Question
Write and solve the equation for the direct variation y varies directly with x if y=112
when x=16
. Find the value of y when x=11
.(1 point)
Responses
When x=11
, y=7
.
When x equals 11 , y equals 7 .
When x=11
, y=162.8
.
When x equals 11 , y equals 162.8 .
When x=11
, y=77
.
When x equals 11 , y equals 77 .
When x=11
, y=11/7
.
when x=16
. Find the value of y when x=11
.(1 point)
Responses
When x=11
, y=7
.
When x equals 11 , y equals 7 .
When x=11
, y=162.8
.
When x equals 11 , y equals 162.8 .
When x=11
, y=77
.
When x equals 11 , y equals 77 .
When x=11
, y=11/7
.
Answers
Answer
Which line is parallel to the line y=−3/2x−5
and goes through the point (0,2)
?(1 point)
Responses
y=−3/2(x+10/3)
y equals negative Start Fraction 3 over 2 End Fraction open parenthesis x plus start fraction 10 over 3 end fraction close parenthesis
y=−3/2x
y equals negative Start Fraction 3 over 2 End Fraction x
3y=−2x−15
3 y equals negative 2 x minus 15
y=−3/2x+2
and goes through the point (0,2)
?(1 point)
Responses
y=−3/2(x+10/3)
y equals negative Start Fraction 3 over 2 End Fraction open parenthesis x plus start fraction 10 over 3 end fraction close parenthesis
y=−3/2x
y equals negative Start Fraction 3 over 2 End Fraction x
3y=−2x−15
3 y equals negative 2 x minus 15
y=−3/2x+2
Answer
Which line is perpendicular to the line 5x−4y=12
and goes through the point (5,2)
?(1 point)
Responses
y=5/4x+6
y equals Start Fraction 5 over 4 End Fraction x plus 6
y=−4/5x−3
y equals negative Start Fraction 4 over 5 End Fraction x minus 3
y=4/5x+6
y equals Start Fraction 4 over 5 End Fraction x plus 6
y=−4/5x+6
and goes through the point (5,2)
?(1 point)
Responses
y=5/4x+6
y equals Start Fraction 5 over 4 End Fraction x plus 6
y=−4/5x−3
y equals negative Start Fraction 4 over 5 End Fraction x minus 3
y=4/5x+6
y equals Start Fraction 4 over 5 End Fraction x plus 6
y=−4/5x+6
Answer
Which statement is true about the slope of a line parallel to the x-axis?(1 point)
Responses
A line parallel to the x-axis has an undefined slope.
A line parallel to the x -axis has an undefined slope.
A line parallel to the x-axis has a negative slope because it is decreasing.
A line parallel to the x -axis has a negative slope because it is decreasing.
A line parallel to the x-axis has zero slope.
A line parallel to the x -axis has zero slope.
A line parallel to the x-axis has a positive slope because it is increasing.
Responses
A line parallel to the x-axis has an undefined slope.
A line parallel to the x -axis has an undefined slope.
A line parallel to the x-axis has a negative slope because it is decreasing.
A line parallel to the x -axis has a negative slope because it is decreasing.
A line parallel to the x-axis has zero slope.
A line parallel to the x -axis has zero slope.
A line parallel to the x-axis has a positive slope because it is increasing.
Answer
Max is given the graph of the linear inequality 3x−7.2y<0
and asked to determine whether a given point is a solution. She notices that the point falls on the boundary line of the graph. What should she conclude?(1 point)
Responses
The point is a solution to the linear inequality because points on boundary lines are always solutions to linear inequalities.
The point is a solution to the linear inequality because points on boundary lines are always solutions to linear inequalities.
The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.
The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.
The point is a solution to the linear inequality because points on boundary lines are only solutions when the inequality symbol is >
or <
.
The point is a solution to the linear inequality because points on boundary lines are only solutions when the inequality symbol is greater than or less than .
The point is not a solution to the linear inequality because points on boundary lines are only solutions when the inequality symbol is ≥
or ≤
.
and asked to determine whether a given point is a solution. She notices that the point falls on the boundary line of the graph. What should she conclude?(1 point)
Responses
The point is a solution to the linear inequality because points on boundary lines are always solutions to linear inequalities.
The point is a solution to the linear inequality because points on boundary lines are always solutions to linear inequalities.
The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.
The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.
The point is a solution to the linear inequality because points on boundary lines are only solutions when the inequality symbol is >
or <
.
The point is a solution to the linear inequality because points on boundary lines are only solutions when the inequality symbol is greater than or less than .
The point is not a solution to the linear inequality because points on boundary lines are only solutions when the inequality symbol is ≥
or ≤
.
Answered by
GPT-4o mini
To solve this problem, we first express the direct variation relationship mathematically. Since \( y \) varies directly with \( x \), we can write:
\[
y = kx
\]
where \( k \) is the constant of variation.
We know from the problem that \( y = 112 \) when \( x = 16 \). We can use this information to find \( k \):
\[
112 = k \cdot 16
\]
Now, solve for \( k \):
\[
k = \frac{112}{16} = 7
\]
Now we have the direct variation equation:
\[
y = 7x
\]
Next, we need to find the value of \( y \) when \( x = 11 \):
\[
y = 7 \cdot 11 = 77
\]
Thus, when \( x = 11 \), \( y = 77 \).
The correct response is:
**When x equals 11, y equals 77.**
\[
y = kx
\]
where \( k \) is the constant of variation.
We know from the problem that \( y = 112 \) when \( x = 16 \). We can use this information to find \( k \):
\[
112 = k \cdot 16
\]
Now, solve for \( k \):
\[
k = \frac{112}{16} = 7
\]
Now we have the direct variation equation:
\[
y = 7x
\]
Next, we need to find the value of \( y \) when \( x = 11 \):
\[
y = 7 \cdot 11 = 77
\]
Thus, when \( x = 11 \), \( y = 77 \).
The correct response is:
**When x equals 11, y equals 77.**
Answered by
GPT-4o mini
To determine which line is parallel to the line given by \( y = -\frac{3}{2}x - 5 \) and passes through the point \( (0, 2) \), we need to remember that parallel lines have the same slope.
The slope of the given line \( y = -\frac{3}{2}x - 5 \) is \( -\frac{3}{2} \). Thus, any line that is parallel to it will also have the slope of \( -\frac{3}{2} \).
Now, let's find the equation of the line that has a slope of \( -\frac{3}{2} \) and passes through the point \( (0, 2) \). Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), we can plug in the values:
- \( m = -\frac{3}{2} \)
- \( (x_1, y_1) = (0, 2) \)
The equation becomes:
\[
y - 2 = -\frac{3}{2}(x - 0)
\]
Simplifying this:
\[
y - 2 = -\frac{3}{2}x
\]
\[
y = -\frac{3}{2}x + 2
\]
Now, let’s look at the given options:
1. \( y = -\frac{3}{2}(x + \frac{10}{3}) \)
2. \( y = -\frac{3}{2}x \)
3. \( 3y = -2x - 15 \)
4. \( y = -\frac{3}{2}x + 2 \)
From these options, we can see that:
- **Option 4: \( y = -\frac{3}{2}x + 2 \)** is indeed the line that is parallel to the original line and passes through the point \( (0, 2) \).
The correct response is:
**y = -\frac{3}{2}x + 2.**
The slope of the given line \( y = -\frac{3}{2}x - 5 \) is \( -\frac{3}{2} \). Thus, any line that is parallel to it will also have the slope of \( -\frac{3}{2} \).
Now, let's find the equation of the line that has a slope of \( -\frac{3}{2} \) and passes through the point \( (0, 2) \). Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), we can plug in the values:
- \( m = -\frac{3}{2} \)
- \( (x_1, y_1) = (0, 2) \)
The equation becomes:
\[
y - 2 = -\frac{3}{2}(x - 0)
\]
Simplifying this:
\[
y - 2 = -\frac{3}{2}x
\]
\[
y = -\frac{3}{2}x + 2
\]
Now, let’s look at the given options:
1. \( y = -\frac{3}{2}(x + \frac{10}{3}) \)
2. \( y = -\frac{3}{2}x \)
3. \( 3y = -2x - 15 \)
4. \( y = -\frac{3}{2}x + 2 \)
From these options, we can see that:
- **Option 4: \( y = -\frac{3}{2}x + 2 \)** is indeed the line that is parallel to the original line and passes through the point \( (0, 2) \).
The correct response is:
**y = -\frac{3}{2}x + 2.**
Answered by
GPT-4o mini
To find a line that is perpendicular to the line given by the equation \( 5x - 4y = 12 \) and goes through the point \( (5, 2) \), we first need to determine the slope of the original line.
### Step 1: Convert to Slope-Intercept Form
The equation \( 5x - 4y = 12 \) can be rearranged to slope-intercept form \( y = mx + b \).
Starting with the original equation:
\[
5x - 4y = 12
\]
We isolate \( y \):
\[
-4y = -5x + 12
\]
Dividing by -4:
\[
y = \frac{5}{4}x - 3
\]
### Step 2: Determine the Slope
From the converted equation, we can see that the slope \( m \) of the original line is \( \frac{5}{4} \).
### Step 3: Find the Perpendicular Slope
When two lines are perpendicular, the slopes are negative reciprocals of each other. Therefore, if the slope of the original line is \( \frac{5}{4} \), then the slope of the line that is perpendicular to it will be:
\[
m_{perpendicular} = -\frac{4}{5}
\]
### Step 4: Use the Point-Slope Form to Find the Equation
Now, we can use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \). We’ll plug in the slope \( -\frac{4}{5} \) and the point \( (5, 2) \):
\[
y - 2 = -\frac{4}{5}(x - 5)
\]
### Step 5: Simplify the Equation
Distributing the slope:
\[
y - 2 = -\frac{4}{5}x + 4
\]
Now, adding 2 to both sides:
\[
y = -\frac{4}{5}x + 6
\]
### Step 6: Check Given Options
Let's check this resulting equation against the options provided:
1. \( y = \frac{5}{4}x + 6 \)
2. \( y = -\frac{4}{5}x - 3 \)
3. \( y = \frac{4}{5}x + 6 \)
4. \( y = -\frac{4}{5}x + 6 \)
The equation we derived, \( y = -\frac{4}{5}x + 6 \), matches with **Option 4**.
### Conclusion
The correct response is:
**y = -\frac{4}{5}x + 6.**
### Step 1: Convert to Slope-Intercept Form
The equation \( 5x - 4y = 12 \) can be rearranged to slope-intercept form \( y = mx + b \).
Starting with the original equation:
\[
5x - 4y = 12
\]
We isolate \( y \):
\[
-4y = -5x + 12
\]
Dividing by -4:
\[
y = \frac{5}{4}x - 3
\]
### Step 2: Determine the Slope
From the converted equation, we can see that the slope \( m \) of the original line is \( \frac{5}{4} \).
### Step 3: Find the Perpendicular Slope
When two lines are perpendicular, the slopes are negative reciprocals of each other. Therefore, if the slope of the original line is \( \frac{5}{4} \), then the slope of the line that is perpendicular to it will be:
\[
m_{perpendicular} = -\frac{4}{5}
\]
### Step 4: Use the Point-Slope Form to Find the Equation
Now, we can use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \). We’ll plug in the slope \( -\frac{4}{5} \) and the point \( (5, 2) \):
\[
y - 2 = -\frac{4}{5}(x - 5)
\]
### Step 5: Simplify the Equation
Distributing the slope:
\[
y - 2 = -\frac{4}{5}x + 4
\]
Now, adding 2 to both sides:
\[
y = -\frac{4}{5}x + 6
\]
### Step 6: Check Given Options
Let's check this resulting equation against the options provided:
1. \( y = \frac{5}{4}x + 6 \)
2. \( y = -\frac{4}{5}x - 3 \)
3. \( y = \frac{4}{5}x + 6 \)
4. \( y = -\frac{4}{5}x + 6 \)
The equation we derived, \( y = -\frac{4}{5}x + 6 \), matches with **Option 4**.
### Conclusion
The correct response is:
**y = -\frac{4}{5}x + 6.**
Answered by
GPT-4o mini
A line that is parallel to the x-axis is horizontal and does not rise or fall as it moves from left to right. This means:
- The change in \( y \) (vertical change) is zero for any change in \( x \) (horizontal change).
- Therefore, the slope \( m \), which is calculated as \( \frac{\text{change in } y}{\text{change in } x} \), is equal to \( \frac{0}{\text{change in } x} \), which is \( 0 \).
Thus, the correct statement is:
**A line parallel to the x-axis has zero slope.**
- The change in \( y \) (vertical change) is zero for any change in \( x \) (horizontal change).
- Therefore, the slope \( m \), which is calculated as \( \frac{\text{change in } y}{\text{change in } x} \), is equal to \( \frac{0}{\text{change in } x} \), which is \( 0 \).
Thus, the correct statement is:
**A line parallel to the x-axis has zero slope.**
Answered by
GPT-4o mini
To determine whether a point on the boundary line of a linear inequality is a solution, we need to consider the specific type of inequality.
For the inequality \( 3x - 7.2y < 0 \):
- The boundary line is determined by the equation \( 3x - 7.2y = 0 \).
- This line itself is not included in the solution set because the inequality is strictly less than (<), rather than less than or equal to (≤).
Thus, since Max notices that the point falls on the boundary line (and the inequality is strictly less than), the conclusion should be that:
**The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.**
So the correct response is:
**The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.**
For the inequality \( 3x - 7.2y < 0 \):
- The boundary line is determined by the equation \( 3x - 7.2y = 0 \).
- This line itself is not included in the solution set because the inequality is strictly less than (<), rather than less than or equal to (≤).
Thus, since Max notices that the point falls on the boundary line (and the inequality is strictly less than), the conclusion should be that:
**The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.**
So the correct response is:
**The point is not a solution to the linear inequality because points on boundary lines are never solutions to linear inequalities.**
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