Mr. Martin is giving a math test next period. The test, which is worth 100 points, has 29 problems. Each problem is worth either 5 points or 2 points. Write a system of equations that can be used to find how many problems of each point value are on the test.

Answers

Answer 1

Answer: x - the number of 5 points worth problems
y - the number of 2 points worth problems

The test has 29 problems.
$x+y=29$

The test is worth 100 points.
$5x+2y=100$

The system of equations:
$x+y=29 \n5x+2y=100$

The solution:
$x+y=29 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |* (-2) \n5x+2y=100 \n \n-2x-2y=-58 \n\underline{5x+2y=100 \ \ \ \ \ } \n3x=42 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |/ 3 \nx=14 \n \nx+y=29 \n14+y=29 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |-14 \ny=15$

There are 14 problems worth 5 points and 15 problems worth 2 points.

Answer 2

Answer:

Final answer:

The system of equations to find the number of 5-point and 2-point problems on the math test is x + y = 29 and 5x + 2y = 100, where x and y are the numbers of 5-point and 2-point problems respectively.

Explanation:

The question posed involves setting up a system of equations to determine how many of each type of problem are on a math test. There are two types of problems: those worth 5 points and those worth 2 points, with the test having a total of 29 problems and worth 100 points in all. To solve this, we can define two variables, let's say x represents the number of problems worth 5 points and y represents the number of problems worth 2 points.

We can set up the following system of equations:

x + y = 29 (This represents the total number of problems.)
5x + 2y = 100 (This represents the total points for the test.)

Solving this system will give the number of 5-point problems and 2-point problems on the test.

Learn more about System of Equations here:

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Write the equation for each transformation of f(x)= |x| described below.a. translate left 9 units, stretch vertically by a factor of 5, and translate down 23 units.

b. translate left 12 units, stretch horizontally by a factor of 4, and reflect over the x-axis.

need steps don't understand how to do!

Answers

okay, original equation of an absolute value function is:
a. f(x) = a |x-h| + k
a is the stretch or shrink
h is horizontal movement (watch the negative!!)
k is vertical shift

Translate left 9 units means horizontal shift so the h changes. When you move to the left, the numbers become negative so y = a|x-(-9)| + k which becomes
y = a|x+9| + k Then the vertical stretch of 5 becomes y = 5|x+9| + k And then a translation down 23 units means a negative shift down (which is your vertical shift) so:
f(x) = 5(x+9) - 23

b. translate left 12 units meaning a negative horizontal shift. y = a|x-(-12)| + k
so then it becomes y = a|x+12| + k
a stretch horizontally by 4 is your a, so y = 4|x+12| (you can just forget about the k since there is no vertical shift so your k = 0)
a reflection over the x-axis means that your horizontal axis is taken and folded and the reflection from the graph is your new graph. So basically, the whole equation becomes negative.
y = -4|x+12|

Final answer:

For a, the transformed function is 5*|x+9| - 23 after translating 9 units to the left, stretching vertically by a factor of 5, and translating down 23 units. For b, the transformed function is -|(x+12)/4|, after translating 12 units to the left, stretching horizontally by a factor of 4, and reflecting over the x-axis.

Explanation:

The given function is f(x) = |x|. To write the equation for each transformation, you need to understand how they influence the function.

a. To translate the function left 9 units, the value 9 needs to be added inside the absolute value brackets creating f(x) = |x+9|. To stretch it vertically by a factor of 5, we multiply the entire function by 5 - 5 * f(x) = 5*|x+9|. Lastly, to translate down 23 units, we subtract 23 from the entire function, leading us to 5*|x+9| - 23.

b. To translate left 12 units, we change the function to |x+12|. To stretch horizontally by a factor of 4, divide the x inside the absolute value by 4, getting |(x+12)/4|. To reflect over the x-axis, we multiply the entire function by -1, leading to -|(x+12)/4|.

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The graph of which function has an axis of symmetry at x =-1/4 ?f(x) = 2x2 + x – 1

f(x) = 2x2 – x + 1

f(x) = x2 + 2x – 1

f(x) = x2 – 2x + 1

Answers

we know that

The equation of the vertical parabola in vertex form is equal to

$y=a(x-h)^(2)+k$

where

(h,k) is the vertex

The axis of symmetry is equal to the x-coordinate of the vertex

$x=h$ ------> axis of symmetry of a vertical parabola

we will determine in each case the axis of symmetry to determine the solution

case A) $f(x)=2x^(2)+x-1$

Convert to vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)+1=2x^(2)+x$

Factor the leading coefficient

$f(x)+1=2(x^(2)+0.5x)$

Complete the square. Remember to balance the equation by adding the same constants to each side

$f(x)+1+0.125=2(x^(2)+0.5x+0.0625)$

$f(x)+1.125=2(x^(2)+0.5x+0.0625)$

Rewrite as perfect squares

$f(x)+1.125=2(x+0.25)^(2)$

$f(x)=2(x+0.25)^(2)-1.125$

the vertex is the point $(-0.25,-1.125)$

the axis of symmetry is

$x=-0.25=-(1)/(4)$

therefore

the function $f(x)=2x^(2)+x-1$ has an axis of symmetry at $x=-(1)/(4)$

case B) $f(x)=2x^(2)-x+1$

Convert to vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)-1=2x^(2)-x$

Factor the leading coefficient

$f(x)-1=2(x^(2)-0.5x)$

Complete the square. Remember to balance the equation by adding the same constants to each side

$f(x)-1+0.125=2(x^(2)-0.5x+0.0625)$

$f(x)-0.875=2(x^(2)-0.5x+0.0625)$

Rewrite as perfect squares

$f(x)-0.875=2(x-0.25)^(2)$

$f(x)=2(x-0.25)^(2)+0.875$

the vertex is the point $(0.25,0.875)$

the axis of symmetry is

$x=0.25=(1)/(4)$

therefore

the function $f(x)=2x^(2)-x+1$ does not have a symmetry axis in $x=-(1)/(4)$

case C) $f(x)=x^(2)+2x-1$

Convert to vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)+1=x^(2)+2x$

Complete the square. Remember to balance the equation by adding the same constants to each side

$f(x)+1+1=x^(2)+2x+1$

$f(x)+2=x^(2)+2x+1$

Rewrite as perfect squares

$f(x)+2=(x+1)^(2)$

$f(x)=(x+1)^(2)-2$

the vertex is the point $(-1,-2)$

the axis of symmetry is

$x=-1$

therefore

the function $f(x)=x^(2)+2x-1$ does not have a symmetry axis in $x=-(1)/(4)$

case D) $f(x)=x^(2)-2x+1$

Convert to vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)-1=x^(2)-2x$

Complete the square. Remember to balance the equation by adding the same constants to each side

$f(x)-1+1=x^(2)-2x+1$

$f(x)=x^(2)-2x+1$

Rewrite as perfect squares

$f(x)=(x-1)^(2)$

the vertex is the point $(1,0)$

the axis of symmetry is

$x=1$

therefore

the function $f(x)=x^(2)-2x+1$ does not have a symmetry axis in $x=-(1)/(4)$

the answer is

$f(x)=2x^(2)+x-1$

axis of symmetry is the x value of the vertex

for
y=ax^2+bx+c
x value of vertex=-b/2a

first one
-1/2(2)=-1/4
wow, that is right

answer is first one
f(x)=2x^2+x-1

The hypotenuse of a right triangle is 6 inches and one of the legs is √ 6 inches, the exact value of the other leg is _______ inches.A. 15
B. √ 15
C. 30
D. √ 30

Answers

Answer:

The answer is the option D

$√(30)\ in$

Step-by-step explanation:

we know that

In a right triangle

Applying the Pythagoras Theorem

$c^(2) =a^(2)+b^(2)$

where

c is the hypotenuse

a and b are the legs

In this problem we have

$c=6\ in\na=√(6)\ in$

substitute and solve for b

$6^(2) =(√(6))^(2)+b^(2)$

$36 =6+b^(2)$

$b^(2)=30$

$b=√(30)\ in$

The answer is D
Cause it is right triangle, so through Pythagorean theorem, you can know the answer

The area of a rectangular bathroom floor is 53 square feet. The bathroom is 6 feet wide. What is the length?

Answers

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A sports ball has a diameter of 16 cm. Find the volume of the ball.

Answers

Answer:

4/3 pi radius cubed

Step-by-step explanation:

4/3 pi x (8x8x8)

512 x 4/3 pi

2144.660585

PLEASE HELP, CURRENTLY STUCK ON THIS QUESTION ! Find the general term for the sequence -3, 1, 5, 9, ....

Answers

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