How many solutions are there to the following system of equations?

3x + 7y = -10
15x + 35y = -45

2
1
infinitely many
0

The solution to the system of equations are ±2i and ±i

Given the quartic equation

x^4 + 3x^2 + 2 = 0

Let u = x^2 to have:

(x^2)^2 + 3x^2 + 2 = 0

u^2 + 3u + 2 = 0

Factorize

u^2 + u +2u + 2 = 0

u(u + 1) + 2(u + 1) = 0

(u + 2)(u + 1) = 0

u = -2 or -1

If u = x^2

-2 = x^2

x = √-2

x = 2i

Similarly

If u = x^2

-1 = x^2

x = √-1

x = i

Hence the solution to the system of equations are ±2i and ±i

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Final answer:

To solve the equation x^4 + 3x^2 + 2 = 0 using u substitution, substitute x^2 for u and solve the quadratic equation. The resulting solutions in the complex number system are x = i, -i, √2i, and -√2i. There are no real solutions for this equation.

Explanation:

The solutions of the equation x4 + 3x2 + 2 = 0 can be found using u substitution. First, let's substitute u for x2, which transforms the original equation into the form u2 + 3u + 2 = 0. This is a quadratic equation, and we can solve it using the quadratic formula or by factoring. Factoring (u + 1)(u + 2) = 0, we get the solutions for u as u = -1 and u = -2. Since u is a substitute for x2, we then solve for x.

For u = -1, the equation x2 = -1 has no real solutions because the square of a real number cannot be negative. However, in the complex number system, the solutions are x = i and x = -i, where i is the square root of -1.

For u = -2, the equation x2 = -2 also has no real solutions, but the complex solutions are x = √2i and x = -√2i.

Therefore, the full set of solutions for the original equation are x = i, -i, √2i, and -√2i.

The distance from Honolulu to Turtle Bay is approximately 37 miles. To calculate this distance, you can use a map or a GPS tool. The most direct route between the two locations is along the Kamehameha Highway.

The distance from Honolulu to Turtle Bay is approximately 37 miles. To calculate this distance, you can use a map or a GPS tool. The most direct route between the two locations is along the Kamehameha Highway.

It's important to note that the actual distance traveled may vary depending on the specific route taken and any detours or alternate paths chosen.

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The exact value of cos 15° is (√6 - √2)/4.

To find the exact value of cos 15°, we can use the trigonometric identities and special angles to determine its value.

One of the key identities that can be used is the cosine addition formula: cos (A + B) = cos A x cos B - sin A x sin B.

We can rewrite 15° as the sum of two special angles: 15° = 45° - 30°.

Using the cosine addition formula, we have:

cos 15° = cos (45° - 30°) = cos 45° x cos 30° - sin 45° x sin 30°.

Now, we know the exact values of cos 45° (which is (√2)/2), cos 30° (which is (√3)/2), sin 45° (which is (√2)/2), and sin 30° (which is 1/2).

Substituting these values into the equation, we get:

cos 15° = (√2)/2 x (√3)/2 - (√2)/2 x 1/2.

Simplifying further, we have:

cos 15° = (√6 - √2)/4.

This means that cos 15° can be expressed exactly as (√6 - √2)/4 without any decimal approximation.

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Answer:

the answer is check2+check6/4

Step-by-step explanation:

i took an exam and got it right

(i dont know what those check mark things are called or how to put the actual thing on a computer so hopefully you understand the check part)

A scatter plot depicting height and age reveals trends: positive correlation if points slope up, negative if down, none if scattered. Regression lines aid age prediction from height. The plot guides decisions by identifying patterns and outliers, supporting informed conclusions based on observed relationships.

Steps to interpret a scatter plot of a person's height and age:

1. **Scatter Plot Description:**

  - Axes: Height on the y-axis, Age on the x-axis.

  - Each point represents an individual's height and age.

  - Scatter points may form a pattern or cluster.

2. **Trends and Interpretation:**

  - **Positive Correlation:** If the points generally slope upward from left to right, it suggests a positive correlation—taller people tend to be older.

  - **Negative Correlation:** If the points slope downward, it implies a negative correlation—taller people tend to be younger.

  - **No Correlation:** If the points seem randomly scattered, there may be no apparent correlation.

3. **Predicting Age from Height:**

  - Fit a regression line to the data. A linear regression line can be used to predict age given height.

  - For a given height, find the corresponding point on the regression line to estimate the age.

4. **Identifying Trends and Decision Making:**

  - **Patterns:** Identify patterns and correlations between variables.

  - **Outliers:** Notice any data points significantly deviating from the trend.

  - **Decision Support:** Use the trends to make informed decisions. For instance, if there's a correlation between height and age, you might anticipate certain health issues associated with age.

Scatter plots are powerful visual tools to analyze relationships between variables, identify trends, and make predictions or informed decisions based on observed patterns.

Answer:

its 282.6 sq cm

Step-by-step explanation:

AL=2πrh=2·π·3·15≈282.74334

Final answer:

To convert 7 square yards to square meters, you multiply by the conversion factor, resulting in approximately 5.85 square meters. Hence, the correct answer is C. 5.85 square meters.

Explanation:

7 square yards is equal to how many square meters? To convert square yards to square meters, you can use the conversion factor 1 square yard = 0.83612736 square meters. Therefore, to convert 7 square yards to square meters:

7 sq yards * 0.83612736 sq meters/sq yard = 5.85289152 sq meters

When rounding to two decimal places, this is approximately 5.85 square meters. So, the correct answer would be C. 5.85 square meters.

If triangle EFG is congruent to triangle STU, the following statements are true:

* Angle E = angle S.

* Angle F = angle T.

* Side EF = side ST.

* Side FG = side TU.

* Side EG = side SU.

If triangle EFG is congruent to triangle STU, then we can conclude that the following pairs of angles are congruent:

* Angle E is congruent to angle S.

* Angle F is congruent to angle T.

This is because congruent triangles have corresponding angles that are congruent.

We can also conclude that the following pairs of sides are congruent:

* Side EF is congruent to side ST.

* Side FG is congruent to side TU.

* Side EG is congruent to side SU.

This is because congruent triangles have corresponding sides that are congruent.

Therefore, the following statements are true:

* Angle E = angle S.

* Angle F = angle T.

* Side EF = side ST.

* Side FG = side TU.

* Side EG = side SU.

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Final answer:

The values of k that ensure the points (-5, 5) and (k, 0) are [tex]\(\sqrt{29}\)[/tex]units apart are -3 and -7, derived using the Pythagorean theorem and algebraic manipulation.

Explanation:

To find all values of k so that the points (-5, 5) and (k, 0) are \(\sqrt{29}\) units apart, we use the distance formula derived from Pythagoras’ theorem, which is[tex]d = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex]. Here, [tex]\(x_1 = -5\), \(y_1 = 5\), \(x_2 = k\), and \(y_2 = 0\). Substituting the given values, we get:[/tex]

[tex]\(\sqrt{29} = \sqrt{(k + 5)^2 + (0 - 5)^2}\)[/tex]

Squaring both sides to eliminate the square root gives:

[tex]29 = (k + 5)^2 + 25[/tex]

Subtracting 25 from both sides, we get:

[tex]4 = (k + 5)^2[/tex]

Therefore, \(k + 5 = \pm2\), which simplifies to:

[tex]\(k + 5 = 2\) giving \(k = -3\)[/tex]

[tex]\(k + 5 = -2\) giving \(k = -7\)[/tex]

The values of k are -3 and -7, so the correct option is c. -3, -7.

The correct answer is option (c) -3, -7.

How is it so?

To find the values of k such that the given points (-5, 5) and (k, 0) are [tex]\(\sqrt{29}\)[/tex] units apart, use the distance formula between two points in a plane:

[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Given that the distance between the points is [tex]\(\sqrt{29}\)[/tex], we have:

[tex]\[ \sqrt{(k - (-5))^2 + (0 - 5)^2} = \sqrt{29} \][/tex]

[tex]\[ \sqrt{(k + 5)^2 + 5^2} = \sqrt{29}[/tex]]

[tex]\[ \sqrt{k^2 + 10k + 25 + 25} = \sqrt{29} \][/tex]

[tex]\[ \sqrt{k^2 + 10k + 50} = \sqrt{29} \][/tex]

Squaring both sides to eliminate the square root:

[tex]\[ k^2 + 10k + 50 = 29 \][/tex]

[tex]\[ k^2 + 10k + 21 = 0 \][/tex]

Solve this quadratic equation

[tex]\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Where a = 1, b = 10, and c = 21.

[tex]\[ k = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1} \][/tex]

[tex]\[ k = \frac{-10 \pm \sqrt{100 - 84}}{2} \][/tex]

[tex]\[ k = \frac{-10 \pm \sqrt{16}}{2} \][/tex]

[tex]\[ k = \frac{-10 \pm 4}{2} \][/tex]

[tex]\[ k = \frac{-10 + 4}{2} \] or \( k = \frac{-10 - 4}{2} \)[/tex]

[tex]\[ k = \frac{-6}{2} \] or \( k = \frac{-14}{2} \)[/tex]

[tex]\[ k = -3 \] \:or \( k = -7 \)[/tex]

So, the correct answer is option (c) -3, -7.

Answer: Option 'B' is correct.

Step-by-step explanation:

Since we have given that

Dimensions of the photo is given by

[tex]20\ in.\ by\ 24\ in.[/tex]

According to question, the photo is reduced so that the longer dimension is 15 inch.

So, Let the width of the reduced photo be x.

So, it becomes,

[tex]\frac{20}{x}=\frac{24}{15}\\\\x=\frac{15\times 20}{24}\\\\x=12.5\ in.[/tex]

So, the width of the reduced photo is 12.5 inch.

Hence, Option 'B' is correct.

The number 0.2 x 10⁶ in scientific notation is written as 2 x 10⁵.

To write the number 0.2 x 10⁶ in scientific notation, we need to convert it into a format that has one non-zero digit to the left of the decimal point, followed by the appropriate exponent of 10.

The given number, 0.2 x 10⁶, is already close to scientific notation, however, we typically want the first number to be between 1 and 10. In this case, we can express 0.2 as 2 x 10⁻¹. Multiplying this by 10⁶ gives us:

2 x 10⁻¹ x 10⁶ = 2 x 10⁽⁶⁻¹⁾ = 2 x 10⁵.

So, the number 0.2 x 10⁶ in scientific notation is 2 x 10⁵.

Answer/Step-by-step explanation:

149.99+(149.99*0.07)=160.4893. Round to $160.49

Answer:

A) -5

Step-by-step explanation:

The solution is store A

The store which sells the bicycle with a better rate is store A with a price of  $ 40.00 for two bicycles

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the first store be represented as A

Let the second store be represented as B

Now , the equation will be

The price of 2 bicycles at store A = $ 40.00

So ,

The price of 1 bicycle at store A = price of 2 bicycles at store / 2

Substituting the values in the equation , we get

The price of 1 bicycle at store A = 40/2

The price of 1 bicycle at store A = $ 20

Now ,

The price of 5 bicycles at store B = $ 110

So ,

The price of one bicycle at store B = The price of 5 bicycles at store B / 5

Substituting the values in the equation , we get

The price of one bicycle at store B = 110/5

The price of one bicycle at store B = $ 22

And ,

The price of one bicycle at store B > The price of 1 bicycle at store A

22 > 20

Hence , the store A provides better rate for the price of one bicycle

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A polygon can grow from one-dimensional objects, which are lines. Zero-dimensional objects, which are points or vertices, also play a role as they define the positions where lines meet, but a polygon cannot directly grow from three-dimensional objects because polygons are two-dimensional shapes.

A polygon can grow from one-dimensional objects, specifically from lines. In geometric terms, a polygon is a two-dimensional shape that is formed by connecting straight line segments. Each line segment connects to two others to form the polygon's perimeter. Moreover, the line segments join at their endpoints, known as vertices, which are zero-dimensional objects that contain only a single coordinate pair.

The formation of a polygon from lines is analogous to constructing a shape on a piece of paper by drawing lines that connect to form a closed loop. In this process, the lines are the sides of the polygon, and the points where the lines meet are the vertices. It is not possible for a polygon to directly grow from three-dimensional objects, as polygons are inherently two-dimensional.

For example, to represent a soccer ball in a simplified manner such as in geographic information systems (GIS), we can use polygons in the shape of pentagons and hexagons. The soccer ball itself is a three-dimensional object but the representation involving pentagons and hexagons is a two-dimensional schematic, emphasizing the importance of polygons in various mapping and modeling applications.