To solve the system of equations x + 3y = 5 and x - 3y = -1, we add both equations to get 2x = 4, solve for x to find x = 2, and then substitute x back into one of the equations to find y = 1, resulting in the solution (2, 1).
The solution of the system of linear equations given by x + 3y = 5 and x - 3y = -1 involves manipulating the equations to find the values of x and y that satisfy both equations simultaneously. One common method to solve these is to add or subtract the equations, which eliminates one variable, making it possible to solve for the other.
Starting with the addition method, we align the equations and add them together:
x + 3y = 5
x - 3y = -1
Adding these equations, we get 2x = 4, and solving for x gives us x = 2. We can substitute x = 2 back into one of the original equations to find the value of y, yielding y = 1.
The solution to the system is the intersection point of the two lines represented by the equations, which is the point (2, 1).
Find the value of x if A, B, and C are collinear points and B is between A and C.
AB=5x−1,BC=14,AC=25−x
Answer:
x = 2
Step-by-step explanation:
Collinear means that the points A, B and C all lie in a straight line, hence
AB + BC = AC, substitute values
5x - 1 + 14 = 25 - x
5x + 13 = 25 - x ( add x to both sides )
6x + 13 = 25 ( subtract 13 from both sides )
6x = 12 ( divide both sides by 6 )
x = 2
The value of x is:
[tex]x=2[/tex]
Step-by-step explanation:Three points are said to be collinear if they lie along a straight line.
Also, it is given that:
Here A, B, and C are collinear points and B is between A and C.
This means that:
[tex]AB+BC=AC-----------(1)[/tex]
Also we are given:
[tex]AB=5x-1,\ BC=14,\ AC=25-x[/tex]
Using property (1) we have:
[tex]5x-1+14=25-x\\\\i.e.\\\\5x+13=25-x[/tex]
Now on adding x on both the side of the equation we have:
[tex]5x+13+x=25\\\\i.e.\\\\5x+x+13=25\\\\i.e.\\\\6x+13=25[/tex]
Now on subtracting 13 on both the sides of the equation we have:
[tex]6x=25-13\\\\i.e.\\\\6x=12\\\\i.e.\\\\x=\dfrac{12}{6}\\\\i.e.\\\\x=2[/tex]
In a circle with a diameter of 25.2 ft. An arc is intercepted by a central angle of 168°. What is the arch length? Use 3.14 for pi and round your final answer to the nearest hundredth.
if the circle has a diameter of 25.2, its radius is half that or 12.6.
[tex]\bf \textit{arc's length}\\\\ s=\cfrac{\pi \theta r}{180}~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad degrees\\ \cline{1-1} r=12.6\\ \theta =168 \end{cases}\implies s=\cfrac{\pi (168)(12.6)}{180} \\\\\\ s=11.76\pi \implies \implies \stackrel{\pi =3.14~\hfill }{s=36.9264}\implies \stackrel{\textit{rounded up}}{s=36.93}[/tex]
What equation is the inverse of y = 7x2 – 10?
For f (c)=2x+1 and g(x)=x^2 - 7 find (f-g)(x)
Answer:
[tex]\large\boxed{D.\ -x^2+2x+8}[/tex]
Step-by-step explanation:
[tex](f-g)(x)=f(x)-g(x)\\\\f(x)=2x+1,\ g(x)=x^2-7\\\\\text{substitute:}\\\\(f-g)(x)=(2x+1)-(x^2-7)=2x+1-x^2-(-7)=2x+1-x^2+7\\\\=-x^2+2x+(1+7)=-x^2+2x+8[/tex]
Which graph below solves the following system of equations correctly?
y = three over four times x squared minus 3
y = negative three over four times x squared plus 3
A) two quadratic graphs opening up. They intersect at 0 and negative 3.
B)one quadratic graph opening up and one quadratic graph facing down. They intersect at 0, 3.
C) quadratic graph opening up and quadratic graph opening down. They intersect at 0, negative 3.
D) two parabolas one facing down with a vertex at 0, 3 and one facing up with a vertex at 0, negative 3
D) two parabolas one facing down with a vertex at 0, 3 and one facing up with a vertex at 0, negative 3
Step-by-step explanation:First of all, let's rewrite the equations in a mathematical language:
y = three over four times x squared minus 3:[tex]y=\frac{3}{4}x^2-3[/tex]
Since the leading coefficient, the number that accompanies [tex]x^2[/tex] is positive, that is, its value is 3/4, then the parabola opens upward. On the other hand, the vertex can be found as:
[tex](h,k)=\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right) \\ \\ a=3/4 \\ b=0 \\ c=-3 \\ \\ h=-\frac{0}{2(3/4)}=0 \\ \\ k=f(0)=\frac{3}{4}(0)^2-3=-3 \\ \\ \\ \boxed{Vertex \rightarrow (h,k)=0,-3}[/tex]
y = negative three over four times x squared plus 3:[tex]y=-\frac{3}{4}x^2+3[/tex]
Since the leading coefficient is negative, that is, its value is -3/4, then the parabola opens downward. Similarly the vertex can be found as:
[tex](h,k)=\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right) \\ \\ a=-3/4 \\ b=0 \\ c=3 \\ \\ h=-\frac{0}{2(-3/4)}=0 \\ \\ k=f(0)=-\frac{3}{4}(0)^2+3=3 \\ \\ \\ \boxed{Vertex \rightarrow (h,k)=0,3}[/tex]
Both graph are shown below and you can see that the conclusion of our problem is correct.
does anyone know the passwords for the module tests for geometry? i use FLVS and my teacher is Ms. Branch
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What polynomial is equivalent to (x-4)(3x^2-x+3)?
Answer:
3x³ - 13x² + 7x - 12
Step-by-step explanation:
Each term in the second factor is multiplied by each term in the first factor, that is
x(3x² - x + 3) - 4(3x² - x + 3) ← distribute both parenthesis
= 3x³ - x² + 3x - 12x² + 4x - 12 ← collect like terms
= 3x³ - 13x² + 7x - 12
Male and female high school students reported how many hours they worked each week in summer jobs. The data is represented in the following box plots: Identify any values of data that might affect the drastically measures of spread and center.
A) The zero hour mark on both plots prevents the graphs from being balanced.
B) The median is near the center of the IQR for both males and females.
C) There is not enough evidence to see any effects on spread or center.
D) The makes have a suspected significant high outlier.
Answer:
I think the answer is d
Step-by-step explanation:
since the graph is a lot bigger than the females, but the box thing is in about the same spot as the females ( you know what i mean), but i'm not 100% sure, but i think its the safest answer
Devin recorded the number of hours he played a video
game, x, and the levels he achieved, y. The regression
calculator shows the equation for the line of best fit.
Use the equation to interpolate the values and estimate
the time it would take him to get to level 5. Round to
the nearest half hour.
hours
Answer:3 hours
Step-by-step explanation:
i did it one edg
Which is a rational function?
AnswerB. because x is in denominator
Step-by-step explanation:
Whats the answer??? And how do i solve it!
Answer:
[tex]\large\boxed{3^\frac{2}{3}}[/tex]
Step-by-step explanation:
[tex]Use\\\\\sqrt[n]{a^m}=a^\frac{m}{n}\\\\(a^n)^m=a^{nm}\\\\a^n\cdot a^m=a^{n+m}\\\\\bigg(\sqrt[4]{9^{15}}\cdot\sqrt{3^3}\bigg)^\frac{2}{27}=\bigg(\sqrt[4]{(3^2)^{15}}\cdot3^\frac{3}{2}\bigg)^\frac{2}{27}=\bigg(\sqrt[4]{3^{2\cdot15}}\cdot3^{\frac{3}{2}}\bigg)^\frac{2}{27}\\\\=\bigg(3^{\frac{30}{4}}\cdot3^\frac{3}{2}\bigg)^\frac{2}{27}=\bigg(3^{\frac{15}{2}}\cdot3^\frac{3}{2}\bigg)^\frac{2}{27}=\bigg(3^{\frac{15}{2}+\frac{3}{2}}\bigg)^\frac{2}{27}[/tex]
[tex]=\bigg(3^{\frac{18}{2}}\bigg)^\frac{2}{27}=\bigg(3^9\bigg)^\frac{2}{27}=3^{9\cdot\frac{2}{27}}=3^\frac{2}{3}[/tex]
What are the coordinates of the point 3/5 of the way from A(-9,3) to B(21, -2)
Answer: The required co-ordinates of the point are (9, 0).
Step-by-step explanation: We are given to find the co-ordinates of the point that is [tex]\dfrac{3}{5}[/tex] of the way from A(-9,3) to B(21, -2).
Let K be the required point. Then, we mus have
[tex]AK:AB=3:5\\\\\Rightarrow \dfrac{AK}{AK+BK}=\dfrac{3}{5}\\\\\\\Rightarrow 5AK=3AK+3BK\\\\\Rightarrow 2AK=3BK\\\\\Rightarrow AK:BK=3:2.[/tex]
We know that
the co-ordinates of a point that divides the line joining the points (a, b) and (c, d) in the ratio m : n are given by
[tex]\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{2}\right).[/tex]
For the given division, m : n = 3 : 2.
Therefore, the co-ordinates of the point K are
[tex]\left(\dfrac{3\times21+2\times(-9)}{3+2},\dfrac{3\times(-2)+2\times3}{3+2}\right)\\\\\\=\left(\dfrac{63-18}{5},\dfrac{-6+6}{5}\right)\\\\=\left(\dfrac{45}{5},\dfrac{0}{5}\right)\\\\=(9,0).[/tex]
Thus, the required co-ordinates of the point are (9, 0).
Building and solving an equation, it is found that the coordinates are: (9,0).
We are given two points: A(-9,3) and B(21, -2).We also want point C(x,y).C is 3/5 of the way from A to B, thus:
[tex]C - A = \frac{3}{5}(B - A)[/tex]
This is used to find both the x-coordinate and the y-coordinate of C.
First, the x-coordinate, considering [tex]C = x, A = -9, B = 21[/tex].
[tex]C - A = \frac{3}{5}(B - A)[/tex]
[tex]x + 9 = \frac{3}{5}(21 + 9)[/tex]
[tex]x = 18 - 9[/tex]
[tex]x = 9[/tex]
Then, the y-coordinate, considering [tex]C = y, A = 3, B = -2[/tex].
[tex]C - A = \frac{3}{5}(B - A)[/tex]
[tex]y - 3 = \frac{3}{5}(-2 - 3)[/tex]
[tex]y = -3 + 3[/tex]
[tex]y = 0[/tex]
Thus, the coordinates are (9,0).
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what is the sum of the product of a and b and the number c? write as an expression
Step-by-step explanation:
The sum of the product of a and b and the number c:i
[tex]a\cdot b+c=ab+c[/tex]
The sum of the product of a and b and the number c, written as a mathematical expression, is ab + c.
Explanation:The question is asking for the sum of the product of a and b and the number c. In mathematical terms, this can be written as an expression: ab + c. This expression first calculates the product of a and b (represented by 'ab') then adds the number c to this product.
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Simplify
8(x + 6) - 10.
Answer:
8 x + 38 higher it is not available
How do I solve rate of change problems? (With picture) thanks!
**please help me understand the 3 problems by explaining
5)
[tex]\bf \begin{array}{|cc|cccc|ll} \cline{1-6} sodas&x&\underline{24}&28&\underline{32}&36\\ \cline{1-6} cost&y&\underline{18}&21&\underline{24}&27\\ \cline{1-6} \end{array}~\hspace{9em} (\stackrel{x_1}{24}~,~\stackrel{y_1}{18})\qquad (\stackrel{x_2}{32}~,~\stackrel{y_2}{24}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{24-18}{32-24}\implies \cfrac{6}{8}\implies \cfrac{3}{4}[/tex]
6)
[tex]\bf \begin{array}{|cc|cc|ll} \cline{1-4} year&x&0&12\\ \cline{1-4} \$&y&720&1080\\ \cline{1-4} \end{array}~\hspace{10em} (\stackrel{x_1}{0}~,~\stackrel{y_1}{720})\qquad (\stackrel{x_2}{12}~,~\stackrel{y_2}{1080}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{1080-720}{12-0}\implies \cfrac{360}{12}\implies \cfrac{30}{1}\implies 30[/tex]
7)
slope as you should already know is rise/run, or how much something moves in relation something else, namely how much the y-axis go up as the x-axis moves sideways, one moves, the other follows, but the increments will be different, sometimes the same, but usually different.
the y-intercept means, when the graph of the equation touches or intercepts the y-axis, and when that happens x = 0, or the horizontal distance is at bay.
for the slope on 6), 30 or 30/1 means, for every 1 year(x) passed, the worth(y) increased by 30, or jumped by 30 units, so as the x-axis moved 1, the y-axis moved 30. After 12 years 30 * 12 = 360, and we add the initial 720 and we end up with 1080.
the y-intercept, well, as aforementioned is when x = 0, is year 0.
Answer:
Top:
The rate change is 4.
Bottom:
The rate change is 3.
Step-by-step explanation:
24+4= 28 28+4=32 (and) 32+4=36
18+3=21 21+3=24 (and) 24+3=27
I will get back to you on the rest>
Hope this helped tho! :3
emily has earned the following grades A, c+, a-,b- and b+. what grade must she earn in her biology class to keep her 3.2 gpa?
Answer:
B+
Step-by-step explanation:
Grades earned in five subjects are;
A,C+,A-,B- and B+
Remaining subject is biology
Total number of subjects will be=6
3.2 gpa as a percentile =86
For her to maintain 3.2 gpa total sum of percentile in the 6 subjects should be at least
6×86=516
Emily total sum of subjects in percentile is 93+77+90+80+87=427
Find the difference , 516-427=89
89 is grade B+
Emily should earn a grade of B+ to keep her 3.2 gpa
9. Which of the following do you need to know to determine the surface area and volume of a sphere?
A. The radius of the sphere
B. The eccentricity of the sphere
C. The circumference of a great circle on the sphere
D. The radius and eccentricity of the sphere
Answer:
A: radius of the sphere
Step-by-step explanation:
The formula for surface area is A = 4πr^2
With r being the radius and A the surface area.
For volume it is
V= (4/3)πr^3
to determine the surface area and volume of sphere A: the radius of the sphere is required.
The formula for surface area is A = 4πr^2
With r being the radius and A the surface area.
For volume it is
V= (4/3)πr^3
How do you find the radius of the sphere?
The radius is half the diameter, so we use the equation r = D / 2. This is the same method used to calculate the radius from the diameter of a circle. If you have a sphere with a diameter of 16 cm, divide 16/2 to find the radius to make it 8 cm. If the diameter is 42, the radius is 21.
A sphere is defined as a set of all points in distant 3D Euclidean space. ("Radius") From a specific point ("center"). Twice the radius is called the diameter, and the pair of points on the sphere on the opposite side of the diameter is called the antipodes.
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The height of a right cylinder is 3 times the radius of the base. The volume of the cylinder is 242
What is the height of the cylinder?
2 units
4 units
6 units
8 units
Answer:
D
Step-by-step explanation:
Formula
Volume = pi*r^2*h
Givens
r = x
h = 3x
Volume= 242
Solution
242 = pi * x^2 * 3x
242 = 3.14 * 3x^3 Divide by pi
242/3.14 = 3.14 * 3x^3 / 3.14 Do the division
77 = 3x^3 Divide by 3
77/3 = 3x^3 /3
25.69 = x^3 Take the cube root of both sides.
2.95 = x
The height of the cylinder is 3 times that of the radius of the circle (x)
The answer is 8.85. I suppose the closest answer is 8.
Find the volume of the cone. Round your answer to the nearest hundredth.
A.
32.05 cubic feet
B.
38.39 cubic feet
C.
56.54 cubic feet
D.
100.69 cubic feet
Answer:
V=100.63 cubic feet
Step-by-step explanation:
The formula for the volume of the cube is given as
[tex]V=\frac{1}{3}\pi r^{2}h[/tex]
Where r=4.3 ft
And h is height = 5.2 ft
Now we put the value of h and r in the formula for volume
[tex]V=\frac{1}{3}\times \pi \times 4.3^{2} \times 5.2[/tex]
[tex]V=\frac{1}{3}\times \pi \times 96.148[/tex]
[tex]V=\frac{1}{3}\times 3.14 \times 96.148[/tex]
[tex]V=\frac{1}{3}\times 301.904[/tex]
V=100.63 cubic feet
To compute the volume of a cone, one needs to use the formula 1/3πr²h, where 'r' represents the radius of the base and 'h' represents the height of the cone. The specific dimensions of the cone are needed to provide a concrete answer. The given options can't be validated without these details.
Explanation:Without specific parameters such as the radius and height of the cone, we cannot directly calculate the volume of the cone. However, the general formula for finding the volume of a cone is 1/3πr²h, where r represents the radius of the base and h represents the height of the cone. This formula is derived from the geometric principles of a cone's structure. Once the values of r and h are known, they can be substituted into this formula to compute the volume. The answer would then be rounded to the nearest hundredth if necessary. Unfortunately, the choices provided A.32.05 cubic feet, B.38.39 cubic feet, C.56.54 cubic feet, D.100.69 cubic feet aren't helpful without knowledge of the specific cone dimensions.
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The equations y= x^2/2 - 8 and y=2x−2 are graphed below. What are the solutions to the equation x^2/2−8=2x−2
graph is attached
Please choose one answer below
x=−6 and x=10
x=−4 and x=4
x=−2 and x=6
x=−8 and x=−2
Your answer is the third option, x = -2 and x=6
We can see this because the solutions of x^2/2 - 8 = 2x - 2 are going to be where the lines y = x^2/2 - 8 and y = 2x - 2, because this is where the two equations are equal to each other.
Therefore, we can just look on the graph at where the two lines intersect, and see that it happens when x = -2 and x = 6.
I hope this helps!
Answer:
x=-2 and x=6
Step-by-step explanation:
The solution to a graph would be where the two lines intersect. When you look at the graph you plot where they connect
(-2, -6), (6, 10)
The x value is the very first variable in an ordered pair. Therefore, the solution to this graph would be x=-2 and x=6.
The cycling tour has 340 bottles of water for each week.there are 10 people on the tour. How many bottles of water is that for each person per week
Answer:
34 bottles per person per week
Step-by-step explanation:
1) identify info: The cycling tour has 340 bottles of water for each week.there are 10 people on the tour.
2) compile info: 340 bottles for 10 people per week
3) calculate: bottles per person = 340/10 = 34 bottles per person each week
The question asks to figure out how many bottles of water each person on a cycling tour gets per week. By dividing the total bottles of water (340) by the number of people (10), we find that each person receives 34 bottles of water per week.
Explanation:The question is asking us to determine how many bottles of water each person on the cycling tour gets per week. This involves division in mathematics as we are dividing the total number of water bottles by the number of people.
To solve such problems, we start with the total amount given, which is 340 bottles of water here. Then, we divide this total by the number of individuals, which is 10 people in this case. So, the calculation becomes 340 bottles ÷ 10 people.
This calculation gives us 34 bottles. Hence, each person on the cycling tour gets 34 bottles of water per week.
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Construct a polynomial function with the following properties: fifth degree, 2 is a zero of multiplicity 3, -2 is the only other zero, leading coefficient is 2.
f(x)=?
Can someone help?
Answer:
[tex]f(x)=2(x-2)^{3}(x+2)^{2}[/tex]
Step-by-step explanation:
we know that
2 is a zero of multiplicity 3 of the polynomial
so
we have that
x=2 is a solution of the polynomial
A factor of the polynomial is
[tex](x-2)^{3}[/tex] ----> is elevated to the cube because is a multiplicity 3
and the other solution is x=-2
since the polynomial is fifth degree, x=-2 must have a multiplicity 2
so
the other factor of the polynomial is
[tex](x+2)^{2}[/tex] ----> is squared because is a multiplicity 2
therefore
The polynomial is equal to multiply the factors by the leading coefficient
so
[tex]f(x)=2(x-2)^{3}(x+2)^{2}[/tex]
The polynomial function based on the given properties: zeros at 2 (with multiplicity 3) and -2 (with multiplicity 1), leading coefficient of 2, is f(x) = 2(x - 2)³(x + 2).
Explanation:To construct a polynomial with given zeros and multiplicities, we need to set up a product of binomial factors based on the zeros, with each binomial factor raised to the power of its respective multiplicity. The resulting polynomial is the given function f(x).
For the given properties, the roots are 2 and -2. The root 2 has multiplicity 3 and the root -2 has multiplicity 1. Also, the leading coefficient is 2. So, we can set up the polynomial function as follows:
f(x) = 2(x - 2)3(x + 2)
This represents a fifth degree polynomial function that has the zeros and multiplicities listed, with the leading coefficient of 2.
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(02.02LL)
If f(x) = 2(X - 5), find f(8).
Answer:
6
Step-by-step explanation:
You can see how f(x) is now f(8), this implies you have to replace any x's you see with an 8.
So f(8) = 2(8-5) = 2(3) = 6
Find the simplified product: 3 sqrt 2x^5 *3 sqrt 64x^9
Answer:
[tex]\large\boxed{4x^4\sqrt[3]{2x^2}}[/tex]
Step-by-step explanation:
[tex]\sqrt[3]{2x^5}\cdot\sqrt[3]{64x^9}\qquad\text{use}\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\\\\=\sqrt[3]{2}\cdot\sqrt[3]{64}\cdot\sqrt[3]{x^5x^9}\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\\\\=\sqrt[3]2\cdot4\cdot\sqrt[3]{x^2x^3x^9}\\\\=4\sqrt[3]2\cdot\sqrt[3]{x^2x^{12}}\\\\=4\sqrt[3]2\cdot\sqrt[3]{x^2x^{4\cdot3}}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=4\sqrt[3]2\cdot\sqrt{x^2(x^4)^3}\\\\=4\sqrt[3]2\cdot\sqrt[3]{x^2}\cdot\sqrt[3]{(x^4)^3}\qquad\text{use}\ \sqrt[n]{a^n}=a\\\\=4\sqrt[3]{2x^2}\cdot x^4\\\\=4x^4\sqrt[3]{2x^2}[/tex]
Answer: it's c to put it simply
Step-by-step explanation:
distance
Jason took 40 min to drive from East Mall to West Mall. He drove at an average
speed of 60 km/h for the first 25 min. He then increased his speed by 12 km/h for
the remaining journey. What was the distance between the two malls?
Answer: 28 km
Step-by-step explanation: For the first 25 minutes, he drove 60 km/h. Since there is 60 minutes in an hour, we know that he drove 25 km. The rest of the journey is 15 minutes. 15 minutes is 1/4 of an hour, so get 1/4 of 12 km/h. This would be 3 km. Add 25 km and 3 km. The distance between the 2 malls is 28 km.
Kevin wants to buy an area rug for his living room. He would like the area rug to be no smaller that 48 square feet and no bigger than 80 square feet. If the length is 2 feet more than the width, what are the range of possible values for the width?
Answer:
[tex]6\leq W\leq 8[/tex]
[tex]8\leq L\leq 10[/tex]
Step-by-step explanation:
Let the length of the rug be = L
Let the width of the rug be = W
Area =[tex]L\times W[/tex]
The length is 2 feet more than the width, so [tex]L=W+2[/tex]
Area = [tex](W+2)\times W[/tex]
= [tex]W^{2} +2W[/tex]
Now given is that the area of rug to be no smaller that 48 square feet and no bigger than 80 square feet.
This can be modeled as:
[tex]48\leq W^{2} +2W\leq 80[/tex]
Solving it separately:
[tex]48\leq W^{2} +2W[/tex]
=> [tex]0\leq W^{2}+2W \leq -48[/tex]
=> [tex]6\leq W[/tex]
[tex]W^{2} +2W-80\leq 0[/tex]
=> [tex]W\leq 8[/tex]
We have the following result:
[tex]6\leq W\leq 8[/tex]
And length will be :
[tex]6+2\leq L\leq 8+2[/tex]
[tex]8\leq L\leq 10[/tex]
Math 8: Linear Functions, Part 2
Write an equation in point-slope form that passes through the given
point with the given slope.
2. (-1, 8), m = 1
1. (3,5), m = -4
Please help me
Answer:
[tex]\boxed{\text{1. y + 5 = -4(x - 3); \qquad 2. y - 8 = x + 1}}[/tex]
Step-by-step explanation:
Question 1
The point-slope formula for a straight line is
y – y₁ = m(x – x₁)
x₁ = 3; y₁ = -5; m = -4
Substitute the values
[tex]\boxed{\textbf{y + 5 = -4(x - 3)}}[/tex]
The diagram shows the graph of equation 1 (red) with slope -4 passing through (3,-5).
Question 2
x₁ = -1; y₁ = 8; m = 1
Substitute the values
[tex]\boxed{\textbf{y - 8 = x + 1}}[/tex]
The diagram shows the graph of equation 2 (green) with slope 1 passing through (-1,8).
One x-intercept for a parabola is at the point
(2,0). Use the quadratic formula to find the
other x-intercept for the parabola defined by
this equation:
y = 4x2 - 4x – 8
Answer:
(2,0) was already given so (-1,0) is the other one.
Step-by-step explanation:
So we are asked to use the quadratic formula.
To find the x-intercepts (if they exist) is use:
[tex]\text{ If } y=ax^2+bx+c \text{ then the } x-\text{intercepts are } (\frac{-b \pm \sqrt{b^2-4ac}}{2a},0)[/tex].
Let's start:
Compare the following equations to determine the values for [tex]a,b, \text{ and }c [/tex]:
[tex]y=ax^2+bx+c[/tex]
[tex]y=4x^2-4x-8[/tex]
So
[tex]a=4[/tex]
[tex]b=-4[/tex]
[tex]c=-8[/tex]
We are now ready to enter into our formula:
[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\frac{4 \pm \sqrt{(-4)^2-4(4)(-8)}}{2(4)}[/tex]
[tex]x=\frac{4 \pm \sqrt{16+16(8)}}{8}[/tex]
[tex]x=\frac{4 \pm \sqrt{16(1+8)}}{8}[/tex]
[tex]x=\frac{4 \pm \sqrt{16}\sqrt{1+8}}{8}[/tex]
[tex]x=\frac{4 \pm 4\sqrt{9}}{8}[/tex]
[tex]x=\frac{ 4 \pm 4(3)}{8}[/tex]
[tex]x=\frac{4 \pm 12}{8}[/tex]
[tex]x=\frac{4(1\pm 3)}{8}[/tex]
[tex]x=\frac{1(1\pm 3)}{2}[/tex]
[tex]x=\frac{1 \pm 3}{2}[/tex]
[tex]x=\frac{1+3}{2} \text{ or } \frac{1-3}{2}[/tex]
[tex]x=\frac{4}{2} \text{ or } \frac{-2}{2}[/tex]
[tex]x=2 \text{ or } -1[/tex]
So the x-intercepts are (2,0) and (-1,0).
(2,0) was already given so (-1,0) is the other one.
6. PATTERNS Complete the pattern: 5, 7,
10, 14,
Answer:
19 is the next number
Step-by-step explanation:
5+2=7
7+3=10
10+4=14
14+5=19
The next number in the pattern is 19.
To complete the number pattern 5, 7, 10, 14, we observe that the differences between the numbers increase by 1. Continuing this pattern, we add 5 to the last number, which results in 19. Thus, the next number in the pattern is 19
To complete the given number pattern: 5, 7, 10, 14, we need to find the rule that governs the pattern. Let's look at the differences between the numbers:
7 - 5 = 210 - 7 = 314 - 10 = 4We can see that the differences are increasing by 1 each time. If this pattern continues, the next difference should be 5. So we add 5 to the last number in the pattern:
14 + 5 = 19
Therefore, the next number in the pattern is 19.
The solutions of the quadratic equation 0 = (x + 3)(x - 2)
are
0 -6 and 0.
04 and 3.
0-3 and 2
0 -2 and 1.
Answer:
x = -3 or x = 2
Step-by-step explanation:
It is given a quadratic equation,
(x + 3)(x - 2) = 0
To find the solution of given equation
Let (x + 3)(x - 2) 0
⇒ either (x + 3) = 0 or (x - 2) = 0
If (x + 3) = 0 then x = -3
If (x - 2) = 0 then x = 2
Therefore the solution of given equation are
x = -3 or x = 2