N8) solve the system using substitution method and then graph the equations.2x - 4y = -23x + 2y = 3-
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Solution
Given:
2x - 4y = -2
3x + 2y = 3
Substitution method
[tex]\begin{gathered} From\text{ 3x+2y=3} \\ 3x=3-2y \\ x=\frac{3-2y}{3} \end{gathered}[/tex][tex]\begin{gathered} Substitute\text{ }x=\frac{3-2y}{3}\text{ into the first equation} \\ 2x-4y=-2 \\ 2(\frac{3-2y}{3})-4y=-2 \\ \frac{6-4y}{3}-4y=-2 \\ Multiply\text{ }trough\text{ by 3} \\ 6-4y-12y=-6 \\ 6-16y=-6 \\ -16y=-6-6 \\ -16y=-12 \\ y=\frac{-12}{-16} \\ y=\frac{3}{4} \end{gathered}[/tex][tex]\begin{gathered} Substitute\text{ y=}\frac{3}{4}\text{ into }x=\frac{3-2y}{3} \\ x=\frac{3-2(\frac{3}{4})}{3}=\frac{3-\frac{3}{2}}{3}=\frac{\frac{6-3}{2}}{3}=\frac{\frac{3}{2}}{3} \\ x=\frac{3}{6} \\ x=\frac{1}{2} \end{gathered}[/tex][tex]Thus,\text{ x=}\frac{1}{2},y=\frac{3}{4}[/tex]Graphical method:
Plot the graph of the two equations on the same graph
The point of intersection of the two graphs gives the solution to the system of equations
The point of intersection is (0.5, 0.75)
Which in fraction gives (1/2, 3/4)
Thus. x = 1/2, y= 3/4
Related Questions
Determine which of the following are true statements. Check all that apply.
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Substitute in each inequality the given corresponding solution (x,y) and prove if it makes a true math expression:
1.
[tex]\begin{gathered} -5x-9y\ge60 \\ (-3,-5) \\ \\ -5(-3)-9(-5)\ge60 \\ 15+45\ge60 \\ 60\ge60 \end{gathered}[/tex]As 60 is greater than or equal to 60, (-3,-5) is a solution for the inequality.2.
[tex]\begin{gathered} 4x-3y>1 \\ (5,7) \\ \\ 4(5)-3(7)>1 \\ 20-21>1 \\ -1>1 \end{gathered}[/tex]As -1 isn't greater than 1, (5,7) is not a solution for the inequality3.
[tex]\begin{gathered} -10x+8y<12 \\ (-9,-10) \\ \\ -10(-9)+8(-10)<12 \\ 90-80<12 \\ 10<12 \end{gathered}[/tex]As 10 is less than 12, (-9,-10) is a solution for the inequality.4.
[tex]\begin{gathered} 9x+7y\le98 \\ (9,3) \\ \\ 9(9)+7(3)\le98 \\ 81+21\le98 \\ 102\le98 \end{gathered}[/tex]As 102 is not less than or equal to 98, (9,3) is not a solution for the inequalityIf the inflation has been 2.7%, how much more do you have to pay this year foran item that cost $11.50 last year?
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Given data:
The cost of the item is $11.50.
The inflation percentage is 2.7%.
Increase in the price is,
[tex]\begin{gathered} =11.50\times(\frac{2.7}{100}_{}) \\ =11.50\times0.027 \\ =0.3105 \end{gathered}[/tex]Total amount to be paid last year,
[tex]\begin{gathered} =11.50+0.3105 \\ =11.8105 \end{gathered}[/tex]Therefore you will have to pay $ 0.3105 more.
A car can travel 43/1/2 miles on 1/1/4 gallons of gas. What is the unit rate for miler per gallon
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The unit rate for the car is 34.8 miles per gallon.
How to get the unit rate for mile per gallon?
The unit rate will be given by the quotient between the distance traveled and the gallons of gas consumed to travel that distance.
Here we know that the car travels 43 and 1/2 miles on 1 and 1/4 gallons of gas, then the quotient is:
U = (43 + 1/2)/(1 + 1/4) mi/gal = (43.5)/(1.25) mi/gal = 34.8mi/gal
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7. The cylinder shown has a radius of 3inches. The height is three times the radiusFind the volume of the cylinder. Round yoursolution to the nearest tenth.
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Answer:
250 cubic inches
Explanation:
Given that:
Radius of the cylinder, = 3 in.
Height of the cylinder = 3r
= 3(3)
=9 in.
The formula to find the volume of a cylinder is
[tex]V=\pi r^2h[/tex]Plug the given values into the formula.
[tex]\begin{gathered} V=\pi3^29 \\ =81\pi \\ =254.469 \end{gathered}[/tex]Rounding to nearest tenth gives 250 cubic inches, which is the required volume of the cylinder.
use the graph to find the following A) find the slope of the lineB) is the line increasing or decreasingC) estimate the vertical intercept(x y)=
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The Solution.
To find the slope of the line from the given graph:
First, we shall pick two coordinates in the graph, that is
[tex](0,2),(2,-1)[/tex]This implies that
[tex]\begin{gathered} (x_1=0,y_1=2)\text{ and} \\ (x_2=2,y_2=-1) \end{gathered}[/tex]By formula, the slope is given as below:
[tex]\text{ slope=}\frac{y_2-y_1}{x_2-x_1}[/tex]substituting the values in the above formula, we get
[tex]\begin{gathered} \text{ Slope=}\frac{-1-2}{2-0} \\ \\ \text{ Slope =}\frac{-3}{2} \end{gathered}[/tex]So, the slope of the line is -3/2
b. From the graph, and from the slope being a negative value, it is clear that the line graph is Decreasing.
c. To estimate the vertical intercept is to find the y-intercept of the line.
Clearly from the graph, we can see that the vertical intercept is (0,2), that is, the point where the line cut the y-axis.
Therefore, the vertical intercept is (0,2).
whic fracción is equivalente to 8/10
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Given data
*The given fraction is 8/10
[tex]\frac{80}{100}=\frac{8}{10}[/tex]80/100 is the fraction equivalent to 8/10
-10 x is greater than or equal to 2x
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We need to solve the following inequality:
[tex]-10\text{ }\ge2x[/tex]We need to isolate the "x" variable, when we do that the number that is multiplying it should go to the other side with its inverse operation, which is division.
[tex]\begin{gathered} \frac{-10}{2}\ge x \\ -5\ge x \end{gathered}[/tex]Now we need to flip the expression, so the variable is isolated on the left side. Since this is an inequality we also need to flip the inequality signal. This is done below.
[tex]x\leq-5[/tex]For the expression to be true x must be less or equal to -5.
Find the distance between the two points. 15.) (-3, -1), (-1, -5)-use Pythagorean Throrem
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Answer:
4.5 units
Explanation:
First, we need to draw the points (-3, -1) and (-1, -5) as follows
Therefore, the distance between the points is the length of the yellow line. This distance is the hypotenuse of a triangle with legs a and b.
The length of a is 2 and the length of b is 4
Then, using the Pythagorean theorem, we can calculate the length of c as follow
[tex]\begin{gathered} c^2=a^2+b^2 \\ c^2=2^2+4^2 \\ c^2=4+16 \\ c^2=20 \end{gathered}[/tex]So, using the calculator, we get that the value of c is equal to
[tex]\begin{gathered} \sqrt{c^2}=\sqrt{20} \\ c=\sqrt{20} \end{gathered}[/tex]To find an approximate value for c, we will use the following:
We know that √16 = 4 and √25 = 5
Since 20 is between 16 and 25, the square root of 20 is a number between 4 and 5, so we can approximate it to 4.5.
Therefore,
c = 4.5
Lisa is a software saleswoman. Let y represent her total pay (in dollars). Let “x”represent the number of copies of History is Fun she sells. Suppose that “x”and “y”are related by the equation 90x + 2200 =v.Answer the questions below.Note that a change can be an increase or a decrease.For an increase, use a positive number. For a decrease, use a negative number.-Picture includes the questions-
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Explanation
Part A
Given that x and y are related by the by the equation 90x + 2200 =y. We can find the change by comparing the formula with the original equation of a line.
[tex]\begin{gathered} y=mx+c \\ where\text{ m is the slope\lparen change in y for x\rparen} \end{gathered}[/tex]Comparing the above with the given question, m becomes change in Lisa's pay for each copy of history is fun. Therefore,
Answer: 90 dollars
Part B
If Lisa does not sell any copies of history is fun, therefore, the value of x becomes zero. We can then have the total pay as
[tex]\begin{gathered} y=90(0)+2200 \\ y=2200 \end{gathered}[/tex]Answer: 2200 dollars
Shown in the equation are the steps a student took to solve the simple interest formula A=P(1+rt) for r
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Given:
We're given the steps a student took to solve the simple interest formula.
To find:
The algebraic error in student's work.
Step-by-step solution:
Let us first solve the equation and then we will spot the error in the solution:
A = P(1 + rt)
A = p + prt
A - p = prt
A - p / pt = r
Upon comparing both solutions, we can clearly see that the student made a mistake in the second step in the multiplication process.
The student should write A = p + prt in the second step in place of
A = p + rt, because p is multiplied with the whole bracket.
Can someone help me with 7? I’m desperate
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Which expression is equivalent to cot2B(1 – cos-B) for all values of ß for which cot2B(1 - cos2B) is defined?
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From the Pythagorean identity,
[tex]\sin ^2\beta+\cos ^2\beta=1[/tex]we have
[tex]\sin ^2\beta=1-\cos ^2\beta[/tex]Then, the given expression can be rewritten as
[tex]\cot ^2\beta\sin ^2\beta\ldots(a)[/tex]On the other hand, we know that
[tex]\begin{gathered} \cot \beta=\frac{\cos\beta}{\sin\beta} \\ \text{then} \\ \cot ^2\beta=\frac{\cos^2\beta}{\sin^2\beta} \end{gathered}[/tex]Then, by substituting this result into equation (a), we get
[tex]\begin{gathered} \frac{\cos^2\beta}{\sin^2\beta}\sin ^2\beta \\ \frac{\cos ^2\beta\times\sin ^2\beta}{\sin ^2\beta} \end{gathered}[/tex]so by canceling out the squared sine, we get
[tex]\cos ^2\beta[/tex]Therefore, the answer is the last option
Fido ran away from home at a speed of 5 mi/hour. He ran in a straight line. After a while he decided to go back home for dinner so turned around and walked home along the same path he had run on. He walked at 2 mi/hour. The walk home took one hour longer than the run did. How long did Fido run?
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Distance = Speed x time
For the run; speed = 5 mi/hr, time = t
For the walk: speed= 2 mi/hr, time = t + 1
Since he walked on a straight line and he returned following the same path
Distance travelled for the run = distance travelled for the walk
Distance for run: 5 x t = 5t
Distance for walk : 2 (t + 1) =2t + 2
Thus , 5t = 2t + 2
5t - 2t = 2
3t = 2
t = 2/3 hour = 2/3 x 60 minutes = 2x 20 = 40 minutes
He took him 40 minutes to run
Simplify.1,5m^7(-4m^50^2A. -6m^14B. 24m^17C. 24m^14D. 12m^17There is a picture too if you need it.
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The expression can be simplified as,
[tex]\begin{gathered} 1.5m^7(-4m^5)^2 \\ =1.5m^7(16m^{10}) \\ =24m^{17} \end{gathered}[/tex]Thus, option (b) is the correct solution.
what will the y-intercept be if the graph is proportional?
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A graph is proportional if the line intersects at the origin (0, 0)
and the y-intercept will always equal to 0
Since y-intercept is the value of y when x =0, and it passes the origin at x = 0, y= 0.
The answer is 0
help me solve the volume of the cylinder? 20 ft x 17 ft
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Remember that the formula for the volume of a cylinder is:
[tex]V=\pi r^2h[/tex]Where:
• r, is the ,radius, of the base
,• h ,is the height of the cylinder
Notice that the base has a diameter of 20 ft. Therefore, the radius is 10 ft.
Using this data and the formula, we get that:
[tex]\begin{gathered} V=\pi(10^2)(17) \\ \rightarrow V=5340.71 \end{gathered}[/tex]The volume of the cylinder is:
[tex]2540.71ft^3[/tex]how do I find the coefficient the queshtion is the expression -5p+20 factored is __
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Factor the coefficient of the expression.
[tex]-5p+20=-5(p-4)[/tex]So answer is -5(p - 4).
The slope and one point on the line are given. Find the equation of the line (in slope-intercept form).(1/4, -4) ; m = -3 y=
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Answer
y = -3x - 13/4
Step-by-step explanation
Equation of a line in slope-intercept form
[tex]y=mx+b[/tex]where m is the slope and (0, b) is the y-intercept.
Substituting into the general equation with m = -3 and the point (1/4, -4), that is, x = 1/4 and y = -4, and solving for b:
[tex]\begin{gathered} -4=(-3)\cdot\frac{1}{4}+b \\ -4=-\frac{3}{4}+b \\ -4+\frac{3}{4}=-\frac{3}{4}+b+\frac{3}{4} \\ -\frac{13}{4}=b \end{gathered}[/tex]Substituting into the general equation with m = -3 and b = -13/4, we get:
[tex]\begin{gathered} y=(-3)x+(-\frac{13}{4}) \\ y=-3x-\frac{13}{4} \end{gathered}[/tex]If f(x) = x² is vertically stretched by a factor of 18 to g(x), what is the equation of g(x)?
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We need to find the equation of the new function g(x) obtained by vertically stretching the function:
[tex]f\mleft(x\mright)=x²[/tex]Vertically stretching a function by a factor of k corresponds to multiplying the whole expression of function by k:
[tex]g(x)=k\cdot f(x)[/tex]In this problem, we have k = 18. Thus, we obtain:
[tex]g(x)=18\cdot f(x)=18x²[/tex]Answer: C. g(x) = 18x²
A deck of cards contains RED cards numbered 1,2,3 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card. Drawing the Red 1 is an example of which of the following events? Select all correct answers.
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The event Red 1 is an example of these following events:
R and O.E'.E or R.Which events are included into Red 1?Red cards are represented by the letter R, while the number 1, which is odd, is represented by the letter O.
Both events R and O happen in the, hence the event R and O is one of the possible events to this problem, as the card is both red and has an odd number.
The number is not even, hence the event E' is another one of the events in this problem.
The final event is E or R, as the card has a red number, meaning that at least one of the options E or R are satisfied.
Missing informationThe options which the event respect are missing, and are given by the image at the end of the answer.
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7 singles, 13 fives, 4 twenties, and 3 hundred dollar bills are all placed in a hat. If a player is to reach into the hat and randomly choose one bill, what is the fair price to play this game?
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So,
The probability to obtain one bill of the four, is 1/4.
So, the expected value is:
[tex]\begin{gathered} \frac{1}{4}(7)+\frac{1}{4}(13)+\frac{1}{4}(4)+\frac{1}{4}(3) \\ \\ =6.75 \end{gathered}[/tex]Therefore, the fair price to play this game is 6.75.
Writing the equation of a quadratic function given its graph
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Answer:
[tex]y=-(x-1)^2+2[/tex]Step-by-step explanation:
A quadratic function in vertex form is represented as:
[tex]\begin{gathered} y=a(x-h)^2+k \\ \text{where,} \\ (h,k)\text{ is the vertex} \end{gathered}[/tex]Given the vertex (1,2) substitute it into the function:
[tex]y=a(x-1)^2+2[/tex]As you can see, we still do not know the value for ''a'', use the point given (4,-7) substitute it (x,y) and solve for ''a'':
[tex]\begin{gathered} -7=a(4-1)^2+2 \\ -7=a(3)^2+2 \\ -7=9a+2 \\ 9a=-7-2 \\ a=-\frac{9}{9} \\ a=-1 \end{gathered}[/tex]Hence, the equation of the function would be:
[tex]y=-(x-1)^2+2[/tex]I'm not sure how to do this. This is a long one that's why.
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We have the following:
For the area surface:
[tex]As=2\pi rh[/tex]repacing:
r = 1.5 in
h = 7 in
[tex]\begin{gathered} As=2\cdot3.14\cdot1.5^{}\cdot7 \\ As=65.94 \end{gathered}[/tex]The answer is 65.9 in^2
For volume:
[tex]\begin{gathered} V=\pi r^2h \\ V=3.14\cdot1.5^2\cdot7 \\ V=49.455 \end{gathered}[/tex]The answer is 49.5 in^3
Mrs. Navarro has 36 students in her class, 16 boys and 20 girls.Select all ratios below that correctly describe the ratio of boys to girls in Mrs.Navarros's class.
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First, we need to know the ratio of boys to girls in Mrs. Navarro's class. There are 16 boys and 20 girls. The ratio would be 16:20.
From this given, we can choose from the options which rations are equivalent to our given ratio.
8 to 10 is a ratio that is equivalent from our given. If we scale are ratio by 2, we can get 8:10.
5:4, 8:18, and 5 to 9, however, are NOT equivalent to 16:20.
4:5 is equivalent. We just need to scale 16:20 by 4, and we will get 4:5.
10:8 is another ratio that is NOT equivalent to 16:20.
*Scaling ratios are similar to finding the lowest terms of fractions.
Rowan is taking his siblings to get ice cream. They can't decide whether to get a cone or a cup because they want to get the most ice cream for their money. If w = 4 in, x =6 in, y = 6 in, z = 2 in, and the cone and cup are filled evenly to the top with no overlap, which container will hold the most ice cream? Use 3.14 for π, and round your answer to the nearest tenth.
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EXPLANATION:
Given;
We are given two ice cream cups in the shapes of a cone and a cylinder.
The dimensions are;
[tex]\begin{gathered} Cone: \\ Radius=4in \\ \\ Height=6in \\ \\ Cylinder: \\ Radius=3in \\ \\ Height=2in \end{gathered}[/tex]Required;
We are required to determine which of the two cups will hold the most ice cream.
Step-by-step solution;
Take note that the radius of the cylinder was derived as follows;
[tex]\begin{gathered} radius=\frac{diameter}{2} \\ \\ radius=\frac{6}{2}=3 \end{gathered}[/tex]The volume of the cone is given by the formula;
[tex]\begin{gathered} Volume=\frac{1}{3}\pi r^2h \\ \\ Therefore: \\ Volume=\frac{1}{3}\times3.14\times4^2\times6 \\ \\ Volume=\frac{3.14\times16\times6}{3} \\ \\ Volume=100.48 \end{gathered}[/tex]Rounded to the nearest tenth, the volume that the cone can hold will be;
[tex]Vol_{cone}=100.5in^3[/tex]Also, the volume of the cylinder is given by the formula;
[tex]\begin{gathered} Volume=\pi r^2h \\ \\ Volume=3.14\times3^2\times2 \\ \\ Volume=3.14\times9\times2 \\ \\ Volume=56.52 \end{gathered}[/tex]Rounded to the nearest tenth, the volume will be;
[tex]Vol_{cylinder}=56.5in^3[/tex]ANSWER:
Therefore, the results show that the CONE will hold the most ice cream.
Find the one-sided limit (if it exists). (If the limit does not exist, enter DNE.)
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Answer:
0
Explanation:
Let us call
[tex]f(x)=\frac{\sqrt[]{x}}{\csc x}[/tex]The function is continuous on the interval [0, 2pi]; therefore,
[tex]\lim _{x\to\pi^+}f(x)=\lim _{x\to\pi^-}f(x)[/tex]To evaluate the limit itself, we use L'Hopital's rule which says
[tex]\lim _{x\to c}\frac{a(x)}{b(x)}=\lim _{x\to c}\frac{a^{\prime}(x)}{b^{\prime}(x)}[/tex]Now in our case, we have
[tex]\lim _{n\to\pi}\frac{\sqrt[]{x}}{\csc x}=\lim _{n\to\pi}\frac{\frac{d\sqrt[]{x}}{dx}}{\frac{d \csc x}{dx}}[/tex][tex]=\lim _{n\to\pi}\frac{d\sqrt[]{x}}{dx}\div\frac{d\csc x}{dx}[/tex][tex]=\frac{1}{2\sqrt[]{x}}\div(-\frac{\cos x}{\sin^2x})[/tex]since
[tex]\frac{d\csc x}{dx}=-\frac{\cos x}{\sin^2x}[/tex]Therefore, we have
[tex]\lim _{n\to\pi}\frac{\sqrt[]{x}}{\csc x}=\lim _{n\to\pi}\frac{1}{2\sqrt[]{x}}\div(-\frac{\cos x}{\sin^2x})[/tex][tex]=\lim _{n\to\pi}-\frac{1}{2\sqrt[]{x}}\times\frac{\sin^2x}{\cos x}[/tex]Putting in x = π into the above expression gives
[tex]-\frac{1}{2\sqrt[]{x}}\times\frac{\sin^2x}{\cos x}\Rightarrow-\frac{1}{2\sqrt[]{\pi}}\times\frac{\sin^2\pi}{\cos\pi}[/tex][tex]=0[/tex]Hence,
[tex]=\lim _{n\to\pi}-\frac{1}{2\sqrt[]{x}}\times\frac{\sin^2x}{\cos x}=0[/tex]Therefore, we conclude that
[tex]\boxed{\lim _{n\to\pi}\frac{\sqrt[]{x}}{\csc x}=0.}[/tex]which is our answer!
is it true that all whole numbers are rational numbers ? why or why not
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all whole numbers are rational numbers
because we can write 21 as 21/1 in rational form.
.
We can write any whole number (a) into the form of
[tex]\frac{a}{b}[/tex]where b = 1,
so all whole numbers can be written in form of rational numbers.
Colin is playing a video game. He wins 25 points for each gold coin he finds. His goal is to win more than 200 poijts. He wants to know how many gold coins he needs to find.
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25 points for each gold coin
He wants more tha 200 points
Number of coins to get 200 points: = 200/25 = 8
Answer:
He needs to find 8 gold coins or more
>= 8
Calculate the five-number summary of the given data. Use the approximation method.19, 2, 23, 25, 20, 2, 4, 8, 16, 11, 10, 12, 8, 2, 18
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Answer:
Explanation:
Given the data:
19, 2, 23, 25, 20, 2, 4, 8, 16, 11, 10, 12, 8, 2, 18
Step 1: Write in an order (we are writing in an ascending order here)
2, 2, 2, 4, 8, 8, 10, 11, 16, 18, 19, 20, 23, 25,
An animal shelter spends $1.50 per day to care for each cat and $6.50 per day to carefor each dog. Gavin noticed that the shelter spent $97.00 caring for cats and dogs onMonday. Gavin found a record showing that there were a total of 18 cats and dogs onMonday. How many cats were at the shelter on Monday?
Answers
4 cats and 14 dogs.
Explanation:
Data :
Amount of cats : c = ?
Cost per cats : $1.50
Amount of dogs : d = ?
Cost per dogs : $6.50
Total spent for dogs and cats : $97.00
Total number of dogs and cats : 18
Formulas:
1.50c + 6.50d = 97.00
c + d = 18
Solution:
c + d = 18 => c = 18 - d
1.50(18 -d) + 6.50d = 97.00
27 - 1.50d + 6.50d = 97.00
27 - 1.50d + 6.50d - 27 = 97 -
