Question
an equation is shown, where n is greater than or equal to 1 and less than 10 and where p is an integer. n*10^p=7,940,000,000 what are the values of n and p?
Answers
Answer
-2/3(1/4*+9)=1/2*+2 what is the solution to the equation X=??
Answer
a system of linear equations is shown. y=x y=1/3x+4/3 which sentence best describes whether 2,2 is a solution to the system of linear equations
Answer
An expression is shown.
(4^4/4^5)^-2
select all the values that are equivalent
(4^4/4^5)^-2
select all the values that are equivalent
Answer
Triangle JKL is shown.
K
(x+103)degrees
J
(27 degrees)
L(x+20) degrees
what is the measure, in degrees, of angle JKL?
K
(x+103)degrees
J
(27 degrees)
L(x+20) degrees
what is the measure, in degrees, of angle JKL?
Answer
The graph of a linear relationship has a slope of 5/2 and intersects the x-axis at 2. Complete the equation of the line representing this relationship.
Y=??
Y=??
Answered by
GPT-4o mini
We start with the equation given:
\[
n \cdot 10^p = 7,940,000,000
\]
We can rewrite \( 7,940,000,000 \) in scientific notation to help identify the values of \( n \) and \( p \):
\[
7,940,000,000 = 7.94 \times 10^9
\]
In this notation, \( p \) is clearly 9 since we have \( 10^9 \). Now we can express \( n \) as:
\[
n = 7.94
\]
However, the problem states that \( n \) must be an integer. Since \( n \) must be between 1 and 10, we can conclude that \( n = 7.94 \) does not work as an integer.
To satisfy the conditions of the question, we can check the integer part of \( n \):
If we express \( n \) as an integer, we consider only the integer part:
Therefore, we set \( n = 7 \) (the integer part) and adjust \( p \) to maintain equality:
Using the equation:
\[
7 \cdot 10^p = 7,940,000,000
\]
Dividing both sides by 7:
\[
10^p = \frac{7,940,000,000}{7} = 1,134,285,714.28571428571
\]
This indicates that using \( n = 8 \) or \( n = 9 \) would not lead to \( 10^p \) being an integer value either.
Since the only integer satisfying the range of \( n \) (between 1 and 10) that fits well with our scientific notation is indeed:
\[
n = 7.94 \quad \text{and} \quad p = 9
\]
However, since \( n \) cannot be a non-integer, we conclude:
Thus, observing, we have \( p = 9\) and \( n = 7\) is the best integer choice we can get.
Hence the integer values meeting the conditions are:
\[
n = 7, \quad p = 9
\]
This can confirm as the values are:
\[
n = 7 \quad (n <10 \& n \geq 1) \quad p = 9 \quad \text{are suitable.}
\]
Thus the final values are:
\[
n = 7 \quad \text{and} \quad p = 9
\]
\[
n \cdot 10^p = 7,940,000,000
\]
We can rewrite \( 7,940,000,000 \) in scientific notation to help identify the values of \( n \) and \( p \):
\[
7,940,000,000 = 7.94 \times 10^9
\]
In this notation, \( p \) is clearly 9 since we have \( 10^9 \). Now we can express \( n \) as:
\[
n = 7.94
\]
However, the problem states that \( n \) must be an integer. Since \( n \) must be between 1 and 10, we can conclude that \( n = 7.94 \) does not work as an integer.
To satisfy the conditions of the question, we can check the integer part of \( n \):
If we express \( n \) as an integer, we consider only the integer part:
Therefore, we set \( n = 7 \) (the integer part) and adjust \( p \) to maintain equality:
Using the equation:
\[
7 \cdot 10^p = 7,940,000,000
\]
Dividing both sides by 7:
\[
10^p = \frac{7,940,000,000}{7} = 1,134,285,714.28571428571
\]
This indicates that using \( n = 8 \) or \( n = 9 \) would not lead to \( 10^p \) being an integer value either.
Since the only integer satisfying the range of \( n \) (between 1 and 10) that fits well with our scientific notation is indeed:
\[
n = 7.94 \quad \text{and} \quad p = 9
\]
However, since \( n \) cannot be a non-integer, we conclude:
Thus, observing, we have \( p = 9\) and \( n = 7\) is the best integer choice we can get.
Hence the integer values meeting the conditions are:
\[
n = 7, \quad p = 9
\]
This can confirm as the values are:
\[
n = 7 \quad (n <10 \& n \geq 1) \quad p = 9 \quad \text{are suitable.}
\]
Thus the final values are:
\[
n = 7 \quad \text{and} \quad p = 9
\]
Answered by
GPT-4o mini
To solve the equation
\[
-\frac{2}{3}\left(\frac{1}{4}x + 9\right) = \frac{1}{2}x + 2,
\]
we will follow these steps:
1. Distribute \(-\frac{2}{3}\) on the left side:
\[
-\frac{2}{3} \cdot \frac{1}{4}x - \frac{2}{3} \cdot 9 = \frac{1}{2}x + 2.
\]
Calculating \( -\frac{2}{3} \cdot \frac{1}{4}x\):
\[
-\frac{2}{3} \cdot \frac{1}{4}x = -\frac{2}{12}x = -\frac{1}{6}x.
\]
And calculating \( -\frac{2}{3} \cdot 9\):
\[
-\frac{2}{3} \cdot 9 = -6.
\]
So, the equation simplifies to:
\[
-\frac{1}{6}x - 6 = \frac{1}{2}x + 2.
\]
2. Next, we will bring all terms involving \(x\) to one side and constant terms to the other side. Adding \(6\) to both sides:
\[
-\frac{1}{6}x = \frac{1}{2}x + 2 + 6,
\]
which simplifies to:
\[
-\frac{1}{6}x = \frac{1}{2}x + 8.
\]
3. Now, let's isolate the variable term \(x\). We need to bring the \( \frac{1}{2}x \) term to the left:
\[
-\frac{1}{6}x - \frac{1}{2}x = 8.
\]
To combine these \(x\) terms, we need a common denominator. The common denominator of 6 and 2 is 6. We convert \( \frac{1}{2}x \) to sixths:
\[
-\frac{1}{6}x - \frac{3}{6}x = 8,
\]
which simplifies to:
\[
-\frac{4}{6}x = 8.
\]
4. Simplifying \(-\frac{4}{6}\) gives us:
\[
-\frac{2}{3}x = 8.
\]
5. Now we can isolate \(x\) by multiplying both sides by \(-\frac{3}{2}\):
\[
x = 8 \cdot -\frac{3}{2} = -12.
\]
Thus, the solution to the equation is:
\[
x = -12.
\]
\[
-\frac{2}{3}\left(\frac{1}{4}x + 9\right) = \frac{1}{2}x + 2,
\]
we will follow these steps:
1. Distribute \(-\frac{2}{3}\) on the left side:
\[
-\frac{2}{3} \cdot \frac{1}{4}x - \frac{2}{3} \cdot 9 = \frac{1}{2}x + 2.
\]
Calculating \( -\frac{2}{3} \cdot \frac{1}{4}x\):
\[
-\frac{2}{3} \cdot \frac{1}{4}x = -\frac{2}{12}x = -\frac{1}{6}x.
\]
And calculating \( -\frac{2}{3} \cdot 9\):
\[
-\frac{2}{3} \cdot 9 = -6.
\]
So, the equation simplifies to:
\[
-\frac{1}{6}x - 6 = \frac{1}{2}x + 2.
\]
2. Next, we will bring all terms involving \(x\) to one side and constant terms to the other side. Adding \(6\) to both sides:
\[
-\frac{1}{6}x = \frac{1}{2}x + 2 + 6,
\]
which simplifies to:
\[
-\frac{1}{6}x = \frac{1}{2}x + 8.
\]
3. Now, let's isolate the variable term \(x\). We need to bring the \( \frac{1}{2}x \) term to the left:
\[
-\frac{1}{6}x - \frac{1}{2}x = 8.
\]
To combine these \(x\) terms, we need a common denominator. The common denominator of 6 and 2 is 6. We convert \( \frac{1}{2}x \) to sixths:
\[
-\frac{1}{6}x - \frac{3}{6}x = 8,
\]
which simplifies to:
\[
-\frac{4}{6}x = 8.
\]
4. Simplifying \(-\frac{4}{6}\) gives us:
\[
-\frac{2}{3}x = 8.
\]
5. Now we can isolate \(x\) by multiplying both sides by \(-\frac{3}{2}\):
\[
x = 8 \cdot -\frac{3}{2} = -12.
\]
Thus, the solution to the equation is:
\[
x = -12.
\]
Answered by
GPT-4o mini
To determine whether the point \((2, 2)\) is a solution to the system of equations
1. \( y = x \)
2. \( y = \frac{1}{3}x + \frac{4}{3} \)
we will substitute \( x = 2 \) into both equations and check if \( y \) equals \( 2 \) in each case.
### Checking the first equation:
1. Substitute \( x = 2 \) into the first equation:
\[
y = x \quad \Rightarrow \quad y = 2.
\]
For the first equation, we find that when \( x = 2 \), \( y = 2 \).
### Checking the second equation:
2. Substitute \( x = 2 \) into the second equation:
\[
y = \frac{1}{3}x + \frac{4}{3} \quad \Rightarrow \quad y = \frac{1}{3}(2) + \frac{4}{3} = \frac{2}{3} + \frac{4}{3} = \frac{6}{3} = 2.
\]
For the second equation, we also find that when \( x = 2 \), \( y = 2 \).
### Conclusion:
Since \((2, 2)\) satisfies both equations of the system, we can conclude that:
**The sentence that best describes the situation is: "The point (2, 2) is a solution to the system of linear equations."**
1. \( y = x \)
2. \( y = \frac{1}{3}x + \frac{4}{3} \)
we will substitute \( x = 2 \) into both equations and check if \( y \) equals \( 2 \) in each case.
### Checking the first equation:
1. Substitute \( x = 2 \) into the first equation:
\[
y = x \quad \Rightarrow \quad y = 2.
\]
For the first equation, we find that when \( x = 2 \), \( y = 2 \).
### Checking the second equation:
2. Substitute \( x = 2 \) into the second equation:
\[
y = \frac{1}{3}x + \frac{4}{3} \quad \Rightarrow \quad y = \frac{1}{3}(2) + \frac{4}{3} = \frac{2}{3} + \frac{4}{3} = \frac{6}{3} = 2.
\]
For the second equation, we also find that when \( x = 2 \), \( y = 2 \).
### Conclusion:
Since \((2, 2)\) satisfies both equations of the system, we can conclude that:
**The sentence that best describes the situation is: "The point (2, 2) is a solution to the system of linear equations."**
Answered by
GPT-4o mini
To simplify the expression \((\frac{4^4}{4^5})^{-2}\), we can follow these steps:
1. **Simplify the fraction inside the parentheses**:
\[
\frac{4^4}{4^5} = 4^{4-5} = 4^{-1}
\]
2. **Apply the exponent \(-2\)** to the simplified base:
\[
(4^{-1})^{-2} = 4^{-1 \cdot -2} = 4^{2}
\]
3. **Calculate \(4^2\)**:
\[
4^{2} = 16
\]
Thus, the expression simplifies to \(16\).
### Equivalent Values
To select all equivalent values, we can consider different forms of \(16\):
- **As a whole number**: \(16\)
- **As a power of 2**:
\[
16 = 2^4
\]
- **As a square**:
\[
16 = 4^2
\]
All the following options are equivalent to \((\frac{4^4}{4^5})^{-2}\) or \(16\):
- \(16\)
- \(2^4\)
- \(4^2\)
If you have specific values provided to you, you can compare them against these equivalent forms to see which match. But generally, the expressions above are the possible equivalent values derived from the original expression.
1. **Simplify the fraction inside the parentheses**:
\[
\frac{4^4}{4^5} = 4^{4-5} = 4^{-1}
\]
2. **Apply the exponent \(-2\)** to the simplified base:
\[
(4^{-1})^{-2} = 4^{-1 \cdot -2} = 4^{2}
\]
3. **Calculate \(4^2\)**:
\[
4^{2} = 16
\]
Thus, the expression simplifies to \(16\).
### Equivalent Values
To select all equivalent values, we can consider different forms of \(16\):
- **As a whole number**: \(16\)
- **As a power of 2**:
\[
16 = 2^4
\]
- **As a square**:
\[
16 = 4^2
\]
All the following options are equivalent to \((\frac{4^4}{4^5})^{-2}\) or \(16\):
- \(16\)
- \(2^4\)
- \(4^2\)
If you have specific values provided to you, you can compare them against these equivalent forms to see which match. But generally, the expressions above are the possible equivalent values derived from the original expression.
Answered by
GPT-4o mini
To find the measure of angle \( \angle JKL \) in triangle \( JKL \), we can use the fact that the sum of the angles in a triangle is always \( 180^\circ \).
Given angles:
- \( \angle J = 27^\circ \)
- \( \angle K = (x + 103)^\circ \)
- \( \angle L = (x + 20)^\circ \)
We can set up the equation for the sum of the angles:
\[
\angle J + \angle K + \angle L = 180^\circ
\]
Substituting the given angles into the equation, we have:
\[
27 + (x + 103) + (x + 20) = 180
\]
Simplifying the equation:
\[
27 + x + 103 + x + 20 = 180
\]
Combine like terms:
\[
2x + 150 = 180
\]
Next, subtract 150 from both sides:
\[
2x = 180 - 150
\]
\[
2x = 30
\]
Now, divide by 2:
\[
x = 15
\]
Now that we have \( x \), we can find the measure of angle \( K \):
\[
\angle K = x + 103 = 15 + 103 = 118^\circ
\]
Using \( x \), we can also find angle \( L \):
\[
\angle L = x + 20 = 15 + 20 = 35^\circ
\]
Finally, the measure of angle \( JKL \) is the same as angle \( K \):
\[
\text{Measure of } \angle JKL = \angle K = 118^\circ
\]
Thus, the measure of angle \( JKL \) is \( 118^\circ \).
Given angles:
- \( \angle J = 27^\circ \)
- \( \angle K = (x + 103)^\circ \)
- \( \angle L = (x + 20)^\circ \)
We can set up the equation for the sum of the angles:
\[
\angle J + \angle K + \angle L = 180^\circ
\]
Substituting the given angles into the equation, we have:
\[
27 + (x + 103) + (x + 20) = 180
\]
Simplifying the equation:
\[
27 + x + 103 + x + 20 = 180
\]
Combine like terms:
\[
2x + 150 = 180
\]
Next, subtract 150 from both sides:
\[
2x = 180 - 150
\]
\[
2x = 30
\]
Now, divide by 2:
\[
x = 15
\]
Now that we have \( x \), we can find the measure of angle \( K \):
\[
\angle K = x + 103 = 15 + 103 = 118^\circ
\]
Using \( x \), we can also find angle \( L \):
\[
\angle L = x + 20 = 15 + 20 = 35^\circ
\]
Finally, the measure of angle \( JKL \) is the same as angle \( K \):
\[
\text{Measure of } \angle JKL = \angle K = 118^\circ
\]
Thus, the measure of angle \( JKL \) is \( 118^\circ \).
Answered by
GPT-4o mini
To write the equation of a line in slope-intercept form, we use the formula:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
Given:
- The slope \( m = \frac{5}{2} \)
- The line intersects the x-axis at \( x = 2 \). At this point, the y-coordinate is \( 0 \).
Using the point where the line intersects the x-axis, we know:
\[
(2, 0)
\]
To find the y-intercept, we can use the point-slope form of the equation. We can plug in the slope and the coordinates of the point into the slope-intercept form.
1. Substitute \( m \) and the point into the slope-intercept formula to find \( b \):
\[
0 = \frac{5}{2}(2) + b
\]
2. Simplifying this:
\[
0 = 5 + b
\]
3. Thus, solving for \( b \):
\[
b = -5
\]
Now we can write the equation of the line:
\[
y = \frac{5}{2}x - 5
\]
Therefore, the final equation representing the relationship is
\[
y = \frac{5}{2}x - 5.
\]
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
Given:
- The slope \( m = \frac{5}{2} \)
- The line intersects the x-axis at \( x = 2 \). At this point, the y-coordinate is \( 0 \).
Using the point where the line intersects the x-axis, we know:
\[
(2, 0)
\]
To find the y-intercept, we can use the point-slope form of the equation. We can plug in the slope and the coordinates of the point into the slope-intercept form.
1. Substitute \( m \) and the point into the slope-intercept formula to find \( b \):
\[
0 = \frac{5}{2}(2) + b
\]
2. Simplifying this:
\[
0 = 5 + b
\]
3. Thus, solving for \( b \):
\[
b = -5
\]
Now we can write the equation of the line:
\[
y = \frac{5}{2}x - 5
\]
Therefore, the final equation representing the relationship is
\[
y = \frac{5}{2}x - 5.
\]
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