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Which graph shows the solution of −3x−y=3 and y=−3x+4 ?(1 point) Responses 1: Two intersecting lines are graphed across a coord...Question
Which graph shows the solution of −3x−y=3
and y=−3x+4
?(1 point)
Responses
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Image with alt text: Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
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and y=−3x+4
?(1 point)
Responses
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Image with alt text: Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
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Answers
Answer
Use the image to answer the question.
One line, graphed as both solid and a dotted, is graphed on a coordinate plane. The x-axis ranges from negative 10 to 10 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
In math class, Emma and Angelle pair as a group. Emma’s task is to graph (72)x−(92)=2y
and Angelle’s task is to graph 4y=7x−9
. After they complete their graphs, they compare their work. They both get exactly the same graph. What is the solution of (72)x−(92)=2y
and 4y=7x−9
?
(1 point)
Responses
an infinite number of solutions
an infinite number of solutions
(1.286,0)
left parenthesis 1.286 comma 0 right parenthesis
(1.286,0)
and (0,−2.25)
left parenthesis 1.286 comma 0 right parenthesis and left parenthesis 0 comma negative 2.25 right parenthesis
no solution
no solution
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One line, graphed as both solid and a dotted, is graphed on a coordinate plane. The x-axis ranges from negative 10 to 10 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
In math class, Emma and Angelle pair as a group. Emma’s task is to graph (72)x−(92)=2y
and Angelle’s task is to graph 4y=7x−9
. After they complete their graphs, they compare their work. They both get exactly the same graph. What is the solution of (72)x−(92)=2y
and 4y=7x−9
?
(1 point)
Responses
an infinite number of solutions
an infinite number of solutions
(1.286,0)
left parenthesis 1.286 comma 0 right parenthesis
(1.286,0)
and (0,−2.25)
left parenthesis 1.286 comma 0 right parenthesis and left parenthesis 0 comma negative 2.25 right parenthesis
no solution
no solution
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Answer
Use the table to answer the question.
x-value y-value if y=2x−5
y-value if y=x−3
0 −5
−3
1 −3
−2
2 −1
−1
3 1 0
Jian made a table showing some x- and y-values in a system of equations. Use his table to identify the solution for the system.
(1 point)
Responses
There is an infinite number of solutions.
There is an infinite number of solutions.
(3,0)
left parenthesis 3 comma 0 right parenthesis
There is no solution.
There is no solution.
(2,−1)
x-value y-value if y=2x−5
y-value if y=x−3
0 −5
−3
1 −3
−2
2 −1
−1
3 1 0
Jian made a table showing some x- and y-values in a system of equations. Use his table to identify the solution for the system.
(1 point)
Responses
There is an infinite number of solutions.
There is an infinite number of solutions.
(3,0)
left parenthesis 3 comma 0 right parenthesis
There is no solution.
There is no solution.
(2,−1)
Answer
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 5 to 5 and its y-axis ranges from negative 3 to 7, both in 1-unit increments. A parabola and a line are plotted. The line and parabola intersect at two points. 1 point of intersection is in quadrant 1 and the other is in quadrant 2.
Which of the following shows the solution(s) for this system of equations?
(1 point)
Responses
(−2,4)
and (2,4)
left parenthesis negative 2 comma 4 right parenthesis and left parenthesis 2 comma 4 right parenthesis
There are no solutions.
There are no solutions.
(−3,0)
and (3,0)
left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 3 comma 0 right parenthesis
(0,8)
left parenthesis 0 comma 8 right parenthesis
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A coordinate plane's x-axis ranges from negative 5 to 5 and its y-axis ranges from negative 3 to 7, both in 1-unit increments. A parabola and a line are plotted. The line and parabola intersect at two points. 1 point of intersection is in quadrant 1 and the other is in quadrant 2.
Which of the following shows the solution(s) for this system of equations?
(1 point)
Responses
(−2,4)
and (2,4)
left parenthesis negative 2 comma 4 right parenthesis and left parenthesis 2 comma 4 right parenthesis
There are no solutions.
There are no solutions.
(−3,0)
and (3,0)
left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 3 comma 0 right parenthesis
(0,8)
left parenthesis 0 comma 8 right parenthesis
Skip to navigation
Answer
Which equation describes an infinite solution system with the equation y=4x−3
?(1 point)
Responses
−2y+8x=−6
negative 2 y plus 8 x equals negative 6
2y−8x=−6
2 y minus 8 x equals negative 6
2y−8x=6
2 y minus 8 x equals 6
2y+8x=−6
?(1 point)
Responses
−2y+8x=−6
negative 2 y plus 8 x equals negative 6
2y−8x=−6
2 y minus 8 x equals negative 6
2y−8x=6
2 y minus 8 x equals 6
2y+8x=−6
Answered by
GPT-4o mini
To determine what the graphs of the given equations look like, we can analyze the two equations:
1. **Equation 1: \(-3x - y = 3\)**
- To express this in slope-intercept form (\(y = mx + b\)), we can rearrange it:
\[
-y = 3 + 3x \implies y = -3x - 3
\]
- This line has a slope of \(-3\) and a y-intercept at \(-3\).
2. **Equation 2: \(y = -3x + 4\)**
- This is already in slope-intercept form with a slope of \(-3\) and a y-intercept at \(4\).
Now comparing the two lines:
- Both equations have the same slope of \(-3\), which means the lines are parallel.
- The y-intercept of the first line is \(-3\), while the y-intercept of the second line is \(4\). Since they have different y-intercepts, they will never intersect.
Thus, the correct description of the graph is that of **two parallel lines**. Since one is described as solid and the other as dotted, this reinforces that they do not intersect.
Therefore, the correct answer is:
**Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.**
1. **Equation 1: \(-3x - y = 3\)**
- To express this in slope-intercept form (\(y = mx + b\)), we can rearrange it:
\[
-y = 3 + 3x \implies y = -3x - 3
\]
- This line has a slope of \(-3\) and a y-intercept at \(-3\).
2. **Equation 2: \(y = -3x + 4\)**
- This is already in slope-intercept form with a slope of \(-3\) and a y-intercept at \(4\).
Now comparing the two lines:
- Both equations have the same slope of \(-3\), which means the lines are parallel.
- The y-intercept of the first line is \(-3\), while the y-intercept of the second line is \(4\). Since they have different y-intercepts, they will never intersect.
Thus, the correct description of the graph is that of **two parallel lines**. Since one is described as solid and the other as dotted, this reinforces that they do not intersect.
Therefore, the correct answer is:
**Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.**
Answered by
GPT-4o mini
To find the solution of the two equations \((72)x - (92) = 2y\) and \(4y = 7x - 9\), we first need to simplify each equation.
1. **Equation 1: \((72)x - (92) = 2y\)**
- Rearranging it into slope-intercept form \(y = mx + b\):
\[
2y = (72)x - (92) \implies y = \frac{72}{2}x - \frac{92}{2} \implies y = 36x - 46
\]
2. **Equation 2: \(4y = 7x - 9\)**
- Rearranging it into slope-intercept form:
\[
y = \frac{7}{4}x - \frac{9}{4}
\]
Now let's compare the two equations:
- The first equation is \(y = 36x - 46\).
- The second equation is \(y = \frac{7}{4}x - \frac{9}{4}\).
Since the two equations are clearly not in the same form and have different slopes (36 and \(\frac{7}{4}\)), they represent different lines.
However, if Emma and Angelle both ended up with the exact same graph, it suggests that their equations, though initially different, are equivalent. This would mean every point on one line is also on the other line.
In this case, both lines being equivalent means they share all their points in common.
Therefore, the correct response to the solution of the two equations is:
**an infinite number of solutions.**
1. **Equation 1: \((72)x - (92) = 2y\)**
- Rearranging it into slope-intercept form \(y = mx + b\):
\[
2y = (72)x - (92) \implies y = \frac{72}{2}x - \frac{92}{2} \implies y = 36x - 46
\]
2. **Equation 2: \(4y = 7x - 9\)**
- Rearranging it into slope-intercept form:
\[
y = \frac{7}{4}x - \frac{9}{4}
\]
Now let's compare the two equations:
- The first equation is \(y = 36x - 46\).
- The second equation is \(y = \frac{7}{4}x - \frac{9}{4}\).
Since the two equations are clearly not in the same form and have different slopes (36 and \(\frac{7}{4}\)), they represent different lines.
However, if Emma and Angelle both ended up with the exact same graph, it suggests that their equations, though initially different, are equivalent. This would mean every point on one line is also on the other line.
In this case, both lines being equivalent means they share all their points in common.
Therefore, the correct response to the solution of the two equations is:
**an infinite number of solutions.**
Answered by
GPT-4o mini
To find the solution for the system of equations from the table provided, we need to look at the x- and y-values corresponding to both equations:
1. **Equation 1: \(y = 2x - 5\)**
2. **Equation 2: \(y = x - 3\)**
From the table, the relevant x and y values for both equations give the following:
- For \(x = 0\):
- \(y = 2(0) - 5 = -5\)
- \(y = 0 - 3 = -3\)
- For \(x = 1\):
- \(y = 2(1) - 5 = -3\)
- \(y = 1 - 3 = -2\)
- For \(x = 2\):
- \(y = 2(2) - 5 = -1\)
- \(y = 2 - 3 = -1\)
- For \(x = 3\):
- \(y = 2(3) - 5 = 1\)
- \(y = 3 - 3 = 0\)
Now we look for where the y-values from both equations are equal (that is, where the two lines intersect):
- At \(x = 2\), both equations yield \(y = -1\).
- Therefore, the point \((2, -1)\) is a solution to the system of equations.
Let's check the other values:
- At \(x = 3\), the first equation gives \(y = 1\) and the second gives \(y = 0\), which are not equal.
- No other x-values yield equal y-values.
Thus, the solution to the system is \((2, -1)\).
Therefore, the correct answer is:
**(2,−1)**.
1. **Equation 1: \(y = 2x - 5\)**
2. **Equation 2: \(y = x - 3\)**
From the table, the relevant x and y values for both equations give the following:
- For \(x = 0\):
- \(y = 2(0) - 5 = -5\)
- \(y = 0 - 3 = -3\)
- For \(x = 1\):
- \(y = 2(1) - 5 = -3\)
- \(y = 1 - 3 = -2\)
- For \(x = 2\):
- \(y = 2(2) - 5 = -1\)
- \(y = 2 - 3 = -1\)
- For \(x = 3\):
- \(y = 2(3) - 5 = 1\)
- \(y = 3 - 3 = 0\)
Now we look for where the y-values from both equations are equal (that is, where the two lines intersect):
- At \(x = 2\), both equations yield \(y = -1\).
- Therefore, the point \((2, -1)\) is a solution to the system of equations.
Let's check the other values:
- At \(x = 3\), the first equation gives \(y = 1\) and the second gives \(y = 0\), which are not equal.
- No other x-values yield equal y-values.
Thus, the solution to the system is \((2, -1)\).
Therefore, the correct answer is:
**(2,−1)**.
Answered by
GPT-4o mini
If there is a parabola and a line intersecting at two points, it indicates that the system of equations has two solutions. Since you mentioned that one point of intersection is in quadrant 1 and the other is in quadrant 2, this suggests that the y-values at those intersection points are both positive (in quadrant 1) and that one of the x-values is negative (in quadrant 2).
Now, let's analyze the possible answers provided:
1. **(−2,4) and (2,4)**:
- This suggests two points, one to the left of the y-axis (in quadrant 2) and one to the right of the y-axis (in quadrant 1) at the same y-value of \(4\). This is a plausible scenario for the line and parabola intersecting at two points.
2. **There are no solutions.**:
- This option is incorrect since you've stated there are two points of intersection.
3. **(−3,0) and (3,0)**:
- This suggests two points on the x-axis. Although this describes two solutions, they are on the x-axis (where \(y=0\)), which does not match with the earlier clues stating that one intersection is in quadrant 1 (where \(y\) is positive).
4. **(0,8)**:
- This only provides a single point which does not match your description of two intersection points.
Given this analysis, the best answer is the first option, as it describes two intersection points with appropriate coordinates across both quadrants.
Therefore, the correct response is:
**(−2,4) and (2,4)**.
Now, let's analyze the possible answers provided:
1. **(−2,4) and (2,4)**:
- This suggests two points, one to the left of the y-axis (in quadrant 2) and one to the right of the y-axis (in quadrant 1) at the same y-value of \(4\). This is a plausible scenario for the line and parabola intersecting at two points.
2. **There are no solutions.**:
- This option is incorrect since you've stated there are two points of intersection.
3. **(−3,0) and (3,0)**:
- This suggests two points on the x-axis. Although this describes two solutions, they are on the x-axis (where \(y=0\)), which does not match with the earlier clues stating that one intersection is in quadrant 1 (where \(y\) is positive).
4. **(0,8)**:
- This only provides a single point which does not match your description of two intersection points.
Given this analysis, the best answer is the first option, as it describes two intersection points with appropriate coordinates across both quadrants.
Therefore, the correct response is:
**(−2,4) and (2,4)**.
Answered by
GPT-4o mini
To determine which equation describes a system that has infinitely many solutions with the equation \(y = 4x - 3\), we need to check if any of the provided equations are equivalent to the original equation.
An equivalent equation to \(y = 4x - 3\) will be a rearrangement of this line. For the systems of equations to have infinite solutions, the equations must represent the same line.
Let's start by rearranging the given equation into the standard form \(Ax + By = C\).
### Original Equation:
1. Start with:
\[
y = 4x - 3
\]
2. Rearranging gives:
\[
-4x + y = -3 \implies 4x - y = 3
\]
Now, let’s check each provided equation to see if it is equivalent to \(4x - y = 3\):
1. **Equation: \(-2y + 8x = -6\)**
- Rearranging gives:
\[
-2y = -8x - 6 \implies y = 4x + 3
\]
- This does NOT match \(y = 4x - 3\).
2. **Equation: \(2y - 8x = -6\)**
- Rearranging gives:
\[
2y = 8x - 6 \implies y = 4x - 3
\]
- This IS equivalent to the original equation.
3. **Equation: \(2y - 8x = 6\)**
- Rearranging gives:
\[
2y = 8x + 6 \implies y = 4x + 3
\]
- This does NOT match \(y = 4x - 3\).
4. **Equation: \(2y + 8x = -6\)**
- Rearranging gives:
\[
2y = -8x - 6 \implies y = -4x - 3
\]
- This does NOT match \(y = 4x - 3\).
### Conclusion:
The equation that describes an infinite solution system with the equation \(y = 4x - 3\) is:
**\(2y - 8x = -6\)**.
An equivalent equation to \(y = 4x - 3\) will be a rearrangement of this line. For the systems of equations to have infinite solutions, the equations must represent the same line.
Let's start by rearranging the given equation into the standard form \(Ax + By = C\).
### Original Equation:
1. Start with:
\[
y = 4x - 3
\]
2. Rearranging gives:
\[
-4x + y = -3 \implies 4x - y = 3
\]
Now, let’s check each provided equation to see if it is equivalent to \(4x - y = 3\):
1. **Equation: \(-2y + 8x = -6\)**
- Rearranging gives:
\[
-2y = -8x - 6 \implies y = 4x + 3
\]
- This does NOT match \(y = 4x - 3\).
2. **Equation: \(2y - 8x = -6\)**
- Rearranging gives:
\[
2y = 8x - 6 \implies y = 4x - 3
\]
- This IS equivalent to the original equation.
3. **Equation: \(2y - 8x = 6\)**
- Rearranging gives:
\[
2y = 8x + 6 \implies y = 4x + 3
\]
- This does NOT match \(y = 4x - 3\).
4. **Equation: \(2y + 8x = -6\)**
- Rearranging gives:
\[
2y = -8x - 6 \implies y = -4x - 3
\]
- This does NOT match \(y = 4x - 3\).
### Conclusion:
The equation that describes an infinite solution system with the equation \(y = 4x - 3\) is:
**\(2y - 8x = -6\)**.
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