Can a system of equations have more than one solution

The correct expression representing the larger number is [tex]\( 2x + 3 \)[/tex]. So option (b) is correct.

To solve this problem, let's represent the smaller number as x and the larger number as y. According to the given conditions:

1. y is 3 more than 2 times ( x ), so ( y = 2x + 3 ).

2. The sum of the two numbers is 27, so ( x + y = 27 ).

Now, let's substitute the value of ( y ) from the first equation into the second equation:

[tex]\[ x + (2x + 3) = 27 \][/tex]

Combine like terms:

[tex]\[ 3x + 3 = 27 \][/tex]

Subtract 3 from both sides:

[tex]\[ 3x = 27 - 3 \][/tex]

[tex]\[ 3x = 24 \][/tex]

Divide both sides by 3:

[tex]\[ x = \frac{24}{3} \][/tex]

[tex]\[ x = 8 \][/tex]

Now, we can find ( y ) by substituting the value of ( x ) into the equation for ( y ):

[tex]\[ y = 2(8) + 3 \][/tex]

[tex]\[ y = 16 + 3 \][/tex]

[tex]\[ y = 19 \][/tex]

So, the smaller number ( x ) is 8, and the larger number ( y ) is 19.

Now, let's check which expression represents the larger number:

1. [tex]\( 3x + 2 = 3(8) + 2 = 24 + 2 = 26 \)[/tex]

2. [tex]\( 2x + 3 = 2(8) + 3 = 16 + 3 = 19 \)[/tex] - This matches the value we found for the larger number.

3. [tex]\( 2(x + 3) = 2(8 + 3) = 2(11) = 22 \)[/tex]

So, the correct expression representing the larger number is [tex]\( 2x + 3 \)[/tex].

Final answer:

The orange juice provides approximately 7.14% of the RDA of niacin.

Explanation:

To find the percentage of the Recommended Daily Allowance (RDA) of niacin provided by the orange juice, we need to compare the amount of niacin in the orange juice to the RDA. Given that 8 ounces of orange juice contain 1 mg of niacin and the RDA for niacin is 14 mg, we can calculate the percentage using the formula:

Percentage = (Niacin in orange juice / RDA) x 100

Substituting the values, we get:

Percentage = (1 mg / 14 mg) x 100 = 7.14%

Therefore, the orange juice provides approximately 7.14% of the RDA of niacin.

Answer:

I think it is D

Step-by-step explanation:

Option D is correct, "Use the straightedge to draw a line through points X and Y."

A perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point.

To draw perpendicular bisector of line segment AB. There are following steps:

1) Draw arcs from points X and Y on the both sides of XY.

2) Name the intersection points as A and B.

3) Use the straightedge to draw a line through points A and B.

4) Name the point as O

Hence we have construct perpendicular bisector AB of XY by using straightedge to draw a line through points X and Y.

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The statement is true; a function defined only for a set of discrete numbers, such as whole numbers, is indeed called a discrete function.

The statement that a function defined only for a set of numbers that can be listed, such as the set of whole numbers, is called a discrete function is true. A discrete function is, in fact, characterized by a domain that consists of distinct and separate elements. For example, the number of phone calls you receive on each day is a representation of discrete data because the counts can only be whole numbers, demonstrating the nature of a discrete function. This is opposed to continuous functions, which have a domain that may include all real numbers, allowing for an infinite number of values between any two different numbers.

Additionally, the concept of a function is that it relates each element of its domain to a single and unique value in its range. If the domain is composed of discrete elements like whole numbers, then the function that relates these elements to values is considered a discrete function.

Keep in mind that a characteristic function or membership function can also be considered a discrete function since it maps elements to specific values, often representing the presence (1) or absence (0) within a set.

Answer:  The required solution is   x = -2,  2.

Step-by-step explanation:  We are given to solve the following quadratic equation by square root method :

[tex]x^2-14=-10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

In square root method, we take the square of the unknown variable on one side and the constant term on the other side. Then, we take the square roots on both sides.

From equation (i), we have

[tex]x^2-14=-10\\\\\Rightarrow x^2=-10+14\\\\\Rightarrow x^2=4\\\\\Rightarrow x=\pm\sqrt{4}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}]\\\\\Rightrarow x=-2,2.[/tex]

Thus, the required solution is   x = -2,  2.

The equation y = 7x + 16 represents how long his jog will be after x additional weeks option (B) is correct.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

It is given that:

Gordon recently started jogging. Last week he jogged for 16 minutes.

Let x be the number of weeks

and y represents the total number of minutes Gordon walks

From the given data the linear equation can be framed as follows:

y = 7x + 16

Thus, the equation y = 7x + 16 represents how long his jog will be after x additional weeks option (B) is correct.

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Answer:

C. is correct

Step-by-step explanation:

The equation equivalent to y - 6 = -12(x + 4) is c. y = -12x - 42.

In mathematics, an equation is a formula that expresses the equality of two expressions by connecting them with the equal sign = .

Now the given equation is,

y - 6 = -12(x + 4)

Expanding the bracket we get,

y - 6 = -12x - 48

Adding 6 both the side we get,

y - 6 +  6 = -12x - 48 +  6

Solving we get,

y = -12x - 42

which is equivalent to the given equation.

Thus, the equation equivalent to y - 6 = -12(x + 4) is c. y = -12x - 42.

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Given:

The temperature is given in Fahrenheit (13 degree F).

To Find:

The correct temperature in Celsius.

Answer:

The temperature in Celsius is approximately -11 degree C.

Kyle has got an incorrect answer because instead of subtracting 32 form the temperature, he has added it.

Step-by-step explanation:

Any temperature can be changed from F to C by using the formula

[tex]C=(\frac{5}{9})(F-32)\\\\C=\frac{5(F-32)}{9}[/tex]

So, we will convert the given temperature in F to C by performing three simple steps.

Step 1 - We will subtract 32 from the Fahrenheit temperature.

Step 2 - Multiply this number by 5.

Step 3 - Divide the resulting number by 9.

So,

Step 1: 13-32 = -19

Step 2: -19*5 = -95

Step 3: -95/9 = -10.55 approx

Thus, the temperature in Celsius is approximately -11 degree C.

Kyle has got an incorrect answer because instead of subtracting 32 form the temperature, he has added it.

Kyle has got an incorrect answer because instead of subtracting 32 form the temperature, he has added it.

Given:

Kyle is converting 13°F to °C.

The formula used to convert F to C is

C= 5/9 (F- 32)

Now, Substitute F= 13 into the formulas as

C=  5/9 (13 - 32)

C = 5/9 (-19)

C= -95/ 9

C= -10.55

Thus, the temperature in Celsius is -11 degree C.

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Answer:

40%

Step-by-step explanation:

To find the maximum safe angle of elevation for a 110-foot ladder used at a height of 100 feet, we use the tangent inverse function with the ladder's maximum safe height as the opposite side and the ladder's hypothetical base length as the adjacent side. The angle of elevation is calculated using θ = tan⁻¹(100/adjacent). The specific calculation of the adjacent side isn't needed as the angle of elevation will be the same regardless of the actual length of the ladder along the ground.

The question is asking to find the maximum safe angle of elevation at which a 110-foot fire department ladder can be used, given that it can be safely used for rescues up to 100 feet off the ground. This is a trigonometry problem that involves the use of right-angled triangles.

To solve this problem, we will use the trigonometric function tangent (tan), which relates the angle of elevation θ to the opposite side of the triangle (the height of the building) and the adjacent side (the length of the ladder along the ground). The formula is θ=tan⁻¹(opposite/adjacent).

In this case, the opposite side is the maximum safe height of 100 feet, and the adjacent side would be the length of the ladder laid on the ground when the ladder reaches 100 feet high. We can find the length of the adjacent side using the Pythagorean theorem. However, we do not need to calculate it, as the angle of elevation does not depend on the horizontal distance but only on the ratio of the opposite side to the adjacent side.

The calculation would be: θ = tan⁻¹(100/adjacent).

Since the ladder length is greater than the height it needs to reach, the ladder base (adjacent) will be inside the triangle formed by the ladder and the building. The maximum safe angle of elevation θ can be calculated using the inverse tangent function of the height over the hypothetical base length, which is less than 110 feet.

PR = QR

5n = 32 +n

 subtract 1 n from each side

4n = 32

divide both sides by 4

n = 32/4 = 8

n=8

PR = 5n = 5*8 = 40

PR = 40

The measure of PR from the diagram given is 40

The given diagram is an isosceles triangle, For an isosceles triangle, two of sides and angles are equal. hence;

PR = QR

5n = 32 + n

Collect the like terms;

5n - n = 32

4n = 32

n = 32/4

n = 8

Get the measure of PR;

PR = 5n

PR = 5(8)

PR = 40

Hence the measure of PR from the diagram given is 40

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Answer:

4.012 to 3 decimal places

Step-by-step explanation:

The division of decimal numbers can be done easily by multiplying the numerator and denominator with a number whose power of 10 completely removes the decimal.

For 7.984 ÷ 1.99

= (7.984 × 10^3) ÷ (1.99 × 10^3)

= 7984/1990

= 4.012060302

≈ 4.012 to 3 decimal places

Answer:

D.Rotate 180 about the origin (0,0)

Step-by-step explanation:

Answer:

D) 4 7/10

Step-by-step explanation:

Any fraction may be written as a mixed number. This type of number is formed by one or more whole parts plus a fractional part.

4.7 = 4 + 0.7

4 + 7/10

4 7/10

Following this same reasoning we can turn a mixed number into improper fraction and improper fraction into mixed number.

Answer: [tex]y+7=25(x+4)[/tex]

Step-by-step explanation:

We know that the point slope form of a line that passing through (a,b) and has slope m is given by :-

[tex](y-b)=m(x-a)[/tex]

Given : Slope: m = 25   ; Point : (a,b) =(−4, −7)

Then, the point slope form of the line with slope 25 that passes through the point (−4, −7) will be :-

[tex](y-(-7))=25(x-(-4))\\\\\Rightarrow\ y+7=25(x+4)[/tex]

To find the safe load of a 10-foot long beam, first calculate the constant of proportionality k using the safe load and length of the known beam, and then multiply k by the new length. The safe load for a beam of length 10 feet would be 1333.3 pounds.

First, calculate the constant of proportionality k using the information provided: k = 2000 pounds / 15 feet = 133.33 pounds/foot.

Next, find the safe load y for a beam of length 10 feet: y = k * x = 133.33 pounds/foot * 10 feet.

Calculate the result: y = 1333.3 pounds.

Therefore, the safe load for a beam of length 10 feet would be 1333.3 pounds.

Graphing an inequality is a great way to visualize a range of possible solutions and steps for graphing the inequality are shown below.

We have to give that,

Graph of inequality.

Here's a step-by-step guide to help graph an inequality:

Step 1:

Understand the inequality. Start by reading and understanding the given inequality. Is it greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) inequality? This will determine the type of graph you will create.

Step 2:

Convert the inequality into slope-intercept form if possible. Rewrite the inequality in the form y = mx + b, where m represents the slope and b represents the y-intercept.

Step 3:

Plot the y-intercept. Identify the y-intercept from the equation obtained in Step 2, and mark it on the graph. This is the point where the line crosses the y-axis.

Step 4:

Determine the slope. If the inequality is in slope-intercept form, the coefficient of x will be your slope. If not, rearrange the inequality to isolate y and find the coefficient of x to determine the slope.

Step 5:

Plot additional points. Use the slope to find one or two more points on the line. To do this, start from the y-intercept and move up or down based on the slope (rise over run). Connect these points to form a line.

Step 6: Determine the shading. Determine whether the inequality represents a region above or below the line. If the inequality is greater than or greater than or equal to, shade the region above the line. If the inequality is less than or less than or equal to, shade the region below the line. You can use dots or arrows to indicate whether the line is included or not.

Step 7: Finalize the graph. Extend the line beyond the plotted points, making sure it remains parallel. Then, label the graph and include any necessary units or context.

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Answer:

Therefore the roots of equation 1 are 1,3 1/2,-i,+i

Step-by-step explanation:

in this question we have given

[tex]f(x) = 2x^5 - 9x^4 + 12x^3 - 12x^2 + 10x - 3 = 0[/tex].......1

we have to find the roots of this equation

put x=1 in equation 1

[tex]f(1)=2-9+12-12+10-3\\f(1)=0[/tex]

It means x=1 is root  of equation 1

now put x=3 in equation 1

[tex]f(3)=2\times3^5-9\times3^4+12\times3^3-12\times3^2+10\times3-3\\f(3)=0[/tex]

therefore,x=3 is root of equation 1

Now put x=[tex]\frac{1}{2}[/tex] in equation 1

[tex]f(\frac{1}{2})=2\times (\frac{1}{2} )^5-9\times(\frac{1}{2})^4+12\times(\frac{1}{2})^3-12*(\frac{1}{2})^2+10\times \frac{1}{2}-3\\f(\frac{1}{2})=0[/tex]

It means x=[tex]\frac{1}{2}[/tex] is also root of equation 1

therefore equation one can be written as

[tex]f(x)=(x-1)(x-3)(2x-1)(x^2+1)[/tex]

therefore remaining 2 roots can be calculated by solving

[tex]x^2+1=0\\x=\sqrt(-1)\\x=i,-i[/tex]

Therefore the roots of equation 1 are 1,3 1/2,-i,+i

To evaluate (4/9)^1/2, you find the square root of 4, which is 2, and the square root of 9, which is 3, hence the result is 2/3.

The student has asked to evaluate the expression (4/9)^1/2. To do this, you take the square root of both the numerator and the denominator separately. The square root of 4 is 2 and the square root of 9 is 3, therefore, the evaluation of the expression (4/9)^1/2 is equivalent to 2/3.

Nearsighted. It is believed that nearsightedness affects about 8% of all children. In a random sample of 194 children, 21 are nearsighted.

(a) Construct hypotheses appropriate for the following question: do these data provide evidence that the 8% value is inaccurate?

(b) What proportion of children in this sample are nearsighted?

(c) Given that the standard error of the sample proportion is 0.0195 and the point estimate follows a nearly normal distribution, calculate the test statistic (the Z-statistic).

(d) What is the p-value for this hypothesis test?

(e) What is the conclusion of the hypothesis test?