56 units are in the sum of the lengths of the three altitudes in a triangle with sides $7,$ $24,$ and $25$
Properties of triangles are:
A triangle has three sides and three angles.
The sum of the angles of a triangle is always 180 degrees.
The exterior angles of a triangle always add up to 360 degrees.
The sum of consecutive interior and exterior angle is supplementary.
How to solve?Given:
Height: ha = 7
Height: hb = 24
Height: hc = 25
Sum to sides is:
25 +7 + 24 =56 units.
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Expand (x – 2)5 using the Binomial Theorem and Pascal’s triangle. Show all necessary steps.
Answer:
[tex](x-2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32[/tex]
Step-by-step explanation:
Binomial Theorem
[tex](a+b)^n=a^n+\dfrac{n!}{1!(n-1)!}a^{n-1}b+\dfrac{n!}{2!(n-2)!}a^{n-2}b^2+...+\dfrac{n!}{r!(n-r)!}a^{n-r}b^r+...+b^n[/tex]
Given expression:
[tex](x-2)^5[/tex]
Compare with the Binomial Theorem:
[tex]a=x, \quad b=-2, \quad n=5[/tex]
Substitute the values into the formula:
[tex]\begin{aligned}(x-2)^5 & = x^5+\dfrac{5!}{1!4!}x^{4}(-2)+\dfrac{5!}{2!3!}x^{3}(-2)^2+\dfrac{5!}{3!2!}x^{2}(-2)^3+\dfrac{5!}{4!1!}x^{1}(-2)^4+(-2)^5\\\\& =x^5+\dfrac{120}{24}x^{4}(-2)+\dfrac{120}{12}x^{3}(4)+\dfrac{120}{12}x^{2}(-8)+\dfrac{120}{24}x(16)-32\\\\& =x^5-\dfrac{240}{24}x^{4}+\dfrac{480}{12}x^{3}-\dfrac{960}{12}x^{2}+\dfrac{1920}{24}x-32\\\\& =x^5-10x^{4}+40x^{3}-80x^{2}+80x-32\end{aligned}[/tex]
Pascal's Triangle
Pascal's triangle is an arrangement of binomial coefficients in triangular form. Each row has 1 at both ends, and each remaining number in the row is found by adding together the two numbers directly above it.
[tex]\phantom{-b)))))))}1\\\phantom{-b)))))}1 \:\quad 1\\\phantom{-b)))}1 \:\quad 2\: \quad 1\\\phantom{-b)}1 \:\quad 3\: \quad 3\: \quad 1\\\phantom{))}1 \quad 4 \:\: \quad 6 \quad \;4 \: \quad 1\\1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1[/tex]
(Note: The initial row with the single number 1 is row zero)
Each term in the binomial expansion using Pascal's triangle is the product of a coefficient, a with an exponent, and b with an exponent.
As we move through each term from left to right, the exponent of a decreases from the exponent of the expression down to zero, and the exponent of b increases from zero up to the exponent of the expression.
The coefficients of each term are the numbers which appear in the number of the row that corresponds with the exponent of the expression.
From the given expression, the power that we are expanding the bracket to is 5, so we need to look at line 5 of Pascal's triangle, which is:
1 5 10 10 5 1
Therefore, the expansion using Pascal's triangle to the power 5 of is:
[tex](a+b)^5=1a^5b^0+5a^4b^1+10a^3b^2+10a^2b^3+5a^1b^4+1a^0b^5[/tex]
Substituting the values of a and b:
[tex]\begin{aligned}(x-2)^5 & = 1x^5(-2)^0 + 5x^4(-2)^1 + 10x^3(-2)^2 + 10x^2(-2)^3 + 5x^1(-2)^4 + 1x^0(-2)^5\\\\ & = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32\end{aligned}[/tex]
From the triangle below, if AD = 5 and CD = 20, find the length of side BD.
Select one:
a.
10
b.
28
c.
7
d.
25
Answer: 10
Step-by-step explanation:
By the geometric mean theorem,
[tex]\frac{BD}{5}=\frac{20}{BD}\\\\(BD)^2 = 100\\\\BD=10[/tex]
The length of side BD, given that AD = 5 and CD = 20 is 10 (Option A)
Data obtained from the questionLength of side AD = 5 Length of side CD = 20Length of side BD =?How to determine the length of side BDSince the triangles are similar, we can obtain the length of side BD as illustrated below:
CD / BD = BD / AD
20 / BD = BD / 5
Cross multiply
BD × BD = 20 × 5
BD² = 100
Take the square root of both sides
BD = √100
BD = 10
Thus, the length of side BD is 10
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68. REAL-WORLD APPLICATIONS
At noon, a barista notices that she has $20 in her tip jar. If she makes an average of $0.50 from each customer, how much will she have in her tip jar if she serves n more customers during her shift?
Answer:
The linear equation for tip barista gets in a day is f(n)=0.5n+20.
Step-by-step explanation:
It is given, at a noon a barista notice that she has $20 in tip jar, if she makes an average of $0.50 from each customer,
It is required to find how much will she have in her tip jar if she serves n more customers during her shift. m=0.5. Bases on this write the equation for n more customers.
Step 1 of 1
It is given, at a noon a barista notice that she has $20 in tip jar, if she makes an average of $0.50 from each customer,
To solve this let f(n) is the total tip barista has in her tip jar after a day.
When n is equal to 0 , that barista has $20,
Then if barista make 0.5 from each customer, then for the linear function m=0.5,
Bases on this write the equation for n more customers,
[tex]$$f(n)=0.5 n+20 .$$[/tex]
Ted borrowed $5,600 from his bank for 4 months with interest at 9%. ted paid the note in full on its due date. how much was the check he gave to the bank for payment?
Ten gave the check of $5768 to the bank for payment.
What is simple interest?Simple interest is calculated based on a loan's principal or the initial deposit into a savings account. Simple interest doesn't compound, so a borrower will never have to pay interest on the interest already accumulated because a creditor will only pay interest on the principal amount.
Formula for simple interest is;-
Simple interest = (principal amount×rate of interest×time duration in years)/100
S.I = (P×R×T)/100
Calculation of the total amount paid by Ted to the bank-
The principal amount is $5,600.
The rate of interest is 9%.
Time duration is 4 months equals 4/12 = 1/3 years.
S.I = (5,600×9×1)/(3×100)
S.I = 168
The simple interest on amount is $168.
Thus, the total amount is S.I + principal amount
Total amount = 168 + 5600
= 5768
Therefore, the total amount paid by the Ted to the bank is $5768.
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What's the range of the graph? I have a menu that I could choose from and this is the option that I have.
Answer:
-∞ < x < 4
Step-by-step explanation:
→ The graph touches the y axis at 4, therefore 4 part of the range. As h ( x ) has a vertical asymptote it will never touch the line x = 1, therefore -∞.
What is the equation, in point-slope form, of the line
that is parallel to the given line and passes through the
point (-3, 1)?
Oy-1= -(x+3)
Oy-1= -(x+3)
Oy-1= (x+3)
Oy - 1 = (x+3)
Answer:
The answer is B.
Step-by-step explanation:
I learned this last year.
Solve x2 – 6x + 9 = 0 by completing the square.
A)
x = 3
B)
x = –3
C)
x = 3 and x = –3
D)
x = 1 and x = 9
Answer:
A.) x = 3
Step-by-step explanation:
This equation is an example of a perfect square quadratic. This means that the quadratic can be simplified into two of the same factors.
How can you factor the equation? The simplest way is to ask yourself, which two numbers multiply to 9 while also adding to -6? The two numbers which satisfy this question are -3 and -3. Knowing this, you can construct your factors, simplify, and then solve for "x".
x² - 6x + 9 = 0 <----- Expanded equation
(x - 3)(x - 3) = 0 <----- Factored equation
(x - 3)² = 0 <----- Combine both factors
x - 3 = 0 <----- Take square root of both sides
x = 3 <----- Add 3 to both sides
An equilateral triangle with side lengths of 8.7 centimeters is shown. An apothem has a length of a and the radius has a length of 5 centimeters. The apothem and radius form a triangle with a base length of b.
Which statements about finding the area of the equilateral triangle are true? Select three options.
The apothem can be found using the Pythagorean theorem.
The apothem can be found using the tangent ratio.
The perimeter of the equilateral triangle is 15 cm.
The length of the apothem is approximately 2.5 cm.
The area of the equilateral triangle is approximately 65 cm2.
Options A,B and D. The correction about the area of the triangle are
The apothem can be found using the Pythagorean theorem.The apothem can be found using the tangent ratio.The length of the apothem is approximately 2.5 cm.How to solve for the length of the Apothem using the Pythagoras theoremFirstly this is an equilateral triangle
3 of the sides have the length of 8.7cm
b = half of 8.7
= 4.35cm
To get A we have
a² + b² = c²
a² + 4.35² = 5²
When we solve this out we would have
a² = 25 - 18.9225
a ≈ 2.5 cm.
Hence we have proved that the first option is correct.
For B,
In an equilateral triangles, each of the angles = 60 degrees
60/2 = 30 is the angle that is made from the base
using tangent ratiosa / 4.35 = tan 30
then a = 4.35 tan 30
This would also give us an approximate of 2.5
For D
The calculations we have done based on the statements in A and B shows that D is correct as 2.5.
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New flowers are being planted
in the school garden. There
are 45 dandelions for each row
in the 15 row garden. How
many dandelions are being
planted?
Answer:
45 dandelion × 15 rows= 675
help asap!!!!!
Select the correct answer from each drop-down menu.
Consider polygon JKLMNO on the coordinate grid.
first: 12.5, 25, 17,07
second: 20, 30, 50
third: 47, 37, 61, 68.5
Answer:
12.5
30
68.5
Step-by-step explanation:
also need help with this question
Answer:
C
Step-by-step explanation:
In option c, the first block is divided into 3 pieces where 2 parts are shaded whereas the second block is divided into 6 pieces where 3 parts are shaded
For both to be equivalent, both of them should be the same
2/3 = x/6
x = 4
4 pieces should be shaded in in the second block to be equal to the first one, in the diagram only 3 parts are shaded thus the answer is c
Which function has the given properties below?
The domain is the set of all real numbers.
One x-intercept is (2 pi, 0).
The amplitude is 4.
The point (StartFraction pi over 2 EndFraction, negative 4 EndFraction) is on the graph.
The y-intercept is (0, 0).
The function that has the given properties below is y = -4sin(x).
How to illustrate the function?It should be noted that a function can be used to illustrate the relationship between variables.
In this case, the domain is the set of all real numbers, the amplitude is 4, and one x-intercept is (2 pi, 0).
The appropriate function is y = -4sin(x).
The graph is attached.
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F(x) = x^2 + 4x. G(x) = 1 + e^2x. As a single logarithm find fg(x) = 21
Answer:
1/2 log2
Step-by-step explanation:
I hope you understand
Answer:
[tex]x & = \dfrac{1}{2} \ln 2[/tex]
Step-by-step explanation:
Given functions:
[tex]\begin{cases}f(x)=x^2+4x\\g(x)=1+e^{2x}\end{cases}[/tex]
Function composition is an operation that takes two functions and produces a third function.
Therefore, the given composite function fg(x) means to substitute the function g(x) in place of the x in function f(x):
[tex]\begin{aligned}f[g(x)] & = [g(x)]^2 + 4[g(x)]\\& = \left(1+e^{2x}\right)^2+4\left(1+e^{2x}\right)\\& = 1+2e^{2x}+(e^{2x})^2+4+4e^{2x}\\& = e^{4x}+6e^{2x}+5\end{aligned}[/tex]
[tex]\textsf{Let }u=e^{2x}[/tex]
[tex]\implies e^{4x}+6x^{2x}+5=u^2+6u+5[/tex]
Set the composite function to 21 and factor:
[tex]\begin{aligned}f[g(x)] & = 21\\\implies u^2+6u+5 & = 21\\u^2+6u-16 & = 0\\u^2+8u-2u-16 & = 0\\u(u+8)-2(u+8) & = 0\\(u-2)(u+8) & = 0\\\end{aligned}[/tex]
Apply the zero-product property:
[tex]\implies (u-2)=0 \implies u=2[/tex]
[tex]\implies (u+8)=0 \implies u=-8[/tex]
[tex]\textsf{Substitute back in }u=e^{2x}:[/tex]
[tex]\implies e^{2x}=2[/tex]
[tex]\implies e^{2x}=-8[/tex]
As we cannot take logs of negative numbers, the only solution is:
[tex]\begin{aligned}e^{2x} & = 2\\\ln e^{2x} & = \ln 2\\2x \ln e & = \ln 2\\2x(1) & = \ln 2\\2x & = \ln 2\\\implies x & = \dfrac{1}{2} \ln 2\end{aligned}[/tex]
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A university is interested in promoting graduates of its honors program by establishing that the mean GPA of these graduates exceeds 4.20. A sample of 37 honors students is taken and is found to have a mean GPA equal to 4.30. The population standard deviation is assumed to equal 0.40. The parameter to be tested is __________.
The parameter to be tested is z.
We would set up the hypothesis test.
For the null hypothesis,
µ ≥ 4.2
For the alternative hypothesis,
µ < 4.2
It is a left tailed test.
Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is
z = (x - µ)/(σ/√n)
where
x = sample mean
µ = population mean
σ = population standard deviation
n = number of samples
From the information given,
µ = 4.2
x = 4.3
σ = 0.4
n = 37
z = (4.3 - 4.2)/(0.4/√37) = 1.53
Looking at the normal distribution table, the probability corresponding to the z score is 0.933
The p value gotten is to the right of the normal curve. Since it is a left tailed test, we need the p value to the left of the curve. Therefore,
p = 1 - 0.933 = 0.067
Since alpha, 0.05 < than the p value, 0.067, then we would fail to reject the null hypothesis. Therefore, At a 5% level of significance, there is no significant evidence that mean GPA of these graduates does not exceed 4.30
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Let u = <-4, 3>. Find the unit vector in the direction of u, and write your answer in component form. (2 points)
Answer: < -4/5, 3/5>
This is equivalent to writing < -0.8, 0.6 >
======================================================
Explanation:
Draw an xy grid and plot the point (-4,3) on it. Draw a segment from the origin to this point. Then draw a vertical line until reaching the x axis. See the diagram below.
We have a right triangle with legs of 4 and 3. The hypotenuse is [tex]\sqrt{4^2+3^2} = \sqrt{16+9} = \sqrt{25} = 5[/tex] through use of the pythagorean theorem.
We have a 3-4-5 right triangle.
Therefore, the vector is 5 units long. This is the magnitude of the vector.
Divide each component by the magnitude so that the resulting vector is a unit vector pointing in this same direction.
Therefore, we go from < -4, 3 > to < -4/5, 3/5 >
This is equivalent to < -0.8, 0.6 > since -4/5 = -0.8 and 3/5 = 0.6
Side note: Unit vectors are useful in computer graphics.
Juan, Mary, Celia, and Mark work as information specialists at a high tech company. They each spent a certain amount of time fixing a problem with the database. Out of the total number of hours all the workers spent, what percent of that time is Celia's?
A) 17%
B) 18%
C) 21%
D) 25%
Answer:
B) 18%
Step-by-step explanation:
100% - 36% - 29% - 17% = 18%
Answer:
the answer would be B) 18%
A line has a y-intercept of 0 and a slope of 7. What is its equation in slope-intercept form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answer:
y=7x
Step-by-step explanation:
The equation of the with a slope of m and a y-intercept of b is y=mx+b.
Brandon is a high school basketball player. In a particular game, he made some two point shots and some three point shots. Brandon scored a total of 16 points and made twice as many three point shots as two point shots. Write a system of equations that could be used to determine the number of two point shots Brandon made and the number of three point shots he made. Define the variables that you use to write the system
The system of equations are x+ y =16 and 2y=x.
What is Algebra?Algebra is the part of mathematics that helps represent problems or situations in the form of mathematical expressions
let the two points shots be x
let the three points shots be y.
According to question,
x+ y =16
and,
2y=x
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In the diagram, O is the centre of the circle, BD = DC and PAB is a straight line. Prove that AD bisects the angle CAP.
Answer:
BAC=BDC(BDX)=30°
Step-by-step explanation:
We know that BD=OD.
But OD=OB= Radius of the circle.
Therefore
BD=OD=OB
BDO is equilateral triangle.
Angle DBO= 60°
Now let us take the intersecting point of CD and AB as X.
In triangle BDX,
BXD= 90°(BXD+BXC=180°, BXD+90°=180°, BXD=90°)
BXD+DBX+BDX=180°{Angle Sum Property}
90°+60°+BDX= 180°
BDX= 30°
We also know that,
BDC(BDX)= BAC (Angles lie on the same arc{BC} are equal in measure.
Therefore,
BAC=BDC(BDX)=30°
The expressionx^2-9/x2+6x+8 is equal to zero when x = .
Hello !
Answer:
When x = -3 or x = 3
Step-by-step explanation:
[tex] \frac{ {x}^{2} - 9}{ {x}^{2} + 6x + 8} = 0[/tex]
[tex]\Leftrightarrow \frac{ {x}^{2} - 9 }{ {x}^{2} + 6x + 8} \times ( {x}^{2} + 6x + 8) = 0 \times ( {x}^{2} + 6x + 8)[/tex]
[tex]\Leftrightarrow {x}^{2} - 9 = 0[/tex]
[tex]\Leftrightarrow(x - 3)(x + 3) = 0[/tex]
[tex]\Leftrightarrow \: x - 3= 0 \: or \: x + 3 = 0[/tex]
[tex]\Leftrightarrow \: x = 3 \: or \: x = - 3[/tex]
[tex]\boxed{S = {-3 \: \: or \: \: 3}}[/tex]
What is the midpoint of the segment shown below?
A. (-15, -15)
B. (-15,-15)
c. (-1/5, -15)
D. (-15-15)
The midpoint of the segment is (-15/2, -15/2)
How to determine the midpoint?The complete question is in the attached image
The points are given as:
(-8, -7) and (-7, -8)
The midpoint is calculated as:
(x,y) = 1/2 * (x1 + x2, y1 + y2)
So, we have:
(x,y) = 1/2 * (-8 - 7, -7 - 8)
Evaluate the difference
(x,y) = 1/2 * (-15, -15)
Evaluate the product
(x,y) = (-15/2, -15/2)
Hence, the midpoint of the segment is (-15/2, -15/2)
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is 2(x+4)=4x+3-2x+5 no solution or infinite solutions?
Answer:
This is always true so there are infinite solutions
Step-by-step explanation:
2(x+4)=4x+3-2x+5
To find the solution, the first step is to distribute
2x+8 =4x+3-2x+5
Combine like terms
2x+8= 2x+8
Subtract 2x from each side
2x+8 -2x = 2x+8-2x
8 = 8
This is always true so there are infinite solutions
Find the missing lengths in the triangle above
ER=
IE=
Step-by-step explanation:
IR = 9√3
E = 60°
I = 30°
R = 90°
• Find ER.
ER/sin(I) = IR/sin(E)
ER/sin(30°) = 9√3/sin(60°)
ER/(1/2) = 9√3/(1/2 √3)
ER/(1/2) = (9√3 . 2)/√3
ER/(1/2) = 18√3/√3
ER/(1/2) = 18 → IR/sin(E)
ER = 18 . 1/2
ER = 9 is the answer
• Find IE (There are 2 ways).
First, using a Sinus Rule.
IE/sin(R) = IR/sin(E)
IE/sin(90°) = 9√3/sin(60°)
IE/1 = 9√3/(1/2 √3)
IE = (9√3 . 2)/√3
IE = (18√3)/√3
IE = 18 is the answer
Second, using a Pythagoras Theorem.
IE = √(IR² + ER²)
IE = √((9√3)² + 9²)
IE = √(81.3 + 81)
IE = √(243 + 81)
IE = √324
IE = 18 is the answer
(10 - d)^0
for d = 11
Answer:
1
Step-by-step explanation:
any number (except zero) raised to the 0th power is equal to 1.
Answer:
1
Step-by-step explanation:
(10 - 11)⁰ = ( -1)⁰ = 1
We have a⁰ = 1
I will give brainiest to who ever can copy and answer this math problem first : 12342345791 + 3214567349=
Answer:
15556913140
Step-by-step explanation:
Intervals of increasing and decreasing are written in terms of the_____value, and you will use the symbol for_______with the____–coordinate from the vertex.
The Intervals of increasing and decreasing are written in terms of the 'x' value, and you will use the symbol for 'x' with the 'y' coordinate from the vertex.
How to denote the termsThe increasing and decreasing intervals are written in the terms of the 'x' value of the function
They are represented thus;
Increasing interval = f (x) ≤ f (y)
Decreasing interval = f (x) ≥ f (y)
Thus, the Intervals of increasing and decreasing are written in terms of the 'x' value, and you will use the symbol for 'x' with the 'y' coordinate from the vertex.
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The letters $A, B$ and $C$ are used to form every possible three letter ``word.'' When these ``words'' are arranged in alphabetical order and numbered so that $AAA$ is Word $1$ and $CCC$ is Word $27$, what number will correspond to the position of word $BAB$ on the list?
The number which would correspond to the position of the word; $BAB$ is; $2×1×2$ = 4.
What number will correspond to the position of word $BAB$ on the list?As evident in the task content, by observation, if follows that since; $AAA$ is Word $1 and $CCC$ is Word $27$.
Then, it can be inferred that A = 1 , B = 2 and C = 3.
On this note, the number which would correspond to the position of the word; $BAB$ is; $2×1×2$ = 4.
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The cube of the sum of a number and 5.
Answer:
[tex](x+5)^{3}[/tex]
Step-by-step explanation:
Hope this helps
The expression for the word problem is ( x + 5 )².
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
The given word problem is the cube of the sum of a number and 5. The expression will be formed as below:-
Let the unknown number be x.
E = ( x + 5 )²
Therefore, the expression for the word problem is ( x + 5 )².
The complete question is given below:-
The cube of the sum of a number and 5. Write the mathematical expression.
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Help me please I need a hand
The statements that are true are
The sum of A and C is negativeA + D = BThe quotient of D and B is -10/3 The product of A and B > the product of A and CNumber lineFrom the question, we are to determine the statements that are true
In the given number line,
A = -2 1/6
B = - 1/2
C = 1/3
D = 1 2/3
B - C = 1/6B - C = -1/2 - 1/3
B - C = -5/6
∴ B - C ≠ 1/6
The sum of A and C is negativeA + C = -2 1/6 + 1/3
A + C = -1 5/6
∴ The sum of A and C is negative
A + D = BA + D = -2 1/6 + 1 2/3
A + D = -1/2
B = -1/2
∴ A + D = B
Quotient of D and B1 2/3 ÷ - 1/2 = -10/3
∴ The quotient of D and B is -10/3
Product of A and B
A×B = -2 1/6 × -1/2
A×B = 13/12
Product of A and C
A×C = -2 1/6 × 1/3
A×C = -13/18
∴ The product of A and B > the product of A and C
|B| < |C||B| = 1/2
|C| = 1/3
|B| > |C|
|B| is NOT less than |C|
Hence, the statements that are true are
The sum of A and C is negativeA + D = BThe quotient of D and B is -10/3 The product of A and B > the product of A and CLearn more on Number line here: https://brainly.com/question/1545224
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What is the solution to the system of equations?
Plug the second equation (the one already solved for y) into the first equation
2x - (2x + 3) = 7
2x - 2x - 3 = 7
-3 = 7
Since the 2x's canceled out and left a false statement, this system of equations has no solution.
Hope this helps!