Write the equation of each line

2. Point = (-9,3) Slope = - 2/3

4. With y-intercept = -3 and parallel to y = 5x - 2

5. With y-intercept = 9 and perpendicular to y = 1/2x + 1

Using an exponential function, it is found that 14.75 mCi of Krypton -85 will be present in the atmosphere after 12,532 days.

A decaying exponential function is modeled by:

[tex]A(t) = A(0)(0.5)^\frac{t}{h}[/tex]

In which:

In this problem, the parameters are:

A(0) = 135, h = 10.75, t = 12532/365 = 34.334.

Hence the amount is:

[tex]A(t) = A(0)(0.5)^\frac{t}{h}[/tex]

[tex]A(t) = 135(0.5)^\frac{34.334}{10.75}[/tex]

A(t) = 14.75.

14.75 mCi of Krypton -85 will be present in the atmosphere after 12,532 days.

More can be learned about exponential functions at https://brainly.com/question/25537936

#SPJ1

Taking into account definition of probability, the probability of those who like both is 0.07 or 7%.

Probability is the greater or lesser possibility that a certain event will occur.

In other words, the probability is the possibility that a phenomenon or an event will happen, given certain circumstances. It is expressed as a percentage.

The union of events, AUB, is the event formed by all the elements of A and B. That is, the event AUB is verified when one of the two, A or B, or both occurs. AUB is read as "A or B".

The probability of the union of two compatible events is calculated as the sum of their probabilities subtracting the probability of their intersection:

P(A∪B)= P(A) + P(B) -P(A∩B)

where the intersection of events, A∩B, is the event formed by all the elements that are, at the same time, from A and B. That is, the event A∩B is verified when A and B occur simultaneously.

A complementary event, also called an opposite event, is made up of the inverse of the results of another event. That is, That is, given an event A, a complementary event is verified as long as the event A is not verified.

The probability of occurrence of the complementary event A' will be 1 minus the probability of occurrence of A:

P(A´)= 1- P(A)

In first place, let's define the following events:

Then you know:

In this case, considering the definition of union of events, the probability that a chef likes carrots and broccoli is calculated from:

P(C∪B)= P(C) + P(B) -P(C∩B)

Then, the probability that a chef likes carrots and broccoli is calculated as:

P(C∩B)= P(C) + P(B) -P(C∪B)

In this case, considering the definition of the complementary event and its probability, the probability that a chef likes NEITHER of carrots and broccoli is calculated as:

P [(C∪B)']= 1- P(C∪B)

In this case, the probability of those who like neither is 0.22

0.22= 1 - P(C∪B)

Solving

0.22 - 1= - P(C∪B)

-0.78= - P(C∪B)

- (-0.78)= P(C∪B)

0.78= P(C∪B)

Now, remembering that P(C∩B)= P(C) + P(B) -P(C∪B), you get:

P(C∩B)= 0.13 + 0.72 -0.78

Solving:

P(C∩B)= 0.07=  7%

Finally, the probability of those who like both is 0.07 or 7%.

Learn more about probability:

brainly.com/question/25839839

brainly.com/question/26038361

#SPJ1

Answer:

Step-by-step explanation:

The relation between intercepted arcs and angles at crossing chords can be used to find the angles of interest. That relation tells you the angle where the chords cross is half the sum of the intercepted arcs.

The arcs intercepted by the chords making angle 1 are given as 53° and 47°. Half their sum is the measure of angle 1:

  ∠1 = (53° +47°)/2 = 100°/2

  ∠1 = 50°

Angles 1 and 2 form a linear pair, so angle 2 is the supplement of angle 1.

  ∠2 = 180° -∠1 = 180° -50°

  ∠2 = 130°

Based on the calculations, the measures of angles 1 and 2 are 50° and 135° respectively.

The theorem of intersecting chord states that when two (2) chords intersect inside a circle, the measure of the angle formed by these chords is equal to one-half (½) of the sum of the two (2) arcs it intercepts.

By applying the theorem of intersecting chord to circle U shown in the image attached below, we can infer and logically deduce that angle 1 will be given by this formula:

m∠1 = ½(53 + 47)

m∠1 = ½(100)

m∠1 = 50°.

Since angles 1 and 2 are linear pair, they are supplementary angles. Thus, we have:

m∠1 + m∠2 = 180°

m∠2 = 180 - m∠1

m∠2 = 180 - 50

m∠2 = 130°.

Read more on intersecting chords here: https://brainly.com/question/27251228

#SPJ1

Answer:

AC = 10

Step-by-step explanation:

Formula

(x + 1) + (x + 3) + 2x = 40                Remove the brackets

Solution

x + 1 + x + 3 + 2x = 40                    Collect like terms

4x + 4 = 40                                     Subtract 4 from both sides

4x +4 -4 = 40 - 4                             Combine

4x = 36                                            Divide both sides by 4

4x/4 = 36/4

x = 9

AC = x + 1

AC = 9 + 1

AC = 10

The total weight of the package was 20 pounds, given the total shipping cost was $3.45, using the total cost function, C(P) = 1.65 + 0.12(P - 5), where P is the weight of the package in pounds.

In the question, we are given that a package weighs P pounds, P being a whole number. To ship this package by express costs $1.65 for the first five pounds and 12c for each additional pound. The total shipping cost was $3.45.

We are asked for the package weight in pounds.

First, we design a general cost function.

The cost is given to be $1.65 for the first 5 pounds, and 12c for each additional pound.

The weight of the package is given to be P pounds.

We assume the total cost function to be C(P), a function of the weight of the package, P pounds.

Thus, the total cost function can be shown as the sum of $1.65 and 12c or $0.12 multiplied with (P - 5), as it is the cost for additional pounds after 5 pounds.

Thus, the total cost function, C(P) = 1.65 + 0.12(P - 5).

Now, we are given that the total shipping cost was $3.45.

Thus, to find the weight of the package, we equate the total cost function, C(P) to 3.45, to get:

1.65 + 0.12(P - 5) = 3.45,

or, 1.65 + 0.12P - 0.6 = 3.45,

or, 0.12P + 1.05 = 3.45,

or, 0.12P = 3.45 - 1.05,

or, 0.12P = 2.40,

or, P = 2.40/0.12 = 20.

Thus, the total weight of the package was 20 pounds, given the total shipping cost was $3.45, using the total cost function, C(P) = 1.65 + 0.12(P - 5), where P is the weight of the package in pounds.

Learn more about the total cost function at

https://brainly.com/question/25109150

#SPJ4

Answer: 3/2

Step-by-step explanation:

[tex]f^{-1}(8)=k \implies f(k)=8\\\\\therefore 2k+5=8\\\\2k=3\\\\k=\frac{3}{2}[/tex]

The algebraic expression for the ice cream is 3x + 2y + 0.75z.

The cost of Rafael ice cream is 12 dollars

The algebraic expression that represent the cost of an ice cream is as follows:

let

number of scoop = x

number of topping = y

number of added syrup = z

Hence,

cost of the ice cream = 3x + 2y + 0.75z

Therefore,

The cost of ice cream Rafael ordered is as follows:

cost of ice cream = 3(3) + 0.75(4)

cost of ice cream = 9 + 3

cost of the ice cream = 12

learn more on algebraic expression here: https://brainly.com/question/15591224

#SPJ1

Answer:

que necesitas? acaso necesitas algo fan de Sparta o no

Step-by-step explanation:

ok

The value of √(7 * 23 - 1)/8 is 4.47, the values of a, b and c are -14/11, -10/11 and 3, respectively and the area of the shape is 5√5 + 5 square meters

The question goes thus:

If √5 = 2.236, evaluate √(7 * 23 - 1)/8

We have:

√(7 * 23 - 1)/8

Evaluate the product of 7 and 23

√(7 * 23 - 1)/8 = √(161 - 1)/8

Evaluate the difference of 161 and 1

√(7 * 23 - 1)/8 = √160/8

Evaluate the quotient of 160 and 8

√(7 * 23 - 1)/8 = √20

Express 20 as the product 4 and 5

√(7 * 23 - 1)/8 = √(4 * 5)

Expand the product

√(7 * 23 - 1)/8 = √4 * √5

Express √4 as 2

√(7 * 23 - 1)/8 = 2 * √5

Substitute √5 = 2.236

√(7 * 23 - 1)/8 = 2 * 2.236

Evaluate the product

√(7 * 23 - 1)/8 = 4.472

Approximate

√(7 * 23 - 1)/8 = 4.47

Hence, the value of √(7 * 23 - 1)/8 is 4.47

The expression is given as:

(3√2 + 5√6)/(3√2 - 5√6)

Rationalize the above expression

(3√2 + 5√6)/(3√2 - 5√6) * (3√2 + 5√6)/(3√2 + 5√6)

Evaluate the product

(3√2 + 5√6)²/((3√2)² - (5√6)²)

Simplify the denominator

(3√2 + 5√6)²/(18 - 150)

This gives

[(3√2)² + (5√6)² + 2 *(3√2) * (5√6)]/(-132)

Simplify the numerator

[168 + 120√3]/(-132)

Simplify the fraction

-14/11 - 10√3/11

Hence, the values of a, b and c are -14/11, -10/11 and 3, respectively

The area is calculated as:

A = 1/2 * (Sum of parallel bases) * Height

So, we have:

A = 1/2 * (4 + 3√5 + 6 - √5) * √5

Evaluate the like terms

A = 1/2 * (10 + 2√5) * √5

Evaluate the product

A = (5 + √5) * √5

Evaluate the product

A = 5√5 + 5

Hence, the area of the shape is 5√5 + 5 square meters

Read more about rational expressions at:

https://brainly.com/question/8008240

#SPJ1

Yes, the non-violent, civil disobedience tactics used by civil right leaders were effective to a great extent at creating change and ending injustice.

How were the civil disobedience tactics effective?

Martin Luther King, Jr.'s and other civil right leaders' leadership and vision were shaped by their steadfast faith in the efficacy of nonviolence and civil disobedience. Although it permitted civil rights protesters to avoid harsher legal penalties, it also had a deeper significance. America only learned about the effectiveness of nonviolent protest as a result of the work of Martin Luther King, Jr. and his civil disobedience allies.

Civil Disobedience Tactics

The Civil Rights Act of 1964 and the Voting Rights Act of 1965 were primarily made possible thanks to King. The Civil Rights Act outlawed discrimination on the basis of "race, color, religion, or national origin" in the workplace and in public places. African Americans' right to vote is safeguarded by the Voting Rights Act.

Inspired by Gandhi, King and other non-violent leaders, utilized civil disobedience to pressure governments to change. It manifested as widespread, nonviolent defiance of official orders. Civil disobedience is the act of refusing to comply with governmental orders or demands and remaining unaffected by the arrest and punishment that may follow.

Learn more about civil disobedience here:

https://brainly.com/question/12041258

#SPJ4

Answer:

option 2 {(12, 3), (11,2), ...}

Step-by-step explanation:

For functions, multiple x-values can have the same y-value but each y-value must have a unique x-value. The second option matches this criterion.

Answer:

5/13

Step-by-step explanation:

sine=opposite/hypotenuse

cosine=adjacent/hypotenuse

opposite=-12 (I know that this is not possible)

hypotenuse=13

These are ratios, not real lengths

using the Pythagorean theorem, the other leg is 5

cos (theta)=5/13

If Sa=2πrh+2π[tex]r^{2}[/tex]v=π[tex]r^{2}h[/tex] then the surface area is π[tex]r^{2} h[/tex] and volume is

(rh-2h)/2r.

Given Sa=2πrh+2π[tex]r^{2} v[/tex]=π[tex]r^{2}h[/tex].

We have to find surface area and volume from the given expression.

Volume is basically amount of substance a container can hold in its capacity.

First we will find the value of v from the expression. Because they are in equal to each other, we can easily find the value of v.

2πrh+2π[tex]r^{2}[/tex]v=π[tex]r^{2}[/tex]h

Keeping the term containing v at left side and take all other to right side.

2π[tex]r^{2}[/tex]v=π[tex]r^{2}h[/tex]-2πrh

v=(π[tex]r^{2}[/tex]h-2πrh)/2π[tex]r^{2}[/tex]

v=π[tex]r^{2} h[/tex]/2π[tex]r^{2}[/tex]-2πrh/2π[tex]r^{2}[/tex]

v=h/2-h/r

v=h(r-2)/2r

Put the value of v in Sa=2πrh+2π[tex]r^{2} v[/tex]

Sa=2πrh+2π[tex]r^{2}[/tex]*h(r-2)/2r

=2πrh+2πrh(r-2)/2

=2πrh+πrh(r-2)

=2πrh+π[tex]r^{2}[/tex]h-2πrh

=π[tex]r^{2}[/tex]h

Hence surface area is π[tex]r^{2}[/tex]h and volume is h(r-2)/2.

Learn more about surface area at https://brainly.com/question/16519513

#SPJ4

The solution to the inequality 6x + 3y ≤ 84 is the dark region shown and the point (9, 10) satisfies this inequality.

An equation is an expression that shows the relationship between two or more variables and numbers.

Let x represent the number of rides Darren can ride and y represent the number of exhibits he can view. Hence:

6x + 3y ≤ 84

The solution to the inequality 6x + 3y ≤ 84 is the dark region shown and the point (9, 10) satisfies this inequality.

Find out more on equation at: https://brainly.com/question/2972832

#SPJ1

It was A! I got it right on the Edmentum/Plato test!! YW!!!

The coordinate notation is (x-3, y+1).

How can be vector converted into coordinate notation?

Let <a, b> be the vector then the coordinate notation will be

(x + a, y + b)

And if <-a, -b> be the vector then the coordinate notation will be

(x - a, y - b)

We can convert vector into coordinate notation as shown below:

For converting given vector into coordinate, we will use above formula

that is <a, b> = (x + a, y + b>

here a = -3, and b = 1

So, <-3, 1> = (x-3, y+1)

Hence, the coordinate notation is (x-3, y+1).

Learn more about Coordinates here:

https://brainly.com/question/28050024

#SPJ1

The cost per person if number of students is 100 given the inverse variation is $46.75

C = k / n

where,

If n = 55 and C = $85

C = k / n

85 = k / 55

85 × 55 = k

4,675 = k

Find C if n = 100

C = k / n

= 4,675 / 100

C = $46.75

Learn more about inverse variation:

https://brainly.com/question/13998680

#SPJ1

The equations that have the same solution as the given equation are 2.3p – 10.1 = 6.49p – 4 and 230p – 1010 = 650p – 400 – p

From the question, we are to determine the equations that have the same solution as the given equation

The given equation is

2.3p – 10.1 = 6.5p – 4 – 0.01p

Simplifying,

2.3p – 10.1 = 6.5p – 0.01p – 4

2.3p – 10.1 = 6.49p – 4

Thus, 2.3p – 10.1 = 6.49p – 4 is equivalent to the given equation

Also,

If we multiply both sides of the given equation by 100

That is,

100 × (2.3p – 10.1) = 100 × (6.5p – 4 – 0.01p)

230p – 1010 = 650p – 400 – p

∴ 230p – 1010 = 650p – 400 – p is equivalent to the given equation

Hence, the equations that have the same solution as the given equation are 2.3p – 10.1 = 6.49p – 4 and 230p – 1010 = 650p – 400 – p

Learn more on Writing equations here: https://brainly.com/question/86348

#SPJ1

For the following pair of functions, the vertex of g(x) 2 units to the right of f(x),

f(x) = x² and g(x) = (x - 2)²

What are functions?

Defining a relationship between an independent variable and a dependent variable using mathematical expressions, rules, or laws is known as a function. Mathematics is rife with functions, and the sciences depend on them for constructing physical relationships.

Determining the Correct Pair of Functions

For a given function f(x), its horizontal translation would be f(x + a). Here, a is the distance translated, owing to which f(x + a) is generated.

If a > 0, then it indicates a right translation of a units.

If a < 0, then it indicates a left translation of a units.

The required functions are to be such that the vertex of g(x) is 2 units to the right of the vertex of f(x).

Thus, if f(x) = x², then g(x) = (x - 2)²

Learn more about functions here:

https://brainly.com/question/12431044

#SPJ4

The function is f(x) = 86(1.01)^7x; grows approximately at a rate of 1% daily

The function is given as:

f(x) = 86(1.08)^x

There are 7 days in a week.

This means that:

1 day = 1/7 week

So, x days is

x day = x/7 week

Substitute x/7 for x in

f(x) = 86(1.08)^(x/7)

Rewrite as:

f(x) = 86(1.08^1/7)^x

Evaluate

f(x) = 86(1.01)^x

In the above, we have:

r = 1.01 - 1

Evaluate

r = 0.01

Express as percentage

r = 1%

Hence, the function is f(x) = 86(1.01)^7x; grows approximately at a rate of 1% daily

Read more about exponential functions at:

https://brainly.com/question/11487261

#SPJ1

Answer: the answer is d

Step-by-step explanation:

The quadratic equation given in the graph can be represented in these two following ways:

The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:

[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]

In which a is the leading coefficient.

In this graph, the roots are [tex]x_1 = 0, x_2 = 4[/tex], hence the factored form of the polynomial is:

f(x) = ax(x - 4).

When x = 5, y = 10, hence the leading coefficient is found as follows:

10 = 5a(5 - 4)

5a = 10

a = 2.

Hence the factored form of the polynomial is:

f(x) = 2x(x - 4).

The equation of a quadratic function, of vertex (h,k), is given by:

y = a(x - h)² + k

In which a is the leading coefficient.

In this problem, the vertex is at point (2,-8), hence h = 2, k = -8 and:

y = a(x - 2)² - 8

When x = 5, y = 10, hence the leading coefficient is found as follows:

10 = a(5 - 2)² - 8

9a = 18

a = 2.

Hence the equation in vertex-form is:

y = 2(x - 2)² - 8

More can be learned about quadratic functions at https://brainly.com/question/24737967

#SPJ1

The polynomial + 8r²s⁴ – 3r³s³ can have a possible first term of -rs⁵

The polynomial 5s⁶t² + 6st⁹ – 8s⁶t² – 6t⁷ has three terms with a degree of 10.

An equation is an expression that shows the relationship between two or more variables and numbers.

Given the polynomial + 8r²s⁴ – 3r³s³. The polynomial is of degree  6, hence the possible first term could be -rs⁵

Also, the polynomial 5s⁶t² + 6st⁹ – 8s⁶t² – 6t⁷

= 6st⁹ - 3s⁶t² - 6t⁷

It has three terms with a degree of 10.

Find out more on equation at: https://brainly.com/question/2972832

#SPJ1

Answer: -1

Step-by-step explanation:

Remember to use the slope formula.

[tex]\frac{y2-y1}{x2-x1}[/tex]

Then plug in your points:

[tex]m=\frac{-1-(4)}{2-(3)}[/tex]

Simplify

[tex]m=\frac{-5}{5}[/tex]

Divide -5 by 5 and your Answer is:

[tex]m=-1[/tex]

Hopes this helps!

Answer:

-1

Step-by-step explanation:

As given in the picture you provided, the slope is defined as: [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] and the reason for this, is because in subtracting y_1 from y_2, you're finding how much the y-value changed, or in other words, the rise. When subtracting x_1 from x_2 you're finding how much the x-value changed, or in other words the run. Another way of expressing the slope that you may have seen is: [tex]\frac{rise}{run}[/tex] which is essentially what this slope formula is doing.

One thing to note is that what I assign to (x_1, y_1) and (x_2, y_1) doesn't matter, as long as they're two different points on the linear line.

So let's just say that: [tex](x_1, y_1) = (-3, 4)[/tex] and that: [tex](x_2, y_2) = (2, -1)[/tex]

Now plugging these values into the equation we get: [tex]\frac{-1-4}{2-(-3)} = \frac{-5}{5} = -1[/tex]

So the slope is -1

The area of the polygon is 4 square units

The vertices are given as:

(1, 2), (3, 2), (3, 0), and (1, 0)

The area is then calculated as:

A = 0.5 * |x1y2 - x2y1 +x2y3 - x3y2 + ....... |

So, we have:

A = 0.5 * |1 * 2 - 3 * 2 + 3 * 0 - 2 * 3 + 3 * 0 - 0 * 1 + 1 * 2 - 0 * 1|

Evaluate

A = 0.5 * |-8|

Remove the absolute bracket

A = 0.5 * 8

This gives

A = 4

Hence, the area of the polygon is 4 square units

Read more about polygon at:

brainly.com/question/14522707

#SPJ1

The matrix A becomes singular when x is equal to ±√8 (positive or negative square root of 8).

We have,

Matrix A becomes singular (i.e., its determinant is zero).

The given matrix A is:

A = | 3x  -6 |

      | -4  x |

Using the determinant formula for a 2x2 matrix, we have:

det(A) = (3x * x) - (-6 * -4)

Simplifying the expression:

det(A) = 3x^2 - 24

To find the value of x for which det(A) = 0, we set the determinant equal to zero:

3x^2 - 24 = 0

Now, we can solve this quadratic equation for x:

3x^2 = 24

x^2 = 8

x = ±√8

Therefore,

The matrix A becomes singular when x is equal to ±√8 (positive or negative square root of 8).

Learn more about matrix here:

https://brainly.com/question/28180105

#SPJ4

The slope intercept form of the line whose points are (1,5) and (-2,-4) is y=3x+2.

Given two points of a line (1,5) and (-2,-4).

We have to form an equation in slop intercept form.

Equation is relationship between two or more variables which are expressed in equal to form.Equations of two variables looks like ax+by=c.

Point slope form of an equation is y=x+mc where m is slope of the line.

From two points the formula of equation is as under:

(y-[tex]y_{1}[/tex])=[tex](y_{2} -y_{1} )/(x_{2} -x_{1} )[/tex]*(x-[tex]x_{1}[/tex])

where [tex](x_{1} ,y_{1} ) and (x_{2} ,y_{2} )[/tex] are the points.

Putting the values of [tex]x_{1}[/tex]=1, [tex]x_{2}[/tex]=-2, [tex]y_{1}[/tex]=5 and [tex]y_{2}[/tex]=-4.

y-5=(-4-5/-2-1)*(x-1)

y-5=-9/-3*(x-1)

y-5=3(x-1)

y-5=3x-3

y=3x-3+5

y=3x+2

Hence the slope intercept form of the line having points (1,5)(-2,-4) is y=3x+2.

Learn more about equation at https://brainly.com/question/2972832

#SPJ4

The probability that all three players obtain an odd sum is 3/14.

The probability is the ratio of possible distributions to the total distributions.

I.e.,

Probability = (possible distributions)/(total distributions)

Given that,

There are nine tiles - 1, 2, 3,...9, respectively.

A player must have an odd number of odd tiles to get an odd sum. That means he can either have three odd tiles, or two even tiles and an odd tile.

In the given nine tiles the number of odd tiles = 5 and the number of even tiles = 4.

The only possibility is that one player gets 3 odd tiles and the other two players get 2 even tiles and 1 odd tile.

So,

One player can be selected in [tex]^3C_1[/tex]  ways.

The 3 odd tiles out of 5 can be selected in [tex]^5C_3[/tex] ways.

The remaining 2 odd tiles can be selected and distributed in [tex]^2C_1[/tex] ways.

The remaining 4 even tiles can be equally distributed in [tex]\frac{4 ! \cdot 2 !}{(2 !)^{2} \cdot 2 !}[/tex] ways.

So, the possible distributions = [tex]^3C_1[/tex] × [tex]^5C_3[/tex] × [tex]^2C_1[/tex] × [tex]\frac{4 ! \cdot 2 !}{(2 !)^{2} \cdot 2 !}[/tex]

⇒ 3 × 10 × 2 × 6 = 360

To find the total distributions,

The first player needs 3 tiles from the 9 tiles in [tex]^9C3=84[/tex] ways

The second player needs 3 tiles from the remaining 6 tiles in [tex]^6C_3=20[/tex] ways

The third player takes the remaining tiles in 1 way.

So, the total distributions = 84 × 20 × 1 = 1680

Therefore, the required probability = (possible distributions)/(total distributions)

⇒ Probability = 360/1680 = 3/14.

So, the required probability for the three players to obtain an odd sum is 3/14.

Learn more about the probability of distributions here:

https://brainly.com/question/2500166

#SPJ4

The conditions required for a random variable X to follow a Poisson process. The probability of success is the sme for aany two intervals of equal length. The probability of two or more successes in any sufficiently small subinterval is 0.

Learn more about continuous function here:

https://brainly.com/question/18102431

#SPJ4

Answer:

Jim is 5

Step-by-step explanation:

word problems can be sort of like translation questions

'times' means multiply

'as old as' means equal or =

'will be' means equal or =

'more than' means plus or +

when I saw

"Dad is 4 times as old as his son Jim is now"

I knew needed to make

Jim = x &

Dad = 4x

when I saw

"In 10 years, Dad's age will be 20 years more than twice Jim's age"

I knew

'Dad's age' means 4x

'In 10 years' means 10 +

'will be' means equal sign or =

'20 years more than' means 20 +

'twice Jim's age' means 2x

sentence can be re written

4x+10=20+2x

'if D = dad's age now' means 4x=D (Dad)

'J = Jim's age now' means x = J (Jim)

I solved for x

4x+10=20+2x

2x=10

x=5

then I checked the math

today

x = Jim = J = 5

4x = Dad = D = 20

'in 10 years'

Dad is 20 + 10 = 30

means

does 30 = 20 years more than twice Jim's age?

does 30 = 20 + 2x ?

does 30 = 20 + 2(5)?

does 30 = 20 + 10?

does 30 = 30?

yes!

it checks out!

Answer + Step-by-step explanation:

Recall That the number of solution

of a quadratic equation ax² + bx + c = 0

depends on the discriminant b² - 4ac :

if b² - 4ac > 0 , the equation has two distinct solutions.

if b² - 4ac = 0 , the equation has only one solution.

if b² - 4ac < 0 , the equation has no solutions.

=======================================

System 1 :

y = x² + 4x + 7  and y = 2

⇔ x² + 4x + 7 = 2

⇔ x² + 4x + 5 = 0

→ b² - 4ac = 4² - 4×1×5 = 16 - 20 = -4 < 0

Then the quadratic equation has no solutions

Therefore the system has no solutions.

System 2 :

y = x² - 2  and y = x + 5

⇔ x² - 2 = x + 5

⇔ x² - x - 7 = 0

→ b² - 4ac = (-1)² + 4×7 = 29 > 0

Then the quadratic equation has two solutions

Therefore the system has two solutions.

System 3 :

y = -x² - 3  and y = 9 + 2x

⇔  -x² - 3 = 9 + 2x

⇔ -x² - 2x - 12 = 0

→ b² - 4ac = (-2)² - 4×(-1)×(-12) = 4 - 48 = -44 < 0

Then the quadratic equation has no solutions

Therefore the system has two solutions.

System 4 :

y = -3x - 6  and y = 2x² - 7x

⇔  -3x - 6 = 2x² - 7x

⇔ 2x² - 4x + 6 = 0

→ b² - 4ac = (-4)² - 4×(2)×(6) = 16 - 48 = -32 < 0

Then the quadratic equation has no solutions

Therefore the system has two solutions.

System 5 :

y = x² and y = 10 - 8x

⇔ x² = 10 - 8x

⇔ x² + 8x - 10 = 0

→ b² - 4ac = 8² - 4×1×(-10) = 64 + 40 = 104 > 0

Then the quadratic equation has two solutions

Therefore the system has two solutions.