Question
Which graph shows the solution of −3x−y=3 and y=−3x+4 ?(1 point)
Answers
Answer
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
(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
an infinite number of solutions
an infinite number of solutions
(1.286,0)
(1 point)
Responses
(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
an infinite number of solutions
an infinite number of solutions
(1.286,0)
Answered by
GPT-4o mini
To find the solution to the equations \(-3x - y = 3\) and \(y = -3x + 4\), we can first convert the first equation into the slope-intercept form (y = mx + b).
Starting with the first equation:
\[
-3x - y = 3
\]
We can isolate \(y\):
\[
-y = 3 + 3x
\]
\[
y = -3x - 3
\]
Now we have the two equations:
1. \(y = -3x - 3\)
2. \(y = -3x + 4\)
Both equations have a slope of \(-3\). This means that the lines are parallel, and they will never intersect.
To find where they diverge, we can evaluate their y-intercepts. The first equation has a y-intercept of \(-3\) (when \(x = 0\)), and the second equation has a y-intercept of \(4\).
The first line slopes down from left to right starting at \(-3\) on the y-axis, while the second line also slopes down but starts at \(4\) on the y-axis.
Since parallel lines do not intersect, there will be no solution for these equations.
If you are looking for a graph representing these lines, you will need a graph that clearly shows two parallel lines, one starting at (0, -3) and the other at (0, 4), both with a downward slope. Please refer to a corresponding graph that accurately represents these characteristics.
Starting with the first equation:
\[
-3x - y = 3
\]
We can isolate \(y\):
\[
-y = 3 + 3x
\]
\[
y = -3x - 3
\]
Now we have the two equations:
1. \(y = -3x - 3\)
2. \(y = -3x + 4\)
Both equations have a slope of \(-3\). This means that the lines are parallel, and they will never intersect.
To find where they diverge, we can evaluate their y-intercepts. The first equation has a y-intercept of \(-3\) (when \(x = 0\)), and the second equation has a y-intercept of \(4\).
The first line slopes down from left to right starting at \(-3\) on the y-axis, while the second line also slopes down but starts at \(4\) on the y-axis.
Since parallel lines do not intersect, there will be no solution for these equations.
If you are looking for a graph representing these lines, you will need a graph that clearly shows two parallel lines, one starting at (0, -3) and the other at (0, 4), both with a downward slope. Please refer to a corresponding graph that accurately represents these characteristics.
Answered by
GPT-4o mini
To determine the solution of the equations \((72)x-(92)=2y\) and \(4y=7x-9\) and see why Emma and Angelle get the same graph, we can convert both equations into slope-intercept form (y = mx + b).
### For the first equation:
\[
(72)x - (92) = 2y
\]
First, we can rearrange it to solve for \(y\):
\[
2y = (72)x - (92)
\]
\[
y = \frac{72}{2}x - \frac{92}{2}
\]
\[
y = 36x - 46
\]
### For the second equation:
\[
4y = 7x - 9
\]
We solve for \(y\):
\[
y = \frac{7}{4}x - \frac{9}{4}
\]
Now we have two equations:
1. \(y = 36x - 46\)
2. \(y = \frac{7}{4}x - \frac{9}{4}\)
Next, we can check if these lines are equivalent or parallel, which would explain why they yield the same graph.
### Finding Solutions
To check for intersection or equivalence, we would set the two \(y\) expressions equal:
\[
36x - 46 = \frac{7}{4}x - \frac{9}{4}
\]
Multiply everything by 4 to eliminate the fraction:
\[
4(36x) - 4(46) = 7x - 9
\]
\[
144x - 184 = 7x - 9
\]
Rearranging gives us:
\[
144x - 7x = -9 + 184
\]
\[
137x = 175
\]
\[
x = \frac{175}{137} \approx 1.286
\]
Substituting \(x\) back into either equation to find \(y\):
Using the first equation:
\[
y = 36\left(\frac{175}{137}\right) - 46
\]
Calculating \(y\):
\[
y = \frac{6300}{137} - 46 \approx \frac{6300 - 6322}{137} \approx \frac{-22}{137} \approx -0.161
\]
However, substituting \(x = \frac{175}{137}\) in the second equation should yield the same or should cross verify for accuracy.
### Interpretation of Results
When both equations are graphed, we identified that the specific point approximately at \((1.286, 0)\) might lead to exact works.
### Conclusion
The unique solution to the systems of equations is at \((1.286, 0)\). Additionally, the reference to "\(\text{left parenthesis }1.286, 0 \text{ right parenthesis}\)" + "\((0,−2.25)\)" indicates distinct points.
Therefore, it appears the closest correct answer to the solution from provided options is:
**(1.286, 0)**.
Thus, the response choice given in your question seems to indicate just one key solution, which is why they yielded an incorrect conclusion in testing for infinite solutions, as they've clearly established a point where both intersected. Hence they cannot have "an infinite number of solutions".
### For the first equation:
\[
(72)x - (92) = 2y
\]
First, we can rearrange it to solve for \(y\):
\[
2y = (72)x - (92)
\]
\[
y = \frac{72}{2}x - \frac{92}{2}
\]
\[
y = 36x - 46
\]
### For the second equation:
\[
4y = 7x - 9
\]
We solve for \(y\):
\[
y = \frac{7}{4}x - \frac{9}{4}
\]
Now we have two equations:
1. \(y = 36x - 46\)
2. \(y = \frac{7}{4}x - \frac{9}{4}\)
Next, we can check if these lines are equivalent or parallel, which would explain why they yield the same graph.
### Finding Solutions
To check for intersection or equivalence, we would set the two \(y\) expressions equal:
\[
36x - 46 = \frac{7}{4}x - \frac{9}{4}
\]
Multiply everything by 4 to eliminate the fraction:
\[
4(36x) - 4(46) = 7x - 9
\]
\[
144x - 184 = 7x - 9
\]
Rearranging gives us:
\[
144x - 7x = -9 + 184
\]
\[
137x = 175
\]
\[
x = \frac{175}{137} \approx 1.286
\]
Substituting \(x\) back into either equation to find \(y\):
Using the first equation:
\[
y = 36\left(\frac{175}{137}\right) - 46
\]
Calculating \(y\):
\[
y = \frac{6300}{137} - 46 \approx \frac{6300 - 6322}{137} \approx \frac{-22}{137} \approx -0.161
\]
However, substituting \(x = \frac{175}{137}\) in the second equation should yield the same or should cross verify for accuracy.
### Interpretation of Results
When both equations are graphed, we identified that the specific point approximately at \((1.286, 0)\) might lead to exact works.
### Conclusion
The unique solution to the systems of equations is at \((1.286, 0)\). Additionally, the reference to "\(\text{left parenthesis }1.286, 0 \text{ right parenthesis}\)" + "\((0,−2.25)\)" indicates distinct points.
Therefore, it appears the closest correct answer to the solution from provided options is:
**(1.286, 0)**.
Thus, the response choice given in your question seems to indicate just one key solution, which is why they yielded an incorrect conclusion in testing for infinite solutions, as they've clearly established a point where both intersected. Hence they cannot have "an infinite number of solutions".