The point on the plane closest to the origin is P(49, 0, 0).
To find the point on the plane that is closest to the origin, we need to minimize the distance from the origin to any point on the plane. Let's call the point on the plane that is closest to the origin P.
We can use the formula for the distance between a point and a plane to set up an equation:
distance = |ax + by + cz - d| / sqrt(a^2 + b^2 + c^2)
where a, b, and c are the coefficients of the plane equation (2, 3, and 6), d is the constant term (98), and x, y, and z are the coordinates of any point on the plane.
Since we want to minimize the distance, we can ignore the absolute value and just focus on the numerator. We can also use the fact that the vector [2, 3, 6] is perpendicular to the plane to simplify the equation:
distance = (2x + 3y + 6z - 98) / sqrt(2^2 + 3^2 + 6^2)
distance = (2x + 3y + 6z - 98) / 7
To minimize this distance, we need to find the point on the plane where (2x + 3y + 6z - 98) is as small as possible. This occurs when the plane equation is satisfied and x, y, and z are as small as possible. Since the plane equation has three variables, we can fix two of them and solve for the third. Let's fix y and z at zero:
2x + 0 + 0 = 98
x = 49
So the point on the plane closest to the origin is P(49, 0, 0).
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Which function is the result of translating y = x2 downward by 3 units and to the left by 4 units?.
The outcome of translating y = x^2 downward by 3 units and to the left by 4 units is y = (x+4)^2-3.
What exactly is Graph?A graph is a mathematical depiction of a network that describes the connection between lines and points.
We must determine which function results from translating y = x^2 downward by 3 units and to the left by 4 units.
A translation is a movement of the graph that is either horizontally or vertically parallel to the axis.
Replace x with x+4 y = (x+4)^2-3 to shift the graph four units to the left.
As a result, y = (x+4)^2-3 is the result of translating y = x^2 by 3 units downward and 4 units to the left.
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The shortest route from London to Bristol is 120 miles.
A lorry is expected to take 2 hours to travel this route.
The lorry actually travels by a different route which increases the distance by 20%, but it still arrives in 2 hours.
By how many more mph than the expected speed does the lorry travel?
Answer:
12 mph
Step-by-step explanation:
First we're going to start by finding the difference in miles from the first route to the second route.
The first route is 120 miles and the second is 20% more than the first. To find the difference you should first multiply the 120 by 20%.
120*0.2=24
You then need to add the additional 24 miles to the original 120 to find the distance of the second route.
120+24=144
After we will begin to find the mph expected speed for both routes.
The first route was 120 miles and it was expected to take 2 hours to get there. This means you need to divide 120 by 2 to find out how many miles they were traveling within one hour.
120/2=60
So the expected mph for the first route would be 60.
We want to repeat the same thing for the second route.
144/2=72
The mph for the second route was 72.
Now you need to find out how much faster the mph was for the second route than the first by subtracting the first from the second.
72-60=12
The second route was 12 mph faster than the first.
What is the difference between X₁ and X₁?
A. X₁ is a random variable, and X₁ is a specific numerical value.
B. X₁ is a random variable, and x₁ is a specific numerical value.
C. There is no difference. They are both random variables.
D. There is no difference. They are both specific numerical values.
The difference between X₁ and X₁ is "X₁ is a random variable, and X₁ is a specific numerical value".
Any variable whose value is uncertain or any function that values each result of an experiment is referred to as a random variable. Random variables are frequently identified by letters and fall into one of two categories: discrete variables, which have definite values, or continuous variables, which can take on any value within a continuous range.
A parameter's estimated value in numbers, specifically. The sample mean is the most accurate estimate of the population mean. A range or interval of values used to gauge the parameter. The value of the parameter under consideration may or may not be included in this estimation.
So we can say the difference between X₁ and X₁ is "X₁ is a random variable, and X₁ is a specific numerical value".
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When a company runs all 50 of its machines, it takes 72 hours to finish a job. If the company only runs 30 of its machines, it takes 120 hours to finish the same job. An equation for the inverse variation between the number of machines, x, the company runs and the time, y, in hours, it takes to finish the job is y=k/x . What is the value of K?
Answer: 3600
Step-by-step explanation: If y = k/x is the formula, solving for k gives xy = k, so 72(50) = 3600; or 30(120) = 3600.
A farmer estimates that he has 9,000 bees producing honey on his farm. The farmer becomes concerned when he realizes the population of bees seems to be decreasing steadily at a rate of 5% per year. If the number of bees in the population after x years is represented by f(x), which statements about the situation are true? Check all that apply.
The correct options regarding given function are (b), (c) and (f).
What is a function?
An association between each element of a non-empty set A and at least one element of a different non-empty set B is called a function.
The set of values that we are permitted to enter into our function is known as the domain of a function.
A function's range is the collection of values it gives as outputs.
The options given are:
a)The function f(x) = 9,000([tex]1.05^{x}[/tex]) represents the situation.
b)The function f(x) = 9,000([tex]0.95^{x}[/tex]) represents the situation.
c)After 2 years, the farmer can estimate that there will be about 8,120 bees remaining.
d)After 4 years, the farmer can estimate that there will be about 1,800 bees remaining.
e)The domain values, in the context of the situation, are limited to whole numbers.
f)The range values, in the context of the situation, are limited to whole numbers.
Now given,
Number of bees initially = 9000
Rate of decay = 5%
f(x) = Number of bee population after x years
So, f(x) = 9000([tex]0.95^{x}[/tex])
After 2 years, number of bees is
f(2) = 9000(0.95²) = 8122.5 ≈ 8120
After 4 years, number of bees is
f(4) = 9000(0.95⁴) = 7330.56≈7330
The domain of the function is number of years(x) which can be whole as well as decimal numbers.
The range of the function is number of bees after x years, which should be a positive integer number. So whole numbers are the domain.
Therefore the correct options regarding given function are (b), (c) and (f).
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What is an example of center of dilation?
As an example, if the centre of dilation were in the centre of a circle with radius 5 and was going through a scale factor 2 dilation, the radius would be 10.
The process of dilation involves growing an object's size without changing its shape. The size of the item may increase or decrease depending on the scale factor. Using dilation math, a square with side 5 units may be made wider to have side 15 units, but the shape of the square remains the same.
Enlargements that generate a bigger picture are called dilations. Reduction describes the dilatation that shrinks the picture. The starting figure is stretched or contracted during a dilatation. The scale factor (or ratio) and the centre of a dilatation are both described. Occasionally, the words "dilate" and words like "magnify," "disptend," "expand," "inflate," and "swell" are used interchangeably. All of these words suggest "expanding in size or volume," but the word "dilate" specifically refers to increased diameter.
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May someone explain this to me in step-by-step? I'd greatly appreciate it!
Thank you.
Answer:
x = 9
Step-by-step explanation:
x = the missing side
24/8 = x/3
8(x) = 24(3)
8x = 72
x = 9
====================================================
Explanation:
The slanted left-most sides of the figures are 24 and 8, when moving from left to right.
Let's reverse the direction and move from right to left. The jump from 8 to 24 is "times 3", which is the scale factor when moving from the smaller figure to the larger figure.
The smaller figure has 3 up top. Apply the "times 3" scale factor to get 3*3 = 9 as the top side length of the larger figure.
In other words, every side of the smaller figure is tripled to get the larger figure.
----------------
Here's another way to solve.
24/8 = x/3
24*3 = 8*x .... cross multiplying
72 = 8x
8x = 72
x = 72/8
x = 9 is the final answer.
Ryan invests $7,171 in a savings account with a fixed annual interest rate of 7% compounded 12 times per year. How long will it take for the account balance to reach $11,688.69?
Round your answer to the nearest whole year.
It will take 7 years for the account balance to reach $11,688.69.
To find how long will it take for the account balance to reach $11,688.69
we will use the compound interest formula that is
[tex]$ \text A = \text P(1 + \frac{\text r}{\text n})^{\text {nt}}[/tex]
where,
A = final amount
P = initial principal balance
r = interest rate
n = number of times interest applied per time period
t = number of time periods elapsed
Given that, P = $7,171
A = $11,688.69
r = 7%
n = 12
t = to find
Substitute the details in the formula and we get
[tex]$11688.69 = 7171(1 + \frac{0.07}{12})^{12t}[/tex]
[tex]$ \text t=\frac{\log _{\frac{1207}{1200}}\left(\frac{1168869}{717100}\right)}{12}[/tex]
t = 7.000003 = 7 years
Thus, It will take 7 years for the account balance to reach $11,688.69.
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Answer:
7 years-----------------------------
Use compound interest equation:
A = P(1 + r/n)^(nt), where A - future amount, P- investment, r - interest rate, n- number of compounds, t - time in yearsSubstitute the given to find the value of t:
11688.69 = 7171(1 + 0.07/12)^(12t)1.0058333333^(12t) = 11688.69 /71711.0058333333^(12t) = 1.62999442198ln 1.0058333333^(12t) = ln 1.6299944219812t = ln 1.62999442198 / ln 1.005833333312t = 84 (rounded)t = 7Constant Variance Definition
Constant variance, also known as homoscedasticity, is a statistical property where the variance of a variable is equal across all values of the variable.
This means that the variability of the variable is consistent, no matter what the value of the variable is. Mathematically, this is expressed by the formula:
Var(X) = σ^2
where Var(X) is the variance of the variable X, and σ^2 is the same across all values of X.
For example, if we look at the heights of a group of people, we may find that the variance of their heights is equal to 25 cm, no matter what the heights are. In this case, the variance of the heights is 25 cm, and this variance (25 cm) is the same for all the heights in the group.
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PLEASE HELP! the graph of y=x^3… (picture provided)
Answer:
Step-by-step explanation:
second one
What is the slope of y =- 3x 10?
Answer:
The Slope would be 3
Greta has two pet lizards, Draco and Lizzy. For every 11 crickets Greta feeds Draco, she always feeds Lizzy 8 crickets. Last week, Greta fed Draco 77 crickets.
How many crickets did Greta feed Lizzy?
The answer is 56 crickets.
What is a ratio?The quantitative relation between two amounts shows the number of times one value contains or is contained within the other. for example-"the ratio of computers to students is now 2 to 1"
Given here: Greta has two pet lizards, Draco and Lizzy. For every 11 crickets Greta feeds Draco, she always feeds Lizzy 8 crickets. Thus the ratio is 11:8 Therefore if Draco got 77 crickets ten lizzy should get 8×7=56 crickets.
Hence, The answer is 56 crickets.
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What is the average rate of change of
g
gg over the interval
−
8
≤
x
≤
−
2
−8≤x≤−2minus, 8, is less than or equal to, x, is less than or equal to, minus, 2?
The average rate of change of g over the interval −8 ≤ t ≤ 2 is equal to -0.5 or -1/2.
How to determine the average rate of change?In Mathematics, the average rate of change of f(x) on a closed interval [a, b] is given by this mathematical expression:
Average rate of change = [f(b) - f(a)]/(b - a)
Next, we would determine the average rate of change of the function g(t) over the interval [-8, 2]:
a = -8; g(a) = 1
b = 2; g(b) = -4
Substituting the given parameters into the average rate of change formula, we have the following;
Average rate of change = (-4 - 1)/(2 - (-8))
Average rate of change = -5/10
Average rate of change = -0.5 or -1/2.
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Complete Question:
What is the average rate of change of g over the interval −8 ≤ t ≤ 2?
Anyone who genius math? Help me, ty
Answer:
B) -3
Step-by-step explanation:
(y - y₁) / (x - x₁) = slope
(8-2) / (m-5) = -3/4
cross-multiply to get:
-3(m-5) = 24
distribute -3 to get:
-3m + 15 = 24
subtract 15 from each side to get:
-3m = 9
m = -3
Which of the following is a solution of the linear equation 2x 3y 5?
Step-by-step explanation:
Solution
The correct option is A Yes, it is.
Given Line:
2
x
+
3
y
=
5
Standard Line :
a
x
+
b
y
=
c
Given line has two variables
x
and
y
where both
x
and
y
have power of
1
similar to standard form of line. So,
2×+3y=5 is a line in two variables.
50 POINTS!
Argument A
A chemistry professor claims that preparing a description of reduction-oxidation chemical reactions requires balancing the component half-reactions for oxidation and reduction by adding molecules or electrons into the chemical equation. Based on the professor's expertise in chemistry, we should conclude that reduction-oxidation chemical equations should be balanced in this way.
Argument A is (strong/weak)
Argument A is strong based on the logic.
What is logic?We know that logic is the process whereby we can be able to decode if an argument is true or false when we look at the premises and the conclusion. It is normal that the premise of the argument should be able to give rise to the conclusion.
In this case, we must note that the professor is an authority figure in chemistry and what he says must be taken as trust worthy as shown in the deductive argument that led to this conclusion as shown above.
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For which value the given polynomial x³ 3x² 4x 12 is zero *?
When x = 4, the given polynomial is equal to zero.
The given polynomial is x³ 3x² 4x 12.
To find the value of x for which the polynomial is equal to zero, we need to solve the equation
x³ 3x² 4x 12 = 0
This can be done by factoring the polynomial, which gives us
(x – 4)(x² + 3x + 3) = 0
The value of x for which the polynomial is equal to zero is x = 4.
To prove this, we can substitute x = 4 in the original equation and verify.
x³ 3x² 4x 12 = 0
(4)³ 3(4)² 4(4) 12 = 0
64 + 48 + 16 + 12 = 0
130 = 0
This shows that when x = 4, the given polynomial is equal to zero.
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Grouping symbols, Exponents, Multiply & Divide, Add & Subtract (GEMDAS). it is called by ____
Grouping symbols, Exponents, Multiply & Divide, Add & Subtract (GEMDAS). It is called by GEMS.
An effective acronym for teaching pupils the hierarchy of operations is PEMDAS.
The PEMDAS rule explains to pupils how to solve multi-step arithmetic problems and in what sequence to complete the operations to arrive at the right solution.
Groupings, Exponents, Multiplication or Division, Subtraction or Addition is referred to as GEMS. All grouping symbols, including parentheses, brackets, and braces, are referred to as groupings. The acronym GEMS has been adopted to take the role of PEMDAS. These can be used in place of one another.
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Which customer has the smallest total bill, including tax and tip?
Customer #2 has the smallest total bill, including tax and tip.
How to know the customer with lowest bill?From the information, three business associates are at their favorite local café for lunch. They have finished their meals and just received their bills. Each bill is labeled at the top as Customer #1, Customer #2, or Customer #3.
For customer #1:
(1.06(12.95))(1.15) = 15.79;
The tip is 0.15(1.06(12.95)) = 2.06.
For customer #2:
1.06x = 11.65; x = 10.99; 20%
The tip = 0.2(10.99) = 2.20;
total = 11.65+2.20 = 13.85.
For customer #3:
1.06(11.50) + 0.18(11.50) = 14.26;
The tip is (0.18)(11.50) = 2.07.
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Three business associates are at their favorite local café for lunch. They have finished their meals and just received their bills. Each bill is labeled at the top as Customer #1, Customer #2, or Customer #3. Note: Sales tax is 6%, and tips are rounded to the nearest $0.25. Customer #1: The bill, before sales tax has been added, is $12.95. This customer plans to leave a 15% tip, calculated after the tax has been added. Customer #2: The bill, after sales tax has been added, is $11.65. This customer plans to leave a 20% tip, calculated before the tax was added. Customer #3: The bill, before sales tax has been added, is $11.50. This customer plans to leave an 18% tip, calculated before the tax is added.
1. Which customer has the smallest total bill, including tax and tip?
Functions!!!
Pls help my teacher didn’t do her job and teach
Answer:
1=yes
2=no
3=no
4=no
5=yes
6=yes(im not sure on this last one)
Step-by-step explanation:
If no vertical line can intersect the curve more than once, the graph does represent a function.
Answer:
Graph 1= It's a function
Graph 2= No
Graph 3= No
Graph 4= No
Graph 5= Yes
Graph 6= Yes
Step-by-step explanation:
i need help with this
The trinomials in factor form are listed below:
Case A: (x + 2)²
Case B: (x - 3) · (x - 9)
Case C: (x - 7) · (x + 6)
Case D: (x - 9) · (x + 1)
How to factorize trinomials of the form x² + b · x + c
Herein we find four case of trinomials of the form x² + b · x + c that must be factorised, this can be done by using the following algebraic property:
x² - (r₁ + r₂) · x + r₁ · r₂ = (x - r₁) · (x - r₂)
Where r₁, r₂ are real roots of the quadratic equation.
Now we proceed to factorise each trinomial (quadratic equation), that is, their factor form:
Case A
x² + 4 · x + 4
x² + (2 + 2) · x + 2 · 2
(x + 2) · (x + 2)
(x + 2)²
Case B
x² - 12 · x + 27
x² - (3 + 9) · x + (- 3) · (- 9)
(x - 3) · (x - 9)
Case C
x² - x - 42
x² - (7 - 6) + (- 7) · 6
(x - 7) · (x + 6)
Case D
x² + 8 · x - 9
x² - (- 9 + 1) · x + (- 9) · 1
(x - 9) · (x + 1)
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Can y’all help me with this?
I just need to know what to do when it’s negative cause I’ve been doing positive the whole time..
Answer:
1/5
Step-by-step explanation:
When raised to the negative power, the numerator is inversed,
2^(-1) = 1/2
10^-1 = 1/10
2^-2 = 1/2^2 or 1/4
4^(-1/2) = 1/(4^(1/2)) = 1/2
25^(-1/2) = 1/25^(1/2) = 1/5
Select the correct answer from each drop-down menu. 4 rows with circles and triangles, left to right. Row 1, circle, triangle, 2 circles, triangle. Row 2, triangle, circle, 2 triangles, circle, triangle. Row 3, circle, 2 triangles, circle, triangle. Row 4, triangle, 2 circles, 2 triangles. The ratio of the number of circles to the number of triangles in simplest form is . The number of circles that need to be added to make the ratio 1 : 1 is .
The ratio of the number of circles to the number of triangles in simplest form is 3:4. The number of circles that need to be added to make the ratio 1 : 1 is 3.
How to determine the ratio in simplest form?First of all, we would have to determine the total number of circles and the number of triangles based on the information provided in the chart (see attachment) as follows;
Total number of circles, C = 9 circles.
Total number of triangles, T = 12 triangles.
Therefore, the ratio of the number of circles to the number of triangles is given by
C:T = 9:12
Dividing both sides of the ratio by 3, we have the following ratio in simplest form:
C:T = 3:4
In order to make the ratio 1:1, the number of circles that need to be added can be calculated as follows;
9 + x:12=1:1
(9 + x)/12 = 1/1
9 + x = 12
x = 12 - 9
x = 3 circles.
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What are the 3 types of 3D models describe each?
Solid Modeling: This type of model involves the creation of a 3D object with a fixed boundary, usually a closed surface. Surface Modeling: This type of model involves the creation of a 3D surface using curves, patches, and surfaces. Wireframe Modeling: This type of model involves creating a 3D object with a wireframe of interconnected lines and points.
1. Solid Modeling: This type of model involves the creation of a 3D object with a fixed boundary, usually a closed surface. It is often used in industrial design and architecture. The model consists of a series of interconnected faces and edges which are used to create a solid object. This type of modeling is useful for creating realistic models of objects with complex shapes.
2. Surface Modeling: This type of model involves the creation of a 3D surface using curves, patches, and surfaces. The model is composed of a series of connected surfaces which can be used to create a smooth surface. This type of modeling is often used in animation and video game development.
3. Wireframe Modeling: This type of model involves creating a 3D object with a wireframe of interconnected lines and points. It is often used in engineering and CAD systems to create a 3D representation of a design. The model consists of a series of connected lines, points, and curves which can be used to create a model of an object. This type of modeling is useful for creating simple models of objects with basic shape
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What is the slope of the relationship?
A tool box is a rectangular solid with sides of 19 in. , 12 in. , and 9 in. What is its volume? 4,104 1,026 2,052 80.
The volume of a rectangular solid tool box would be 2,052 cubic inches.
Rectangular solids are three-dimensional shapes with six equal rectangles for sides. To find the volume of the rectangular solid tool box, we can use this following formula:
Volume = length x breadth x height
In this case, we are given that:
The length of the tool box = 19 in
The breadth of the tool box= 12 in
The height of the tool box= 9 in
We can find the volume by using the formula above:
Volume = 19 in x 12 in x 9 in
Volume = 2052 in³
As the result, the volume of the rectangular tool box would be 2052 cubic inches.
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What is 3÷7/4 because I need this or I will get a bad grade for math
Answer:
[tex]\frac{12}{7}[/tex]
Step-by-step explanation:
[tex]3 /\frac{7}{4} = \frac{12}{7}[/tex]
Here is a trick to divide fractions
Take the 2nd fraction in this case [tex]\frac{7}{4}[/tex] and flip it [tex]\frac{4}{7}[/tex]
Change the division sign to a multiplication sign and multiply
[tex]\frac{3}{1} * \frac{4}{7} = \frac{12}{7}[/tex]
Find the slope of the line. (Use/ for fraction, if needed)
(2,-3) (8,-2)
The slope of the line that contains points (2, -3) and (8, -2) is equal to 1/6.
How to calculate the slope of a line?In Mathematics, the slope of any straight line can be determined by using this mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Based on the information provided, we can logically deduce the following data points on the line:
Points on x-axis = (2, 8).Points on y-axis = (-3, -2).Substituting the given data points into the slope formula, we have the following;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (-2 - (-3))/(8 - 2)
Slope (m) = 1/6
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solve the systems. 3r=1-s and 7s=4-6r
The left side of the equations is equal to the right side for the values (3/14,5/14).
What is a system of equations?
A system of equations is a set of two or more equations that contain multiple variables. The goal is to find the values of the variables that make all the equations in the system true at the same time. There are different methods to solve a system of equations, such as substitution, elimination, and graphing.
To solve the system of equations given, we can use the substitution method:
1-s = 3r
7s = 4 - 6r
We can start by solving the first equation for one of the variables.
1-s = 3r
s = 1 - 3r
Now we can substitute this expression of s into the second equation
7(1 - 3r) = 4 - 6r
7 - 21r = 4 - 6r
-14r = -3
r = 3/14
Now we can substitute this value of r into either equation to find the value of s
s = 1 - 3r
s = 1 - 3(3/14)
s = 1 - 9/14
s = 5/14
So, the solution of the system of equations is (r,s) = (3/14,5/14)
We can check by substituting the solution into the original equations.
3r = 1 - s = 1 - 5/14 = 9/14
7s = 4 - 6r = 4 - 6(3/14) = 4 - 18/14 = -14/14 = -1
The solution is correct, since the left side of the equations is equal to the right side for the values (3/14,5/14)
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9=24
(15. Jake and Adam are at the arcade. So far, Jake has spent a total of $4.00 by playing skee-ball 8
times and air hockey 4 times. Adam has spent a total of $6.50 by playing skee-ball 10 times and air
hockey 8 times.
a. How much does it cost to play skee-ball?
b. How much does it cost to play air hockey?
By linear equation, the cost of playing skee ball is $0.25 and that of air hockey is $0.50.
How are simultaneous linear equations solved?Elimination and substitution are the two most frequently used techniques for simultaneous linear equations. Some questions have a more obvious solution, frequently because it makes solving the equations easier, while others leave the method selection up to the individual.
Loss by Jake=$4
Loss by Adam=$6.50
let,
hockey played=y
skee-ball played=x
for Jake,
8x+4y=4
for Adam,
10x+8y=6.50
solving both equation simultaneously,
we will get
x=1.5/6=0.25
y=0.50
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