8.1.2 Exam: Semester 1 Exam Question 36 of 40 Solve 3-212. A. X≥-18 Correct! B. X. The answer is in the picture.
Answers
The solution to the given inequality is x ≤ -18 which is the correct answer that would be an option (D).
The inequality is given in the question as:
3 -x/2 ≥ 12
To solve this inequality, we need to isolate the variable x. To do this, we can start by subtracting 12 from both sides of the inequality:
3 - x/2 ≥ 12
3 - x/2 - 12 ≤ 0
Next, we can simplify the left side of the inequality by combining like terms:
-x/2 - 9 ≤ 0
Finally, we can solve for x by adding 9 to both sides and multiplying both sides by -2:
x ≤ -18
Thus, the solution to the inequality is x ≤ -18.
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Related Questions
Out of 400 plants sold daily at a nursery, 90% of them have flowers. How many of the
plants have flowers?
A. 40
B. 90
C. 310
D. 360
10 points.
Answers
Answer: D; 360
Step-by-step explanation: To find percentages, I like to set up proportional relationships to find the amount. It would look like this:
[tex]\frac{90}{100}[/tex] = [tex]\frac{?}{400}[/tex]
90= part %
100= whole %
?= amount of the %
400= all of the item(s)
So, we would cross multiply the 90 and the 400, then divide it by 100. That would be 36,000. So, that divided by 100 leaves us with 360. I hope this helped!
Prove that if a ≡ b mod m and c ≡ d mod m, then ac ≡ bd mod m. [duplicate]
Answers
The four integers say a,b,c and d. If a and b are congruent to moulo m and c ,d are congruent to modulo m then ac and bd are must be congruent to modulo m .
What is congruent modulo ?Two integers a and b are "congruent modulo n" if their difference is a multiple of n or in other words we say n/(b-a) . For example, 17 and 5 are congruent modulo 3 because 17 - 5 = 12 = 4×3
We have, a ≡ b mod m => m/(a - b)
and c ≡ d mod m => m/(c-d)
Now, a + c - b - d = (a-b) + (c-d)
because, m/( a - b) and m/(c-d)
=> m/( a-b+ c-d )
=> m/(a+c)-(b+d)
=> a+c ≡ b+d mod m
We have to prove that ac ≡ bd mod m.
ac - bd = c(a-b) + b(c- d)
Now, as we see, m/(a-b) => m/c(a-b)
and m/(c-d) => m/b(c-d)
=> m/(c(a-b) + b(c-d))
=> m/(ac - bc + bc - bd)
=> m/(ac - bd)
=> ac ≡ bd mod m
Hence proved.
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Unique buys 918 Spyder Porsche worth $845,000 every year that he owns the car
the car depreciates in value at a rate of 32% per year.
Write a function that models the value of the Porsche after t years.
*
Answers
The function for the value of the Porsche after t years is [tex]y = 845000(1-\frac{32}{100} )^t[/tex].
What is function?Function is a combination of different types of variable and constants in which for the different values of x the value of function y is unique.
Given that,
The initial price of the 918 Spyder Prosche car = $845,000
Since, the price of car depreciated at the rate of 32% per year.
The function for the value of Porsche car, after t years, can be given as,
The price of car after t year is y,
[tex]y = 845000(1-\frac{32}{100} )^t[/tex]
The required function, the models, the value of the Porsche after t years is [tex]y = 845000(1-\frac{32}{100} )^t[/tex].
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25% of what number is 40?
Percent=unknown, Amount = 40, Base = 25
Percent = 25 %. Amount = 40. Base=unknown
Percent = 25 %. Amount = unknown. Base = 40
Percent = 40 %. Amount = 25. Base=unknown
Answers
Answer: percent: 25% amount: 40 base: unknown
Base after finding it: 160
Step-by-step explanation:
40/x = 25%/100
butterfly method
25x/25 = 4000/25
simplify; cancel out
x (aka the base) = 160
Find the distance between the points.
(3,4) and (3,1)
Answers
. To find the distance between two points, use the distance formula.
It will be the square root of (x2-x1)squared + (y2-y1)squared.
Use (3,-4) as x1, y1. Use (5,1) as x2, y2.
We get the square root of (5-3)squared + (1- -4)squared.
That gives us the square root of 2 squared + 5 squared.
Now we have the square root of 4 + 25, = the square root of 29.
That is the distance between the two points.
Which of the following is the equation for a line parallel to the containing the points (0,4) and (8,0) in the xy-plane?
Answers
Answer: y = -1/2x + 4.
The equation of line passes through the points (0, 4) and (8, 0) will be;
⇒ y = -1/2x + 4
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
Two points on the line are (0, 4) and (8, 0).
Now,
Since, The equation of line passes through the points (0, 4) and (8, 0)
So, We need to find the slope of the line.
Hence, Slope of the line is,
m = (y₂ - y₁) / (x₂ - x₁)
m = (0 - 4) / (8 - 0)
m = - 4 / 8
m = -1/2
We know that, The slopes of parallel lines are equal.
Thus, The equation of line with slope -1/2 is,
⇒ y - 4 = -1/2 (x - 0)
⇒ y - 4 = -1/2x
⇒ y = -1/2x + 4
Therefore, The equation of line passes through the points (0, 4) and
(8, 0) will be;
⇒ y = -1/2x + 4
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Find out the missing frequency in the following incomplete distribution
CI F
0-10 3
10-20 ?
20-30 20
30-40 12
40-50 ?
The Value of median and mode are 27 and 26 respectively.
Answers
8 and 7 are missing frequency in the incomplete distribution.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
Class Interval frequency Cumulative frequency
0 - 10 3 3
10 - 20 a 3+a
20 - 30 20 23+a
30 - 40 12 35+a
40 - 50 b 35+a+b
Mode = 26 hence lies in 20 - 30
f of modal class = 20
Mode = 20 + 10( 20 - a) / (20 - a + 20 - 12)
26 = 20 + 10 (20 - a) / (28 - a)
60(28 - a) = 10 ( 20 - a)
168 -6a = 200 - 10a
4a = 32
a = 8
Class Interval frequency Cumulative frequency
0 - 10 3 3
10 - 20 8 11
20 - 30 20 31
30 - 40 12 43
40 - 50 b 43+b
Median = 27
27 = 20 + 10 ( (43 + b)/2 - 11)/ 20
7 = ( (21 + b)/ 4
28 = (21 + b)
b = 7
Hence, the missing frequencies are 8 and 7.
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Find integers a, b that do not have a greatest common divisor. Prove that the pair of numbers that you found are the only pair of integers that do not have a gcd.
Answers
C(n) = min(αn, βn) then gcd(0,b) = p(1)^(c1)∙p(2)^(c2)..................p(n)^(cn). So for all other a, b have gcd exist.
The integer a=0 and b=0 does not have any greatest common divisor.
Suppose n = gcd(0,r)
Then 12|0 and 12|0 also n is the greatest divides both (a=0), (b=0).
But n+1|0, n+1|0 as (n+1)*0 = 0 which is a contradiction to n = gcd(0, r).
Hence, gcd(0, r) does not exist.
But if a≠0 or b≠0.
It has a unique prime factorization that is;
a= p(1)^(α1)∙p(2)^(α2)∙p(3)^(α3)......................p(n)^(αn),
b= p(1)^(β1)∙p(2)^(β2)∙p(3)^(β3)......................p(n)^(βn).
Let C(n) = min(αn, βn) then gcd(0,b) = p(1)^(c1)∙p(2)^(c2)..................p(n)^(cn).
So for all other a, b have gcd exist.
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Convert 284 °F to Celsius
Answers
Hi, To convert a temperature from degrees Fahrenheit to degrees Celsius, you can use the following formula:
C = (F - 32) / 1.8
Substituting the value of F and solving for C, we find that:
C = (284 - 32) / 1.8
= 252 / 1.8
= 140
Therefore, 284 °F is equivalent to 140 °C.
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A biologist is studying a plant blight. At the start of their research, the blight covers a circular area with a radius of 5 kilometers. When they re-examine the blight two years later, the area has grown to a circular area with a radius of5.2kilometers. How rapidly did the radius change over the interval? _____Year/km
Answers
Step-by-step explanation:
Area of circle 1= pie r²
=3.14 x 5²
=3.14x 25
= 78.5km²
Area of circle 2=3.14 x 5.2²
= 3.14 x 27.04
= 84.9km²
change in area= 84.9-78.5
=6.4
radius²= 6.4/ 3.14
r²=2.03
r =√2.03
r=1.42
Suppose 6 adults produce 189.6 lb of garbage in one week. At this rate, how many pounds will 40 adults produce in one week?
Answers
Answer: 1264 pounds of garbage in one week
Step-by-step explanation: This problem is a unit rate problem, so, we need to find the unit rate of this word problem. To do that, you would do the amount of garbage produced (189.6) divided by the number of adults (6), which equals 31.6. So, the unit rate is 31.6 pounds of garbage for one adult. Then, you need to do 31.6 times 40 (for the number of adults) which equals 1264.
Witch rate is equivalent to 15 milesper hours
Answers
Answer: 60 miles per 4 gallons. If that's the question.
Step-by-step explanation:
The polynomial of degree 5,
P(x) has leading coefficient 1, has roots of multiplicity 2 at x=3 and x=0, and a root of multiplicity 1 at x=−5 Find a possible formula for P(x).
Answers
The possible formula for P(x) is x⁵-x⁴-21x³+45x²
What is a polynomial?
Polynomial is formed composed of the phrases Nominal, which means "terms," and Poly, which means "many." An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable). Based on the number of terms present in the expression, it is categorised as monomial, binomial, and trinomial.
P(x) has roots of multiplicity 2 at x=3 and x=0.
The equation stands, P(x) = x²(x-3)²
P(x) has a root of multiplicity 1 at x=-5
The equation stands, P(x) = x²(x-3)²(x+5)
Simplifying the equation,
P(x) = x²(x²+9-6x)(x+5)
⇒ P(x) = (x⁴+9x²-6x³)(x+5)
⇒ P(x) = x⁵+5x⁴-6x⁴-30x³+9x³+45x²
⇒ P(x) = x⁵-x⁴-21x³+45x²
The possible formula for P(x) is x⁵-x⁴-21x³+45x²
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A leasing agreement is for 5 years. A sum of £10,000 must be paid at the beginning of the first year, to be followed by four equal payments of £x at the beginning of years two, three, four and five. At a discount rate of 8 per cent, the present value of the four equal payments is £26,496. Compute the total amount to be paid during the lease period.
Answers
The total amount paid during the lease period is £45,665.64.
To find the total amount paid during the lease period, we need to add the initial payment of £10,000 to the present value of the four equal payments, which can be calculated using the present value formula:
PV = FV / (1 + r)^n
Where PV is the present value, FV is the future value, r is the discount rate, and n is the number of periods. In this case, the present value of the four equal payments is £26,496, the discount rate is 8%, and the number of periods is 4. We can solve for the future value by rearranging the formula:
FV = PV × (1 + r)^n
Plugging in the given values, we get:
FV = £26,496 × (1 + 0.08)^4
FV = £26,496 × 1.34
FV = £35,664.64
This is the total future value of the four payments. To find the total amount paid during the lease period, we need to add this to the initial payment of £10,000. Therefore, the total amount paid during the lease period is:
£35,664.64 + £10,000 = £45,665.64
Therefore the total amount paid during the lease period is £45,665.64.
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1. According to the text, the three IT trends that are having the greatest impact on the IT environment are: A. Privacy management, virus suppression, and enterprise portals B. Grid computing, on-demand computing, and real-time computing C. Digital forensics, the distributed workforce, and grid computing D. Knowledge management, privacy management, and intranets E. Privacy management, enterprise warehouses, and knowledge management
Answers
Computing on demand, on the cloud, and in real time IT Enviroment is most impacted by Are is known as gird computing.
What is Data Analysis?
Data analysis is the methodical application of logical and/or statistical approaches to describe and demonstrate, summarise and assess, and assess data.
Concept:
Step 1: A network of connected computers that act as a single supercomputer to complete complex tasks like modelling or data analysis is known as gird computing.
Step 2 : On-demand computing is a delivery method in which users are provided with computing resources as needed.
Step 3 : Real-time computing refers to a computer system that responds to an event by completing a task in a predetermined amount of time. a system for air traffic management as an example.
Grid computing, on-demand computing, and real-time computing
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7. Heather used a basic fact to complete
the equation below. What is the
missing number? Explain how you
can use a basic fact to find the
missing number.
4,200 + = 700
Answers
The missing value of x = 6
What is value ?
Value in mathematics can refer to a number of closely related ideas. A mathematical value can generally be any specific mathematical object. This is most frequently a number in elementary mathematics, such as a real number like or an integer like 42.
Depending on where it is in the number, each digit has a different value, which is referred to as value. We figure it out by multiplying the digit's place value by its face value. Place value plus face value equals value.
Values are the benchmarks or ideals by which we judge the acts, traits, possessions, or circumstances of others. Values that are embraced by many include those of beauty, honesty, justice, peace, and charity.
4200 / x = 700
If the fraction is defined, the denominator cannot equal 0:
x ≠ 0
Append the inequality as the domain:
x ≠ 0
Multiply both sides of the equation by the common denominator:
400x / x = 700 x
Reduce the fractions:
4200 = 700 x
Swap the sides:
700 x = 4200
Divide both sides of the equation by the coefficient of variable:
x = 4200 / 700
Cross out the common factor:
x = 6
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I need. Help with this question
Answers
Derivative of function is 5(2x + 4) * (x² + 4x + 6)⁴.
What is differentiation?
In arithmetic, the derivative of a function of a true variable measures the sensitivity modification of the perform price with relation to a change in its argument. Derivatives are a elementary tool of calculus.
Main body:
by using product rule;
Let = u = x² + 4x + 6
⇒ du/dx = 2x + 4 .
Now y = u⁵
⇒ dy/dx = 5u⁴ .
dy/dx = dy/dx * du/dx = 5u⁴ * (2x + 4)
= 5 * (x² + 4x + 6)⁴ * (2x + 4)
= 5(2x + 4) * (x² + 4x + 6)⁴
hence , derivative of function is 5(2x + 4) * (x² + 4x + 6)⁴.
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Solve using Substitution Method
x=4-8y
3x +24y = 12
Answers
Observing the provided equation, We can't solve these equations using substitution method or any other methods because these lines are parallel and never intersect each other.
What do you mean by equations?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
What is a substitution method?The algebraic approach to solving simultaneous linear equations is known as substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies.
x=4-8y
3x+24=12
Taking first equation and rearranging it we get,
x+8y=4
multiply this with 3, we get 3x+24y=12 which are identical in nature. Which shows these two lines are parallel in nature. We can't solve them because parallel lines never intersect.
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Observing the provided equation, We can't solve these equations using substitution method or any other methods because these lines are parallel and never intersect each other.
What do you mean by equations?
The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
What is a substitution method?
The algebraic approach to solving simultaneous linear equations is known as substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies.
x=4-8y
3x+24=12
Taking first equation and rearranging it we get,
x+8y=4
multiply this with 3, we get 3x+24y=12 which are identical in nature. Which shows these two lines are parallel in nature. We can't solve them because parallel lines never intersect.
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which of the above diagrams correctly portray a perfectly competitive firm's demand ( d) curve? a. a b. d c. b d. c
Answers
Correct option will be (C) In the short run, firms may incur economic losses or earn economic profits, but in the long run they earn normal profits.
The short run is a concept that states that, within a certain period in the future, at least one input is fixed while others are variable. In economics, it expresses the idea that an economy behaves differently depending on the length of time it has to react to certain stimuli. The short run does not refer to a specific duration of time but rather is unique to the firm, industry or economic variable being studied.
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Analysis of daily output of a factory shows that, on average, the number of units per hour y produced after t hours of production is
y = 102t + 0.5t2 − t3, 0 ≤ t ≤ 10.
(a) Find the critical values of this function. (Assume −[infinity] < t < [infinity]. Enter your answers as a comma-separated list.)
t =
(b) Which critical values make sense in this particular problem? (Enter your answers as a comma-separated list.)
(c) For which values of t, for 0 ≤ t ≤ 10, is y increasing? (Enter your answer using interval notation.)
Answers
The critical values for the function are t = 6 , -5.667 where t = 6 makes sense . At t = 6 , y is increasing
According to the question,
After t hours of production, a plant produces an average of y units per hour, according to analysis of its daily output is
y = 102t + 0.5t² - t³ , 0 ≤ t ≤ 10
(a) To calculate critical values for the given function
We have to differentiate it w.r.t time and then have to put y'=0
=> y = 102t + 0.5t² - t³
Derivating w.r.t time
=> y' = 102 + t - 3t²
Equating to zero,
=> 102 + t - 3t² = 0
Solving quadratic equation,
=> t = 6 , -5.667 are critical values
(b) t = 6 makes sense in particular problem because time cannot be negative.
(c) To check increasing value
Derivate y' w.r.t time again
=> y" = 1 - 6t
Putting critical value,
At t = 6
y" < 0 , Therefore at t = 6 function is increasing
Hence , For t = 6 y is increasing
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Which conjecture, if any, could be made about the sum of two even integers and one odd integer?
The sum will be an even integer.
The sum will be negative.
It is not possible to make a conjecture.
The sum will be an odd integer.
Answers
The sum will be an odd integer.
What is an integer?
An integer is an integer (not a fraction) that is positive, negative, or zero.
In mathematics, integers are collections of integers and negative numbers. Like whole numbers, whole numbers do not contain fractions. Therefore, we can say that integers are numbers that can be positive, negative, or zero, but which are not fractions.
Let p,q,r be three random numbers
Let S be the sum of two even integers and one odd integer
S = 2p + 2q + (2r + 1)
S = 2p + 2q + 2r + 1
S = 2(p+q+r)+1
This shows that S is odd integer for all values of p,q,r
Therefore, the sum will be an odd integer.
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the absolute value of -7-3
negative 7 minus three
Answers
-7-1=-8
-7-2=-9
-7-3=-10
a room three times as long as it is broad and height is 4.6m. st of carpeting its floor at Rs. 240 per sq.m. be Rs. nd the cost of painting the 4 walls at Rs. 24 per sq.m.
Answers
The total cost of painting and carpeting the 4 walls and the floor is C = ₹(720x² + 1324.8x).
What is the area and perimeter of a rectangle? What is a mathematical function, equation and expression? rectangle perimeter : 2{L + B} (L - length, B - breadth)rectangle area : {L x B} (L - length, B - breadth)function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators.equation : A mathematical equation is used to equate two expressions.Given is that a room three times as long as it is broad and its height is 4.6 m. The cost of carpeting its floor at Rs. 240 per square meter be the cost of painting the 4 walls at Rs. 24 per sq. m.
Assume that the breadth to be equivalent to [x] meters. Then its length will be [3x] meters.
Area of floor = {3x} x {x} = 3x² sq. m.
The cost of carpeting the floor = Rs. 240 per sq. m.
For 1 sq. m., the cost is 240 sq. m.
For {3x² sq. m}, the cost will be -
C{f} = 240 x 3x² = 720x²
C{f} = 720x²
The total area of the 4 walls = 4 × (3x) × 4.6 = 55.2x sq. m.
The cost of carpeting the walls = Rs. 24 per sq. m.
For {55.2x sq. m}, the cost will be -
C{w} = 24 x 55.2x = 1324.8x
C{w} = 1324.8x
The total cost of painting and carpeting -
C = C{w} + C{f} = 720x² + 1324.8x
C = ₹(720x² + 1324.8x)
Therefore, the total cost of painting and carpeting the 4 walls and the floor is C = ₹(720x² + 1324.8x).
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5w-15 as equivalent fraction
Answers
The required equivalent expression is given as 5(w - 3).
What is the equation?The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.
Here,
An equivalent expression of any expression is given as the simplifying the expression using various operations such as factors.
So,
Given expression,
= 5w - 15
As in the expression 5 is a common term between 5w and 15.
Take 5 commons and put the rest of the expression in the parenthesis.
= 5(w - 3)
Thus, the required expression is 5(w - 3).
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A video store manager observed that the number of DVDs sold seem to vary inversely as the price per DVD. If the store sells 710 DVDs per week when the price per DVD is $18.90, how many does he expect to sell if he lowers the price to $11.40.
Answers
If the price is reduced to $11.40 the number of DVDs sold will be 1177.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that a video store manager observed that the number of DVDs sold seems to vary inversely to the price per DVD. If the store sells 710 DVDs per week when the price per DVD is $18.90.
The number of DVDs sold for $11.40 will be calculated by forming the inverse equation for the number of DVDs.
N = K / P
Here, N is the number and P is the price and K is the inverse constant.
The value of K will be,
K = N x P
K = 18.90 x 710
K = 13419
The number of DVDs for $11.40 will be,
N = K / P
N = 13419 / 11.40
N = 1177
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Debs and Jamie share sweetss in the ratio 3:7. Debs has 20 less sweets than Jamie. How many sweets are there in total? Show all workings please
Answers
Using ratios we know that the total number of sweet Debs and Jamie has is 50.
What is the ratio?A ratio in mathematics demonstrates how many times one number is present in another.
For instance, if a dish of the fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six.
Two quantities are compared to form a ratio.
An equality of two ratios is a proportion.
How to format a ratio Analyze the ratio to see whether it is part to part or part to whole.
Calculate the whole and the pieces as necessary.
So, we have a ratio of 3:7.
Debs has 20 fewer sweets than Jamie.
Let, Jamie have x sweets and Debs has x - 20.
Now, we get the equation:
3/x - 20 = 7/x
Now, solve as follows:
3/x - 20 = 7/x
3x = 7(x - 20)
3x - 7x - 140
7x - 3x = 140
4x = 140
x = 140/4
x = 35
x - 20 = 35 - 20 = 15
Total: 35 + 15 = 50
Therefore, using ratios we know that the total number of sweet Debs and Jamie has is 50.
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(7x6)÷2[{45÷3(7×2-15+6)}+4)
Answers
Answer:
21/79
Step-by-step explanation:
(7x6)÷2[{45÷3(7×2-15+6)}+4)
(7*6)/2[{15(7*2-15+6)}+4)
(42)/2[{15(14-15+6)}+4)
(42)/2[{15(5))}+4)
(42)/2[{75}+4)
(42)/2[75 + 4]
(42)/2 * 79
42/2 * 79
21/79
the width of a rectangle is 6y - 7.5 feet and the length is 7.5y + 8 feet find the perimeter of the rectangle
Answers
The perimeter of the rectangle is, 3y + 1 feet.
What is perimeter?
The circumference of a shape is known as its perimeter. The lengths of all four sides must be added in order to determine a rectangle's or square's perimeter. In this instance, the rectangle's length is represented by x and its width by y.
P = x + x + y + y represents the perimeter, P.
Let the length of triangle is 7.5y + 8 feet and the width of triangle is 6y - 7.5 feet.
So, the perimeter of rectangle is,
Perimeter = 2(length + width)
= 2(7.5y + 8 + 6y - 7.5)
= 2(1.5y + 0.5)
Perimeter = 3y + 1
Hence, the perimeter of the rectangle is, 3y + 1 feet.
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simplify (5/3-7/4)+√1/16+√1/9
Answers
Answer:
1/2
Step-by-step explanation:
(5/3-7/4)+√1/16+√1/9
= {(5x4-7x3)/12} + 1/4 + 1/3
= -1/12 + (3+4)/12
= (-1+7)/12
= 6/12
= 1/2
If a matrix equation [math]Ax=b[/math] has one unique solution, does that imply the columns of A are linearly independent?
Answers
The matrix equation Ax=b with one unique solution says that column A is linearly independent.
Two vectors are linearly dependent if at least one of them can be expressed as a linear combination of the others. If no vector can be expressed as a linear combination of the others, then two or more vectors are said to be linearly independent.
Given the matrix equation is Ax=b. And this equation has one unique solution. Then, let us assume this unique solution is x₁. Let the column values of A be A₁, A₂, A₃, .... Aₙ.
If there is only one solution to the equation Ax = b for all b, then assume b = 0 and come to the conclusion that there is only one solution to the equation Ax = 0 as well. Consequently, the only solution is straightforward, and the columns of A are linearly independent.
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