now, substitute [tex]x=1[/tex] in [tex]y=-3x+4[/tex]
[tex]y=-3(1)+4\\y=-3+4\\y=1[/tex]
Thus, the solution is [tex](1,1)[/tex]
Answer:
y = -3x + 4
x + 1/3y = 4/3
Substitute the first expression into the second equation for y:
x + 1/3(-3x + 4) = 4/3
x - x + 4/3 = 4/3
4/3 = 4/3
Thus, all real numbers are solutions.
Step-by-step explanation:
HOPE THIS HELPS, GOOD-LCK, and Brainliest, also let me know if i am wrong.
Is 3 a factor of 81? Use divisibility rules to explain
Answer:
Step-by-step explanation:
yes it is a factor because 31 is a factor and if u add there is a one
Final answer:
Yes, 3 is a factor of 81 because the sum of 81's digits (8 and 1) is 9, which is divisible by 3. Also, since 81 is 3 raised to the power of 4, 3 is repeatedly a factor of 81.
Explanation:
To determine if 3 is a factor of 81, we can use the concept of divisibility. A number is divisible by 3 if the sum of its digits is divisible by 3. The number 81 only consists of the digits 8 and 1, and when added together (8 + 1), they equal 9. Since 9 is divisible by 3, so is 81. Therefore, 3 is indeed a factor of 81.
Additionally, the three-factor product rule can confirm this, as combining the factors 3, 3, and 3 (since 3 x 3 x 3 = 27, and 27 x 3 = 81), you end up with 81, which proves that 3 is a factor of 81.
darren got 42 out of the 56 question correct on the test. what percent of the questions did he get correct? please help ....
Answer:
75%
Step-by-step explanation:
42/56 = [tex]\frac{42}{56}[/tex]
Convert to decimal form:
42÷56 = 0.75
= 75%
Darren got 75% of the questions correct on his test, as 42 out of 56 simplifies to 3/4 or 75%. This percentage might place him in a higher percentile, indicating good performance.
Explanation:Darren scored 42 out of 56 questions correctly on his test. To find what percent of the questions he got correct, you set up a fraction of correct answers over the total number of questions and then convert it to a percentage. The fraction for Darren's test is 42/56, which simplifies to 3/4 or 0.75 when divided. To convert this to a percentage, you multiply by 100, giving Darren a score of 75% correct on his test.
Some students might be interested in how this compares to percentile ranks. For example, if the 70th percentile for a test is 16 correct answers out of 20, this means that 70% of the students answered 16 or fewer questions correctly. Darren's score of 75% would likely place him in a higher percentile since he got a larger proportion of the questions correct, which is generally considered good.
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A student has a monthly budget of $120 to spend on either burritos, which cost $6 each, or sodas, which cost $4 each. What is the largest number of burritos that a student could afford to purchase in one month ?
Answer:
Step-by-step explanation:
he has 120 bucks...burritos are 6 and sodas are 4
the largest number of burritos a person can afford to purchase....
120/6 = 20....he can purchase 20 burritos....but then he doesn't get any soda because thats gonna take all his money
The largest number of burritos a person can afford to purchase is 20.
What is an expression?
Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication and division.
It is given that:-
He has 120 bucks...burritos are 6 and sodas are 4The largest number of burritos a person can afford to purchase:-
120 / 6 = 20.
He can purchase 20 burritos....but then he doesn't get any soda because that's gonna take all his money.
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f(-2)=2x-3 ?
lol help me please
This equation means what would f(x) be if x was -2. F(x) is another way to write y. So this equation can be rewritten as y = 2(-2) - 3.
y = -4 - 3
y = -7
Answer:
2(-2)-3 =-4-3 =-7
Step-by-step explanation:
the 17th term of an A.P is 13 times the second term.prove that the twelve term is five term the third term.
Answer:
T_12 = 5(T_3)
Step-by-step explanation
T_17 = 13(T_2)
T_n = a + (n-1)d; where a=first term and d=common difference
T_17 = a + 16d
T_2 = a + d
Therefore,
a + 16d = 13(a + d)
a + 16d = 13a + 13d
16d - 13d = 13a - a
3d = 12a
d = 4a
To prove that T_12 = 5(T_3)
a + 11d = 5(a + 2d)
a + 11d = 5a + 10d
since d = 4a
a + 11(4a) = 5a + 10(4a)
a + 44a = 5a + 40a
45a = 45a
Since LHS = RHS
Therefore; T_12 = 5(T_3)
Show how to solve the problem mentally.
for each sequence, find the first 4 term and the 10th term 2nsquard (algbra) pls can sm1 answer me plis
Answer:
see explanation
Step-by-step explanation:
To find the terms of the sequence substitute n = 1, 2, 3, 4, 10 into the rule.
n = 1 → 2(1)² = 2(1) = 2
n = 2 → 2(2)² = 2(4) = 8
n = 3 → 2(3)² = 2(9) = 18
n = 4 → 2(4)² = 2(16) = 32
n = 10 → 2(10)² = 2(100) = 200
The first 4 terms are 1, 8, 18, 32 and the 10 th term is 200
Answer:
2, 8, 18, 32.
10th term = 200.
Step-by-step explanation:
The first 4 terms of 2n^2 are
2(1)^2, 2(2)^2, 2(3)^2, 2(4)^2
= 2, 2*4, 2*9, 2*16
= 2, 8, 18, 32.
The 10th term = 2(10)^2
= 200.
A. (5, 7) and (-4,-2)
find the slope of a line
Answer: m = 1
Step-by-step explanation: Use the slope formula to find the slope m.
Hope this helps you out! ☺
Answer:
m = 1
Step-by-step explanation:
refer to attached picture for reference
we are given 2 points
(x₁, y₁) = (5,7)
(x₂,y₂) = (-4,-2)
simply substitute these into the given equation for slope
m = (y₂ - y₁)/(x₂-x₁)
= (-2 -7) / (-4 - 5)
= -9 / -9
= 1
3. You can ride your bike 1/5 of a mile per minute. If it takes you 3 1/3 minutes to get
to your friend's house, how many miles away does your friend live?
Answer:
(1/5)(10/3) = 10/15 = 2/3 mile
The friend lives [tex]$\frac{2}{5}$[/tex] of a mile away.
To solve this problem, we need to calculate the total distance based on the speed at which the bike is ridden and the time it takes to reach the friend's house.
First, we determine the speed at which the bike is ridden, which is given as [tex]$\frac{1}{5}$[/tex] of a mile per minute.
Next, we calculate the time it takes to reach the friend's house, which is [tex]$3\frac{1}{3}$[/tex] minutes. To work with this mixed number easily, we convert it to an improper fraction. [tex]$3\frac{1}{3}$[/tex] minutes is the same as [tex]$\frac{10}{3}$[/tex] minutes because [tex]$3\frac{1}{3} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3}$[/tex].
Now, we can find the total distance by multiplying the speed by the time:
[tex]\[ \text{Distance} = \text{Speed} \times \text{Time} \][/tex]
[tex]\[ \text{Distance} = \frac{1}{5} \text{ mile/minute} \times \frac{10}{3} \text{ minutes} \][/tex]
When we multiply these two fractions, we multiply the numerators together and the denominators together:
[tex]\[ \text{Distance} = \frac{1}{5} \times \frac{10}{3} \][/tex]
[tex]\[ \text{Distance} = \frac{1 \times 10}{5 \times 3} \][/tex]
[tex]\[ \text{Distance} = \frac{10}{15} \][/tex]
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[ \text{Distance} = \frac{10 \div 5}{15 \div 5} \][/tex]
[tex]\[ \text{Distance} = \frac{2}{3} \][/tex]
Therefore, the friend lives [tex]$\frac{2}{5}$[/tex] of a mile away.
Melvin walked 5/8 mile to the library. He then walked 3/10 mile from the library to the store. About how far did Melvin walk?
Answer:
37/40 = 0.925 miles
Step-by-step explanation:
The total distance Melvin walked (in miles) was ...
5/8 + 3/10 = 25/40 + 12/40 = 37/40 . . . . miles
__
In decimal, the sum is ...
0.625 + 0.30 = 0.925 . . . . miles
I’m confused on what’s going on here, so if someone could explain it I would be super grateful.
Given AB = 4x + 5, BC = 3x - 2 and AC 9x -5
The sum of the length of any two sides of a triangle is greater than the length of third side.
Case I
AB + BC > AC
⇔4x +5 +3x -2 > 9x -5
⇔7x + 3 >9x - 5
⇔ 7x -9x > -5 -3
⇔-2x > -8
⇔x <4
Case II
if AB+AC > BC
4x +5 +9x -5> 3x -2
⇔13x >3x -2
⇔ 13x -3x> -2
⇔10x > -2
⇔x >[tex]-\frac{1}{5}[/tex]
Case III
AC+BC > AB
⇔9x - 5 +3x - 2 > 4x +5
⇔6x -7 > 4x +5
⇔6x -4x >5+7
⇔2x > 12
⇔x > 6
If you want to make 2 1/2 Batches of cookies and you need one and 1 2/3 cups of flour per cookie batch how much flower do you need
Answer:
you need 4 1/3 cups of flower
(6x - 2y - 5)-(-5 9y - 8x)
At winstead elementary school 10.2% of the 88 boys have blue eyes and 5.3% of the 75 girls have blue eyes. What percent of the school as a whole had blue eyed students
Answer: ≈ 8%
Step-by-step explanation:
Total number of students = total number of boys + total number of girls
Total number of students = 88 + 75
Total number of students = 163
10.2% of the boys have blue eyes , this means that 10.2% of 88 have blue eyes
[tex]\frac{10.2}{100}[/tex] x 88 = 8.976
Since we are dealing with humans , therefore , this means that 9 boys have blue eyes.
Also , 5.3% of the girls have blue eyes means that 5.3% of 75 have blue eyes. That is
[tex]\frac{5.3}{100}[/tex] x 75 = 3.975 , therefore : 4 girls have blue eyes.
In total , 13 students have blue eyes
Therefore , to calculate the % of students that have blue eyes , we have :
13/163 x 100 = 7.975460123
Therefore : ≈ 8% of the students have blue eyes
3 tools federal reserve use to enact monetary policies
The three important tools of Federal Reserve's monetary policies are as follows:
open market operationsthe discount ratereserve requirements.Step-by-step explanation:
The monetary policies of the United States's central bank, Federal Reserve are the acts of the entity to influence money and raise the country's economy. These policies also helps in looking over the aspects of how the money and credits draw affects on credit rates and the overall performance of the U.S. Economy.
The three prime tools of the Federal reserve's monetary policies are the Open Market Operations, Discount Rates and the Reserve Requirements.
Open Market operations
This involves in purchase and selling process of government securities. The primary dealer with which the Reserve deals compete on the basis of prices and thus the dealer gets decided with whom the reserve deal for the day.
Discount Rates
This is the discount rate charged to depository institutions for short term loans by the Federal Reserve.
Reserve Requirements
This is the money or deposit amount the Reserve Bank must sustain in its vault or depository.
Place the following steps in order to complete the square and solve the quadratic equation x2-6x+7=0
Answer:
The solved expression is [tex]x=3+\sqrt{2}[/tex] and [tex]x=3-\sqrt{2}[/tex]
Therefore [tex]x=3\pm\sqrt{2}[/tex]
Step-by-step explanation:
Given quadratic equation is [tex]x^2-6x+7=0[/tex]
To solve the given equation by using completing the square :
[tex]x^2-6x+7=0[/tex]
Rewritting the above equation as below :
[tex]x^2-6x+7+2-2=0[/tex]
[tex](x^2-6x+7+2)-2=0[/tex]
[tex](x^2-6x+9)-2=0[/tex]
[tex](x^2-6x+3^2)-2=0[/tex]
[tex](x^2-2(x)(3)+3^2)-2=0[/tex] ( it is of the form of [tex](a-b)^2=a^2-2ab+b^2[/tex] heere a=x and b=3 )
[tex](x-3)^2-2=0[/tex]
[tex](x-3)^2=2[/tex]
Taking square root on both sides we get
[tex]\sqrt{(x-3)^2}=\pm\sqrt{2}[/tex]
[tex]x-3=\pm\sqrt{2}[/tex]
[tex]x=\pm\sqrt{2}+3[/tex]
[tex]x=3\pm\sqrt{2}[/tex]
Therefore [tex]x=3+\sqrt{2}[/tex] and [tex]x=3-\sqrt{2}[/tex]
triangle ABC has AB=5, BC=7, and AC=9. D is on AC with BD=5. Find the length of DC
Answer:
[tex]DC=\frac{8}{3}\ units[/tex]
Step-by-step explanation:
The picture of the question in the attached figure
we know that
The triangle ABD is an isosceles triangle
because
AB=BD
The segment BM is a perpendicular bisector segment AD
so
In the right triangle ABM
Applying the Pythagorean Theorem
[tex]BM^2=AB^2-AM^2[/tex]
we have
[tex]AB=5\ units\\AM=x\ units[/tex]
substitute
[tex]BM^2=5^2-x^2[/tex]
[tex]BM^2=25-x^2[/tex] -----> equation A
In the right triangle BMC
Applying the Pythagorean Theorem
[tex]BM^2=BC^2-MC^2[/tex]
we have
[tex]BC=7\ units\\MC=AC-AM=(9-x)\ units[/tex]
substitute
[tex]BM^2=7^2-(9-x)^2[/tex]
[tex]BM^2=49-(81-18x+x^2)[/tex]
[tex]BM^2=49-81+18x-x^2[/tex]
[tex]BM^2=-x^2+18x-32[/tex] ----> equation B
equate equation A and equation B
[tex]-x^2+18x-32=25-x^2[/tex]
solve for x
[tex]18x=25+32\\18x=57\\\\x=\frac{57}{18}[/tex]
Simplify
[tex]x=\frac{19}{6}[/tex]
Find the length of DC
[tex]DC=AC-2x[/tex]
substitute the given values
[tex]DC=9-2(\frac{19}{6})[/tex]
[tex]DC=9-\frac{19}{3}\\\\DC=\frac{8}{3}\ units[/tex]
Can someone help me solve these systems using elimination? I have a hard time with these.
8x-6y=-20
-16x+17y=30
and then..
-4y-11x=36
20= -10x-10y
Please show the steps so I can learn too. Thank you!
Question 1:
For this case we must solve the following system of equations by the method of elimination:
[tex]8x-6y=-20\\-16x+17y=30[/tex]
We multiply the first equation by 2:
[tex]16x-12y = -40[/tex]
We have the equivalent system:
[tex]16x-12y = -40\\-16x + 17y = 30[/tex]
We add the equations:
[tex]5y = -10[/tex]
We divide between 5 on both sides of the equation:
[tex]y = - \frac {10} {5}\\y = -2[/tex]
We find the value of the variable "x":
[tex]8x-6 (-2) = - 20\\8x + 12 = -20\\8x = -20-12\\8x = -32[/tex]
We divide by 8 on both sides of the equation:
[tex]x = - \frac {32} {8}\\x = -4[/tex]
Thus, the solution of the system is:
[tex](x, y): (- 4, -2)[/tex]
Answer:
[tex](x, y): (- 4, -2)[/tex]
Question 2:
For this case we must solve the following system of equations by the method of elimination:
[tex]-4y-11x = 36\\20 = -10x-10y[/tex]
We multiply the first equation by 10:
[tex]-40y-110x = 360[/tex]
We multiply the second equation by -4:
[tex]40y + 40x = -80[/tex]
We have the equivalent system:
[tex]-40y-110x = 360\\40y + 40x = -80[/tex]
We add the equations:[tex]-70x = 280[/tex]
We divide by -70 on both sides of the equation:
[tex]x = \frac {280} {- 70}\\x = -4[/tex]
We find the value of the variable "y":
[tex]-4y-11 (-4) = 36\\-4y + 44 = 36\\-4y = 36-44\\-4y = -8[/tex]
We divide between -4 on both sides of the equation:
[tex]y = - \frac {8} {- 4}\\y = 2[/tex]
Thus, the solution of the system is:
[tex](x, y): (- 4,2)[/tex]
Answer:
[tex](x, y): (- 4,2)[/tex]
What is the area of this parallelogram?
A.
2.9 cm2
B.
5.8 cm2
C.
8.7 cm2
D.
6.0 cm2
Answer:
5.8
Step-by-step explanation:
How would you write out 3x6+7x4=46 to make it correct?
Answer:
3x6+7x4=46 it’s correct
we can rewrite it this way : (3x6)+(7x4)=46
Which shows the correct first step to solving the system of equations in the most efficient manner ? 3x+2y=17 x+4y=19
The correct answer is option 1: x = -4y+19
Step-by-step explanation:
Given equations are:
[tex]3x+2y=17\ \ \ Eqn1\\x+4y=19\ \ \ \ Eqn2[/tex]
When we have to solve a system of equations using substitution method we look for a variable in both of equations that is without any co-efficient as it is easy to isolate that variable in the equation and then using substitution method to solve it
We can see that in equation 2, x is without any co-efficient
So,
[tex]x+4y = 19[/tex]
Subtracting 4y from both sides
[tex]x+4y-4y = 19-4y\\x = 19-4y\\[/tex]
Can also be written as:
[tex]x = -4y+19[/tex]
Hence,
The correct answer is option 1: x = -4y+19
Keywords: Linear equations, variables
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Answer:
The correct answer is option 1: x = -4y+19
Step-by-step explanation:
The equation of a line is x + 3y = 14. What is the y-intercept of the line?
Answer:
First of all, before writing the y-intercept you need to make sure that the equation is in slope intercept form.
x + 3y = 14
3y = 14 - x
y = 14/3 -x/3
y= -x/3 + 14/3
14/3 is the y-intercept
The y-intercept of the line with equation x + 3y = 14 is 4.67.
Explanation:The equation of the line is in the form x + 3y = 14. To find the y-intercept, we need to set x = 0 and solve for y. Plugging in x = 0, we get 0 + 3y = 14. Simplifying the equation gives us 3y = 14. Dividing both sides by 3, we find that y = 4.67. Therefore, the y-intercept of the line is 4.67.
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A sales person is paid a weekly salary of $760, plus a commission of 7.5% of all her sales for the week. Express her weekly income in terms of the weekly sales amount. Is this a function? How do you know? What is an appropriate domain?
Answer:
1. We can see that salesperson's weekly income is the sum of her constant weekly salary ($760) and a commission which is variable and depends on her weekly sales.
So, if we say that y is her weekly income and x is her weekly sales, we can write this as:
y = 760 + 0.075x
Note that we had to change percentage to decimal number dividing it by 100.
2 Since for each value of x there is only one corresponding value of y, we can say that this is a function. For any value of x we input there is only one solution we get - that is the main feature of function and a way to tell if something is really a function.
Since this is a function, it can also be written as:
f(x) = 760 + 0.075x
3. Domain of a function is, basically, set of all values of x for which the function can work. That practically means that, since x is weekly sale, it can not be negative (one cannot make -$500 sale, for example). However, it is possible that she doesn't make a sale one week, making it possible for x to be 0. Also, the value of her sales doesn't have to be integer (it is quite possible that she makes $673.50 sale).
All this means that appropriate domain for this function are positive real numbers including 0.
There are 1,657 souvenir paperweights that need to be packed in boxes. Each box will hold 17
paperweights. How many boxes will be needed?
boxes will be needed to hold all the souvenir paperweights.
Answer:
Therefore we can say that 98 boxes will be needed to hold all the paperweights.
Step-by-step explanation:
i) there are 1657 souvenir paperweights that need to be packed in boxes.
ii) each box will hold 17 paperweights
iii) therefore the number of boxes that will be needed are
= [tex]\dfrac{total\hspace{0.15cm} number\hspace{0.15cm} souvenir\hspace{0.15cm} paperweights}{number\hspace{0.15cm} of\hspace{0.15cm} paperweights\hspace{0.15cm} per\hspace{0.15cm} box}[/tex] = [tex]\dfrac{1657}{17}[/tex] = 97.471
iv) Therefore we can say that 98 boxes will be needed to hold all the paperweights.
At any given football game that Burbank plays , the probability that Patrick plays during the first half of the game is 5/6 . he plays during the first half, there is only a 3/7 chance that he'll play again during the second half. (He gets tired if he plays all out in the first half.) If he doesn't play during the first half, there is a 5/ 7 chance that he'll play during the second half . Find the probability that Patrick played in the first half, given that he played in the second half.
Answer:
the probability answer will be 8/49.
Step-by-step explanation:
the answer probability would be equal to multiplication of the odd cases of the given probabilities.Therefore,for him to playin both the halves of the game would be 8/49.
Using Bayesian probability, we find that the probability of Patrick playing in the first half given that he played in the second half is 3/4 or 75%.
This was determined by calculating intermediate probabilities and applying Bayes' theorem.
Let's define the following events:F: Patrick plays during the first half.S: Patrick plays during the second half.We are given the probabilities:P(F) = 5/6P(S|F) = 3/7P(S|FC) = 5/7, where FC is the complement of event F (Patrick does not play in the first half).We need to find P(F|S), the probability that Patrick played in the first half given that he played in the second half, using Bayes' theorem.Bayes' theorem states:Therefore, the probability that Patrick played in the first half given that he played in the second half is 3/4 or 75%.
The middle school golf team is selling golf balls to raise $850 for new equipment. They sell golf balls for two dollars each. A case has 24 packs of golf balls. Each pack has six golf balls. How many cases of golf balls does the team need to sell to raise enough money for new equipment.
Answer:
3 cases
Step-by-step explanation:
Total Money Needed = $850
Each Golf Ball = $2
Total Balls needed = 850/2 = 425 balls
Case = 24 Pack
Pack = 6 balls
So a Case has:
24 * 6 = 144 balls
They need 425 balls to sell and each case has 144 balls, so number of case needed would be:
425/144 = 2.95
We can't have 2.95 cases, so we round up to 3 cases
At the yard sale,3\4 of items for sale were toys and 5\6 of the items for sale were books Were there more toys of books for sale explain
Answer:
No, the books had more sales by 5/6 being bigger (Greater Than) than 3/4.
By converting both fractions to have a common denominator of 12, it becomes clear that there were more books than toys for sale, as 10/12 (books) is greater than 9/12 (toys).
To determine whether there were more toys or books for sale, we need to compare the fractions ⅓ (representing toys) and ⅗ (representing books). To compare these fractions, they must have a common denominator. The least common denominator for 4 and 6 is 12. Converting both fractions to have the denominator of 12, you would get:
Toys: ⅓ = 3/4 = 9/12Books: ⅗ = 5/6 = 10/12When comparing 9/12 (toys) to 10/12 (books), it is clear that there were more books than toys for sale since 10/12 is greater than 9/12.
Find the slope determined by the following pair of points
(19/4,2) , (1,3/4)
Answer:
1/3
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(3/4-2)/(1-19/4)
m=(3/4-8/4)/(4/4-19/4)
m=(-5/4)/(-15/4)
m=(-5/4)(-4/15)
m=20/60
m=2/6
m=1/3
Answer:
1/3
Step-by-step explanation:
Will ran the diagonal distance across a square field measuring 40 yards on each side. James ran the diagonal distance across a rectangular field with a length of 25 yards and a width of 35 yards. Will ran a longer distance. How much longer did he run?
Step-by-step explanation:
Given Will ran the diagonal distance across a square field measuring 40 yards each sides .
∴The distance covered by Will was =[tex]\sqrt{40^{2}+40^2 }[/tex] yards
= [tex]40\sqrt{2}[/tex] yards=56.56 yards
Again James ran diagonal distance across a rectangle field with a length of 25 yards and a width of 35 yards.
So,distance covered by James was =[tex]\sqrt{25^2+35^2}[/tex] yards
=[tex]5\sqrt{74}[/tex] yards = 43.01 yards
∴Will ran longer distance.
∴Will ran 13.55 yards more than James.
To calculate the longer distance Will ran compared to James, find the diagonal distances each ran. Will ran approximately 13.56 yards longer.
To determine how much longer Will ran compared to James, we need to calculate the diagonal distances for each run.
For the square field, diagonal = 40√2 ≈ 56.57 yards.The diagonal for the rectangular field = √(25² + 35²) = √(625 + 1225) = √1850 ≈ 43.01 yards.Will ran 56.57 - 43.01 ≈ 13.56 yards longer than James. Thus, Will ran around 13.56 yards longer than James, as determined by subtracting the diagonal of the rectangular field from that of the square field. This calculation offers a quantitative measure of the difference in running distances, providing insight into their comparative performances.
Solve x^2=2x+3 by factoring
The solution is x = 1 and x = -3
Solution:
Given that we have to solve the given equation by factoring
Given equation is:
[tex]x^2=2x + 3\\\\x^2 - 2x - 3 = 0[/tex]
[tex]\text{Consider the form } x^2+bx+c[/tex]
Find a pair of integers whose product is c and and whose sum is b
[tex]\text{Compare } x^2-2x-3 = 0 \text{ with } x^2+bx +c\\\\\text{We get } b = -2 \text{ and } c = -3[/tex]
Now find, a pair of integers whose sum is -2 and product is -3
The integers that satisfies this condition is -1 and 3
When we add - 1 and 3 we get 2
When multiply -1 and 3 we get -3
Thus the pair of integers are -1 and 3
Write the factored form using these integers.
[tex](x-1)(x+3) = 0[/tex]
The Zero Product Property states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0
Set the factors equal to 0
[tex]x - 1 = 0 \text{ and } x + 3 = 0[/tex]
x = 1 and x = -3
Thus the solution is x = 1 and x = -3