The equations in the red box I need to use substitution. Explaining the steps as I go.
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We are given the following system of equations:
[tex]\begin{gathered} y=-x^2+63x-790,(1) \\ y=3x+10,(2) \end{gathered}[/tex]We will substitute the value of "y" from the first equation into the second equation:
[tex]-x^2+63x-790=3x+10[/tex]Now we subtract "3x" from both sides:
[tex]-x^2+63x-3x-790=10[/tex]Adding like terms:
[tex]-x^2+60x-790=10[/tex]Now we subtract 10 from both sides:
[tex]-x^2+60x-790-10=0[/tex]Adding like terms:
[tex]-x^2+60x-800=0[/tex]Now we multiply by -1 on both sides of the equation:
[tex]x^2-60x+800=0[/tex]Now we factor in the left side. We need two numbers that when multiplied the product is 800 and their algebraic sum is -60. Those numbers are -40 and -20. Therefore, we get:
[tex](x-40)(x-20)=0[/tex]Now we set each factor to zero:
[tex]\begin{gathered} x-40=0 \\ x=40 \end{gathered}[/tex]For the next factor:
[tex]\begin{gathered} x-20=0 \\ x=20 \end{gathered}[/tex]These are the two values of "x" for the solution. To get the corresponding value of "y" we substitute in the second equation. Substituting the first value we get:
[tex]\begin{gathered} y=3(40)+10 \\ y=120+10 \\ y=130 \end{gathered}[/tex]Now we substitute the second value:
[tex]\begin{gathered} y=3(20)+10 \\ y=60+10 \\ y=70 \end{gathered}[/tex]Therefore, the solutions of the system are:
[tex]\begin{gathered} (40,130) \\ (20,70) \end{gathered}[/tex]Related Questions
Given {(-1,4),(-1,9),(-1,15),(-1,0)}Find the following.Domain=Range-Determine if it is a Function or Not?
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Given the relation:
[tex]\left\{ \left(-1,4\right),\left(-1,9\right),\left(-1,15\right),\left(-1,0\right)\right\} [/tex]The domain of the relation is the set of all values of x.
Therefore:
[tex]\text{Domain}=\left\lbrace -1\right\rbrace [/tex]The range of the relation is the set of all values of f(x) or y. The range is:
[tex]\text{Range}=\left\lbrace 0,4,9,15\right\rbrace [/tex]For a relation to be a function, it must have no more than one y value for each x value.
Since the given relation has more than one y value for each x value, it is NOT A FUNCTION.
Let D be the event that a randomly chosen student enjoys drawing. Let S be the event that a randomly chosen studentplays sports. Identify the answer which expresses the following with correct notation: The probability that a randomlychosen student plays sports, given that the student enjoys drawing.
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ANSWER
P(S|D)
EXPLANATION
The notation P(A|B) always reads as "the probability of A given B"
If the events are
D: student enjoys drawing
S: student play sports
The probability that a randomly chosen student plays sports, given that the student enjoys drawing is: P(S|D)
can u help me with this by using inverse trig.ratios. Find angle A and angle B.
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The hypotenuse is 39 because is opposite to the angle of 90 degrees.
So, angle A is given by
[tex]\measuredangle A=\sin ^{-1}\frac{7}{39}[/tex]Then, we have
[tex]\measuredangle A=\sin ^{-1}(0.17948)[/tex]which gives
[tex]\measuredangle A=10.339\text{ degre}es[/tex]Now, angle C is equal to 90 degrees and angle B is given by
[tex]\begin{gathered} \measuredangle B=\cos ^{-1}(\frac{7}{39}) \\ \measuredangle B=\cos ^{-1}(0.17948) \\ \measuredangle B=79.66\text{ degr}ees \end{gathered}[/tex]Therefore, the answer is
[tex]\begin{gathered} \measuredangle A=10.339\text{ degre}es \\ \measuredangle B=79.669\text{ degre}es \\ \measuredangle C=90\text{ degre}es \end{gathered}[/tex]Rewrite the expression 8m^6/4m^9 using the rules of exponents
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Answer:
Rewrite the expression 8m^6/4m^9 using the rules of exponents
Step-by-step explanation:
8 ÷ 4 = 2
m^6 ÷ m^9
exponent rule in a division problem is you subtract them so...
6 - 9 = -3
you can't have a negative exponent so it would be 2 over m^3
If you have a negative exponent the rule is "flip it" meaning you flip the sign but when you do that it becomes a fraction. so... 2/m^3
9. Marty wants to buy a gallon of lemonade, and a gallon jug costs $3.89, while a pint costs $0.59. He wants to know how much money he would save by buying the gallon jug instead of multiple pints. (3 points: Part I - 1 point; Part II - 1 point; Part III - 1 point)Part I: How many pints are in a gallon?Part II: How much money would Marty spend if he bought multiple pints?Part III: How much money would Marty save by buying the gallon jug instead of multiple pints?
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Part I. In a gallon, there are 8 pints.
Part II. If Marty bought multiple pints, he will spend:
[tex]8\text{ pints}\cdot0.59=\text{ \$4.72}[/tex]Part III. Marty would save:
[tex]4.72-3.89=\text{ \$0.83}[/tex]Marty would save $0.83 by buying the gallon jug instead of multiple pints.
Solve for all values of x in simplest form.
|8x| = 64
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Answer: x=8
Step-by-step explanation:
For the following find the Range:{(₁-2, 4), (3,-2), (1,0), (-2, -2), (0, 6)}
Answers
Explanation
The range represents the y-value of the given points
Answer:
[tex]Range=(-2,0,4,6)[/tex]Two squares are shown in the diagram . The larger square has sides of length 5x units. The smaller square has an area equal to ⅑ of the larger square . Find the length of the sides or the smaller square. Give your answer in the form a/b where a,b ∊ ℕ
Answers
From the information given,
length of side of the larger square = 5x units
The area of a square is calculated by the faormula,
area = length of side^2
Thus,
Area of larger square = (5x)^2 = 25x^2
The smaller square has an area equal to ⅑ of the larger square. This means that the area of the smaller square is
Area of smaller square = 1/9 * 25x^2 = 25x^2/9
Length of the side of square = square root of area
Thus,
Length of the sides of the smaller square = square root of (25x^2/9)
Length of the sides of the smaller square = 5x/3
how many killer whales are there when there are 5 million beluga whales
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We know that the number of og beluga whales varies directly as the number of killer whales, that means that we can write a relation of the form:
[tex]y=kx[/tex]where y is the number of beluga whales, x is the number of killer whales and k is the constant of proportionality.
To answer the question we first need to find k, to do that we plug the values x=23 and y=14 in the equation above and solve for k:
[tex]\begin{gathered} 14=k23 \\ k=\frac{14}{23} \end{gathered}[/tex]Now that we know k the relation is:
[tex]y=\frac{14}{23}x[/tex]Finally to find the number of killer whales if there are 5 millions of beluga whales we plug y=5 in the equation above and solve for x:
[tex]\begin{gathered} 5=\frac{14}{23}x \\ x=\frac{5\cdot23}{14} \\ x=8.21 \end{gathered}[/tex]Therefore when there are 5 millions beluga whales we have 8.21 millions of killer whales.
the vertex of this parabola Is at parabola is at (2,-4)
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Answer:
[tex]A=3[/tex]
Explanation: We are given two points, P1 is vertex and P2 is another point on the parabola:
[tex]\begin{gathered} P_1(2,-4) \\ P_2(3,-1) \end{gathered}[/tex]The general form of the equation of a parabola is:
[tex]y(x)=A(x\pm B)^2+C[/tex]Where A is the Coefficient of the parabola function which is responsible for compression and stretch, likewise B is responsible for the translation along the x-axis and C is responsible for translation along the y-axis.
We know that our function is translated 2 units towards the right and 4 units downwards:
Therefore:
[tex]\begin{gathered} B=-2 \\ C=-4 \end{gathered}[/tex]And this turns the parabola equation into:
[tex]y(x)=A(x-2)^2-4[/tex]Using P2 we can find the constant-coefficient as:
[tex]\begin{gathered} y(x)=A(x-2)^2-4_{} \\ P_2(3,-1) \\ \\ \\ \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} y(3)=A(3-2)^2-4=-1\rightarrow A-4=-1 \\ \because\rightarrow \\ A=3 \end{gathered}[/tex]If m<9 = 97° and m<12 = 114°, find the measure of each missing angle.
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1) Since we have a pair of parallel lines cut by transversal ones then we can state the following conclusions about the angles formed
2) Let's set our table
m∠1
m∠2
m∠3
m∠4
m∠5 = 83º ∠9 and ∠10 are supplementary angles
m∠6 = 97º ∠6 and ∠7 are consecutive interior angles (supplementary)
m∠7 = 114º ∠7 and ∠12 are alternate interior angles (congruent)
m∠8 =66º ∠8 and ∠11 are alternate interior angles (congruent)
m∠9 =97º Given
m∠10 = 180 - 97 =83º ∠9 and ∠10 are supplementary angles
m∠11 = 180º -114º = 66º ∠11 and ∠12 are supplementary angles
m∠12 = 114º Given
m∠13 = 83º ∠10 and ∠13 are Vertical angles
m∠14 = 97º ∠9 and ∠14 are Vertical angles
m∠15= 114º ∠12 and ∠15 are Vertical angles
m∠16 = 66º ∠11 and ∠16 are Vertical angles
Select the best answer for the question. 3. What is 996 times 32? O A. 29,880 B. 31,680 C. 31,872 D. 51,792
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First, write the factors 996 and 32 in the following arrangement:
Next, take the last digit of 32, which is 2, and multiply it by 996. To do so, first, multiply 2 times 6:
[tex]2\times6=12[/tex]Write the units below the column of the 6, and save the remaining 10 units to be added in the next step.
Next, multiply 2 by the next digit from right to left of 996, which is 9:
[tex]2\times9=18[/tex]Add 1 to the result, since it was a remainder from the last operation:
[tex]18+1=19[/tex]Write a 9 below the second colum, and save the remaining 10 units to be added on the next step.
Repeat the procedure for the third digit of 996 from right to left, which is 9.
[tex]2\times9=18[/tex][tex]18+1=19[/tex]Since there are no more digits from the upper number, write 19 below the third colum.
Now, repeat the procedure with the next digit from the lower number (32), which is 3. Write the result one place shifted to the left.
[tex]3\times6=18[/tex]Next, move to the next digit from right to left of the upper number, which is 9:
[tex]3\times9=27[/tex][tex]27+1=28[/tex]Next, move to the next and last digit from right to left of the upper number, which is 9:
[tex]\begin{gathered} 3\times9=27 \\ 27+2=29 \end{gathered}[/tex]Fill the blank space at the right of the last row with a 0 and add both numbers:
[tex]1992+29880=31872[/tex]Therefore:
[tex]996\times32=31872[/tex]Can someone help me with this? How do I figure this out?
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We are given the following radical expression:
[tex]\sqrt[]{a+b}[/tex]We are asked to determine the equivalent exponential expression. To do that we will use the following relationship:
[tex]\sqrt[]{x}=x^{\frac{1}{2}}[/tex]Applying the relationship we get:
[tex]\sqrt[]{a+b}=(a+b)^{\frac{1}{2}}[/tex]Therefore, the right answer would be B.
Create an equation could be used to find the volume, V, of the cylindrical tank. DO 0 TT sin cos tan sin-cos'tan а в E 1 = vo 0 < A p CSC sec cot log log in 1 Part B Question
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Given:
V denotes the volume of the water tank.
r represents the radius of the tank.
h represents the length of the water tank.
The volume of cylindrical tank is expressed as,
[tex]V=\pi\times r^2\times h[/tex]Given that the length of tank h = 20 feet.
The equation becomes,
[tex]\begin{gathered} V=\pi\times r^2\times20 \\ V=20\pi(r^2) \end{gathered}[/tex]Answer:
[tex]V=20\pi(r^2)[/tex]Complete the statement with < >, or =. 25 نتانا
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Given
[tex]\frac{3}{2}\text{?}\sqrt[]{\frac{25}{4}}[/tex]Procedure
[tex]\begin{gathered} \sqrt[]{\frac{25}{4}}=\frac{5}{4} \\ so, \\ \frac{3}{2}<\frac{5}{2} \end{gathered}[/tex]if eddie buy carrots and her discount is 35% and the original price of the carrots is $7.40 how much did she pay
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First, find what is 35% of $7.40 equal to. To do so, multiply 7.40 times 35/100:
[tex]7.4\times\frac{35}{100}=2.59[/tex]Then, the discount is equal to $2.59. To find the final price, substract the discount (2.59) from the original price (7.4):
[tex]7.4-2.59=4.81[/tex]Therefore, the price that she paid was:
[tex]\text{ \$4.81}[/tex]A stack of 30 science flashcards includes a review card for each of the following 10 insects, 8 trees, 8 flowers and 4 birds. What is the probability of randomly selecting an insect and then a tree???
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The probability (P) of event A occurring is:
[tex]P(A)=\frac{\text{ number of favorable outcomes to A}}{\text{ total number of outcomes}}[/tex]The probability of 2 consecutive events A and B occur is:
[tex]P=P(A)*P(B)[/tex]Then, let's calculate the probability of selecting an insect:
Favorable outcomes: 10
Total outcomes: 30
[tex]P(insect)=\frac{10}{30}=\frac{1}{3}[/tex]Now, let's calculate the probability of selecting tree:
If the insect card is replaced:
Favorable outcomes: 8
Total outcomes: 30
[tex]P(B)=\frac{8}{30}=\frac{4}{15}[/tex]If the insect card is not replaced:
Favorable outcomes: 8
Total outcomes: 29
[tex]P(B)=\frac{8}{29}[/tex]The probability of randomly selecting an insect and then a tree is:
With replacement:
[tex]\begin{gathered} P=\frac{1}{3}*\frac{4}{15} \\ P=\frac{4}{45} \end{gathered}[/tex]Without replacement:
[tex]\begin{gathered} P=\frac{1}{3}*\frac{8}{29} \\ P=\frac{8}{87} \end{gathered}[/tex]Answer:
With replacement: 4/45
Without replacement: 8/87
I get a whole number so I’m not taking steps correctly
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The sum of the given matrices is not possible to calculate as the order of both the matrix are different.
To add two matrices both matrices must be of the same order.
[3 -8] is of order 1 × 2
[4 -5 -6] is of order 1 × 3
Since the two matrices are of different orders it is not possible for the two matrices to be added.
Although direct sum or Kronecker sum could be used to add the matrices but the correct symbol for them is not used.
Hence the required result is impossible .
A matrix, also spelled "matrices," is a rectangular table or array of characters that are arranged in rows and columns to represent a mathematical object or a property of one.
Matrix representations of linear mappings in linear algebra allow for explicit computations. Because of this, matrices are studied extensively in linear algebra, and most properties and actions of abstract discrete mathematics may be explained in terms of matrices.
To learn more about matrix visit:
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Find the area of the shape shown below. 7 3
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To find the area of the figure you can divide the figure into to smaller shapes
calculate the area of each of the smaller figures.
for the square
[tex]\begin{gathered} A=s\cdot s \\ A=3\cdot3 \\ A=9 \end{gathered}[/tex]For the triangle the base is the difference between the longest side which is 7 and the side on the square which is 3
[tex]7-3=4[/tex]find the area of the triangle
[tex]A=b\cdot\frac{h}{2}[/tex][tex]\begin{gathered} A=4\cdot\frac{3}{2} \\ A=\frac{12}{2} \\ A=6 \end{gathered}[/tex]Add both areas to find the area of the figure
[tex]\begin{gathered} A_T=9+6 \\ A_T=15 \end{gathered}[/tex]Daniel's BAC, B, after drinking can be determined using the formula B - 0,08 - 0.016Nhere N is the number of hours that have elapsed since drinking, What is Daniel's BACNer 3 hours and 15 minutes?
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1) Firstly, We need to convert 3 hours and 15 min into decimal. So, we can write out the following:
[tex]\begin{gathered} 3+\frac{15}{60} \\ 3\frac{1}{4}=3.25 \end{gathered}[/tex]2) As our second step, we can plug into the formula the number we just found:
[tex]\begin{gathered} B=0.08-0.016N \\ B=0.08-0.016(3.25)\Rightarrow B=0.08-0.052 \\ B=0.028 \end{gathered}[/tex]So Daniel's BAC after 3 hours and 15 minutes drinking is
In your own words, describe the location of each point on a coordinate plane. Be specific. 1. (0,4)2.(2,5) 3.(-3, 6)
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Locations on the coordinate plane are described as ordered pairs. An ordered pair tells you the location of a point by relating the point's location along the x-axis (the first value of the ordered pair) and along the y-axis (the second value of the ordered pair)
1. Point ( 0, 4)
The coordinate of ( 0, 4) is on the y-axis, because x = 0 and y = 4.
The point ( 0, 4) is on the y-axis at y = 4
2. Point ( 2, 5 )
Starting from the origin, go along the x-axis 2 units in a positive direction (right) and along the y-axis 5 units in a positive direction (up).
3. P( -3 , 4 )
Starting from the origin, go along the x-axis 3 units in a negative direction (left) and along the y-axis 4 units in a positive direction (up).
6 out of 20 students at a school assembly were first grade students. What percentage of students. at assembly were first grades
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We are given that 6 out of 20 students at a school assembly were first-grade students.
What percentage of students at the assembly were first-grades students?
To find the percentage of students, divide 6 by 20 and then multiply the result by 100.
[tex]\begin{gathered} =\frac{6}{20}\times100\% \\ =0.3\times100\% \\ =30\% \end{gathered}[/tex]Therefore, 30% of students at the assembly were first-grade students.
Choose each conversion factor that relates cups to fluid Ounces.A. 8 fl oz/ 1cB. 8c/ 1 fl ozC. 1 fl oz/ 8 cD. 1 c/ 8 fl oz
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Answers:
A. 8 fl oz/ 1c
D. D. 1 c/ 8 fl oz
Explanation:
1 cup is equal to 8 fluid ounces, so the conversion factors that relate these units are factors that keep the same equivalence. So, the conversion factors are:
A. 8 fl oz/ 1c
D. D. 1 c/ 8 fl oz
A regular polygon has 20 sides. If one of its angles measures (5h − 12)°, what is the value of h?
h = 38.4
h = 34.8
h = 33.6
h = 30
Answers
Answer:
h = 34.8
Step-by-step explanation:
(20 - 2) × 180 = 3240
3240 ÷ 20 = 162
5h - 12 = 162
+ 12 to cancel out the -
5h = 174
÷ 5 to cancel out 5 × h
h = 34.8
Evaluate. Write your answer as an integer or as a decimal rounded to the nearest hundredth. cos 67° = ____
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Answer: 0.39
The decimal form of cos 67° rounded to the nearest hundredth is 0.39.
what is the slope when the rise is -10 and the run is 2
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Slope intercdpt form of line
Then
Rise means elevation over y axis
Run means displacement along X axis
Now the Slope m = ( Rise/Run ) = (-10/2) = -5
So its a line directed down with slope -5
(when slope is negative, line goes down)
You spend 51/2hours at the park this week. You spend 210 fewer minutes at the library than you do at park. How many minutes do you spend at the library?
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Solution
Given:
Time spent at the park =
[tex]\begin{gathered} 5\text{ }\frac{1}{2}\text{ hours} \\ \end{gathered}[/tex]Convert this time to minutes
[tex]\begin{gathered} \text{Recall that } \\ \text{1 hour = }60\text{ minutes} \\ 5\frac{1}{2}\text{ hours = 5}\frac{1}{2}\text{ x 60}=\frac{11}{2}\text{ x 60 = 330 minutes} \\ \\ \end{gathered}[/tex]Time spent at the library is 210 minutes fewer than that spent at the park.
This implies the time spent at the library is
[tex]\begin{gathered} \text{Time spent at the park minus 210 minutes} \\ =330\text{ minutes - 210 minutes} \\ =120\text{ miutes} \end{gathered}[/tex]The answer is 120 minutes
At a town fair, for one of the game booths contestants pick a single card from a standard deck, and payouts are based on the cards chosen. Find the probability of your card being a club and a 6
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Recall,
Probability = number of favorable outcomes/number of total outcomes
The number of cards in a standard deck of cards is 52. There is only one 6 of clubs. Thus, the probability of selecting a a club and a 6 is
1/52
Suppose logex = 3, log y = 7, and logz= -2.Find the value of the following expression.loga42
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Therefore,
[tex]\begin{gathered} \log _ax^3+\log _ay-\log _az^4=\log _a(\frac{x^3y}{z^4}) \\ 3\log _ax^{}+\log _ay-4\log _az=3(3)+7-4(-2)=9+7-8=8 \end{gathered}[/tex]find an equation of the line passing through the pair points. write the equation in the form ax+by=c (-7,5),(-8,-9)
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Given the pair of coordinates;
[tex]\begin{gathered} (-7,5) \\ (-8,-9) \end{gathered}[/tex]We would begin by first calculating the slope of the line.
This is given by the formula;
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]The variables are as follows;
[tex]\begin{gathered} (x_1,y_1)=(-7,5) \\ (x_2,y_2)=(-8,-9) \end{gathered}[/tex]We will now substitute these into the formula for finding the slope as shown below;
[tex]\begin{gathered} m=\frac{(-9-5)}{(-8-\lbrack-7)} \\ \end{gathered}[/tex][tex]\begin{gathered} m=\frac{-14}{-8+7} \\ \end{gathered}[/tex][tex]\begin{gathered} m=\frac{-14}{-1} \\ m=14 \end{gathered}[/tex]The slope of this line equals 14. We shall use this value along with a set of coordinates to now determine the y-intercept.
Using the slope-intercept form of the equation we would have;
[tex]y=mx+b[/tex]We would now substitute for the following variables;
[tex]\begin{gathered} m=14 \\ (x,y)=(-7,5) \end{gathered}[/tex][tex]5=14(-7)+b[/tex][tex]5=-98+b[/tex]Add 98 to both sides of the equation;
[tex]103=b[/tex]We now have the values of m, and b.The equation in "slope-intercept form" would be;
[tex]y=14x+103[/tex]To convert this linear equation into the standard form which is;
[tex]Ax+By=C[/tex]We would move the term with variable x to the left side of the equation;
[tex]\begin{gathered} y=14x+103 \\ \text{Subtract 14x from both sides;} \\ y-14x=103 \end{gathered}[/tex]We can now re-write and we'll have;
[tex]-14x+y=103[/tex]Note that the coefficients of x and y (that is A and B) are integers and A is positive;
Therefore, we would have;
[tex]\begin{gathered} \text{Multiply all through by -1} \\ 14x-y=-103 \end{gathered}[/tex]The equation of the line passing through the points given expressed in standard form is;
ANSWER:
[tex]14x-y=-103[/tex]