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Original Question
Determine the number of solutions to the system of linear equations y=x+5 y=-x+5Question
Determine the number of solutions to this system of linear equations:
y = x + 5
y = -x + 5
y = x + 5
y = -x + 5
Answers
Answer
Solving the system y = -x + 3 and x + y = 8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
Answer
Does the graph show the system of equations x + y = 2 and -x + y = 7? Should the lines for the system be parallel?
Answer
The graph of x + y = 2 is incorrect. The lines should intersect?
Answer
Choose
1. The graph of x + y = 2 is incorrect. The lines should intersect.
2. The graph of -x + y = 2 is incorrect. The lines should intersect.
1. The graph of x + y = 2 is incorrect. The lines should intersect.
2. The graph of -x + y = 2 is incorrect. The lines should intersect.
Answered by
GPT-4o mini
To determine the number of solutions to the given system of linear equations, we can analyze the equations graphically or algebraically.
The two equations are:
1. \( y = x + 5 \)
2. \( y = -x + 5 \)
We can rewrite these equations in standard form for clarity:
1. \( x - y + 5 = 0 \)
2. \( x + y - 5 = 0 \)
Next, we can solve the system algebraically by setting the two expressions for \(y\) equal to each other:
\[
x + 5 = -x + 5
\]
Now, we can solve for \(x\):
\[
x + x = 5 - 5
\]
\[
2x = 0
\]
\[
x = 0
\]
Now that we have \(x\), we can substitute it back into either of the original equations to find \(y\). Using the first equation:
\[
y = 0 + 5 = 5
\]
Thus, we find the solution is:
\[
(x, y) = (0, 5)
\]
To determine the number of solutions, we can also analyze the nature of the lines represented by the two equations. The first equation \(y = x + 5\) is a line with a slope of 1, and the second equation \(y = -x + 5\) is a line with a slope of -1.
Since the slopes of the two lines are different, they intersect at exactly one point. Therefore, the system of equations has one unique solution.
**Conclusion**: The number of solutions to the system of equations is \( \boxed{1} \).
The two equations are:
1. \( y = x + 5 \)
2. \( y = -x + 5 \)
We can rewrite these equations in standard form for clarity:
1. \( x - y + 5 = 0 \)
2. \( x + y - 5 = 0 \)
Next, we can solve the system algebraically by setting the two expressions for \(y\) equal to each other:
\[
x + 5 = -x + 5
\]
Now, we can solve for \(x\):
\[
x + x = 5 - 5
\]
\[
2x = 0
\]
\[
x = 0
\]
Now that we have \(x\), we can substitute it back into either of the original equations to find \(y\). Using the first equation:
\[
y = 0 + 5 = 5
\]
Thus, we find the solution is:
\[
(x, y) = (0, 5)
\]
To determine the number of solutions, we can also analyze the nature of the lines represented by the two equations. The first equation \(y = x + 5\) is a line with a slope of 1, and the second equation \(y = -x + 5\) is a line with a slope of -1.
Since the slopes of the two lines are different, they intersect at exactly one point. Therefore, the system of equations has one unique solution.
**Conclusion**: The number of solutions to the system of equations is \( \boxed{1} \).
Answered by
GPT-4o mini
To analyze the statement about the system of equations:
1. \( y = -x + 3 \)
2. \( x + y = 8 \)
We will first express both equations in a form that can be easily graphed and examine their lines.
**Equation 1:**
The first equation, \( y = -x + 3 \), has:
- y-intercept of 3 (point (0, 3))
- slope of -1 (meaning it decreases one unit in \(y\) for each unit it increases in \(x\))
**Equation 2:**
The second equation can be rearranged to solve for \(y\):
\[
y = 8 - x
\]
This equation has:
- y-intercept of 8 (point (0, 8))
- slope of -1 (similar to the first equation)
Now, we can observe both equations:
- \( y = -x + 3 \): The line intersects the y-axis at (0, 3) and has a downward slope.
- \( y = 8 - x \): The line intersects the y-axis at (0, 8) and also has a downward slope.
Both lines have the same slope of -1, which means they are parallel. Since parallel lines never intersect, they do not share any points in common.
Therefore, these two equations do not have a solution: they are inconsistent.
**Conclusion:** The statement "Solving the system \( y = -x + 3 \) and \( x + y = 8 \) by graphing shows that there is no solution to the system" is **true**. The two lines representing these equations are parallel, confirming that there is no point at which they intersect, indicating no solution exists.
1. \( y = -x + 3 \)
2. \( x + y = 8 \)
We will first express both equations in a form that can be easily graphed and examine their lines.
**Equation 1:**
The first equation, \( y = -x + 3 \), has:
- y-intercept of 3 (point (0, 3))
- slope of -1 (meaning it decreases one unit in \(y\) for each unit it increases in \(x\))
**Equation 2:**
The second equation can be rearranged to solve for \(y\):
\[
y = 8 - x
\]
This equation has:
- y-intercept of 8 (point (0, 8))
- slope of -1 (similar to the first equation)
Now, we can observe both equations:
- \( y = -x + 3 \): The line intersects the y-axis at (0, 3) and has a downward slope.
- \( y = 8 - x \): The line intersects the y-axis at (0, 8) and also has a downward slope.
Both lines have the same slope of -1, which means they are parallel. Since parallel lines never intersect, they do not share any points in common.
Therefore, these two equations do not have a solution: they are inconsistent.
**Conclusion:** The statement "Solving the system \( y = -x + 3 \) and \( x + y = 8 \) by graphing shows that there is no solution to the system" is **true**. The two lines representing these equations are parallel, confirming that there is no point at which they intersect, indicating no solution exists.
Answered by
GPT-4o mini
To determine the relationship between the two equations \(x + y = 2\) and \(-x + y = 7\), we can analyze their slopes and intercepts.
### Step 1: Put equations in slope-intercept form
**Equation 1:**
Starting with \(x + y = 2\), we can rearrange it to slope-intercept form (\(y = mx + b\)):
\[
y = -x + 2
\]
This equation has:
- **Slope (\(m\))**: \(-1\)
- **Y-intercept (\(b\))**: \(2\) (point: \((0, 2)\))
**Equation 2:**
Next, rearranging \(-x + y = 7\) into slope-intercept form:
\[
y = x + 7
\]
This equation has:
- **Slope (\(m\))**: \(1\)
- **Y-intercept (\(b\))**: \(7\) (point: \((0, 7)\))
### Step 2: Compare slopes
We see that the slopes are:
- For \(y = -x + 2\) (slope = -1)
- For \(y = x + 7\) (slope = 1)
Since the slopes are different \((-1 \neq 1)\), this indicates that the lines are **not parallel** and will intersect at some point.
### Conclusion
The lines for the system \(x + y = 2\) and \(-x + y = 7\) **should not be parallel**; instead, they intersect at one point. Therefore, the claim that these lines are parallel is **false**. The graph should show the intersection of these two lines.
If you were to graph these two equations, you would see that they cross each other at a specific point, which represents the unique solution to the system.
### Step 1: Put equations in slope-intercept form
**Equation 1:**
Starting with \(x + y = 2\), we can rearrange it to slope-intercept form (\(y = mx + b\)):
\[
y = -x + 2
\]
This equation has:
- **Slope (\(m\))**: \(-1\)
- **Y-intercept (\(b\))**: \(2\) (point: \((0, 2)\))
**Equation 2:**
Next, rearranging \(-x + y = 7\) into slope-intercept form:
\[
y = x + 7
\]
This equation has:
- **Slope (\(m\))**: \(1\)
- **Y-intercept (\(b\))**: \(7\) (point: \((0, 7)\))
### Step 2: Compare slopes
We see that the slopes are:
- For \(y = -x + 2\) (slope = -1)
- For \(y = x + 7\) (slope = 1)
Since the slopes are different \((-1 \neq 1)\), this indicates that the lines are **not parallel** and will intersect at some point.
### Conclusion
The lines for the system \(x + y = 2\) and \(-x + y = 7\) **should not be parallel**; instead, they intersect at one point. Therefore, the claim that these lines are parallel is **false**. The graph should show the intersection of these two lines.
If you were to graph these two equations, you would see that they cross each other at a specific point, which represents the unique solution to the system.
Answered by
GPT-4o mini
Let’s clarify and examine the equations \(x + y = 2\) and \(-x + y = 7\) carefully to determine their graphical representation, including their intersections.
### Step 1: Convert equations to slope-intercept form.
1. **For the equation \(x + y = 2\)**:
\[
y = -x + 2
\]
- **Slope**: \(-1\)
- **Y-intercept**: \(2\) (point: \((0, 2)\))
2. **For the equation \(-x + y = 7\)**:
\[
y = x + 7
\]
- **Slope**: \(1\)
- **Y-intercept**: \(7\) (point: \((0, 7)\))
### Step 2: Analyze the slopes
- The slope of the first line \(y = -x + 2\) is \(-1\) (a downward slope).
- The slope of the second line \(y = x + 7\) is \(1\) (an upward slope).
Since the slopes are different \((-1\) and \(1\)), the lines are **not parallel** and will intersect at some point.
### Step 3: Find the point of intersection
To find the intersection point, we can solve the system of equations:
1. Start with the two equations:
\[
x + y = 2 \quad (1)
\]
\[
-x + y = 7 \quad (2)
\]
2. We can express \(y\) from equation (1):
\[
y = 2 - x
\]
3. Substitute this expression for \(y\) into equation (2):
\[
-x + (2 - x) = 7
\]
Simplifying this:
\[
-x + 2 - x = 7
\]
\[
-2x + 2 = 7
\]
\[
-2x = 7 - 2
\]
\[
-2x = 5
\]
\[
x = -\frac{5}{2}
\]
4. Now substitute \(x = -\frac{5}{2}\) back into equation (1) to find \(y\):
\[
y = 2 - (-\frac{5}{2}) = 2 + \frac{5}{2} = \frac{4}{2} + \frac{5}{2} = \frac{9}{2}
\]
### Conclusion
The intersection point of the lines is \(\left(-\frac{5}{2}, \frac{9}{2}\right)\).
- Thus, the two lines do indeed intersect, which confirms that the graph of \(x + y = 2\) was not represented incorrectly in the assertion.
- If you have seen a graph where it seems the lines do not intersect, it would be the result of an error in the graph or its interpretation.
**Final Confirmation**: The lines of the system \(x + y = 2\) and \(-x + y = 7\) do intersect at the point \(\left(-\frac{5}{2}, \frac{9}{2}\right)\). Therefore, the statement that the lines should intersect is indeed **true**.
### Step 1: Convert equations to slope-intercept form.
1. **For the equation \(x + y = 2\)**:
\[
y = -x + 2
\]
- **Slope**: \(-1\)
- **Y-intercept**: \(2\) (point: \((0, 2)\))
2. **For the equation \(-x + y = 7\)**:
\[
y = x + 7
\]
- **Slope**: \(1\)
- **Y-intercept**: \(7\) (point: \((0, 7)\))
### Step 2: Analyze the slopes
- The slope of the first line \(y = -x + 2\) is \(-1\) (a downward slope).
- The slope of the second line \(y = x + 7\) is \(1\) (an upward slope).
Since the slopes are different \((-1\) and \(1\)), the lines are **not parallel** and will intersect at some point.
### Step 3: Find the point of intersection
To find the intersection point, we can solve the system of equations:
1. Start with the two equations:
\[
x + y = 2 \quad (1)
\]
\[
-x + y = 7 \quad (2)
\]
2. We can express \(y\) from equation (1):
\[
y = 2 - x
\]
3. Substitute this expression for \(y\) into equation (2):
\[
-x + (2 - x) = 7
\]
Simplifying this:
\[
-x + 2 - x = 7
\]
\[
-2x + 2 = 7
\]
\[
-2x = 7 - 2
\]
\[
-2x = 5
\]
\[
x = -\frac{5}{2}
\]
4. Now substitute \(x = -\frac{5}{2}\) back into equation (1) to find \(y\):
\[
y = 2 - (-\frac{5}{2}) = 2 + \frac{5}{2} = \frac{4}{2} + \frac{5}{2} = \frac{9}{2}
\]
### Conclusion
The intersection point of the lines is \(\left(-\frac{5}{2}, \frac{9}{2}\right)\).
- Thus, the two lines do indeed intersect, which confirms that the graph of \(x + y = 2\) was not represented incorrectly in the assertion.
- If you have seen a graph where it seems the lines do not intersect, it would be the result of an error in the graph or its interpretation.
**Final Confirmation**: The lines of the system \(x + y = 2\) and \(-x + y = 7\) do intersect at the point \(\left(-\frac{5}{2}, \frac{9}{2}\right)\). Therefore, the statement that the lines should intersect is indeed **true**.
Answered by
GPT-4o mini
Based on your question, we need to determine which statement correctly addresses the equations we have been discussing.
1. **The graph of \(x + y = 2\) is incorrect. The lines should intersect.**
2. **The graph of \(-x + y = 2\) is incorrect. The lines should intersect.**
From our previous conversation, we analyzed the system of equations:
1. \(x + y = 2\)
2. \(-x + y = 7\)
### Summary of Findings:
- The first equation, \(x + y = 2\), can be rewritten as \(y = -x + 2\).
- The second equation, \(-x + y = 7\), can be rewritten as \(y = x + 7\).
- These two lines have different slopes (\(-1\) and \(1\)), which means they are not parallel and will intersect at a point.
### Choosing the Correct Statement:
Considering the analysis above:
- **The graph of \(x + y = 2\) is indeed correct, and the lines do intersect. Therefore, the first statement is false.**
- **The second statement mentions \(-x + y = 2\) which was never part of our original discussion; however, it likely refers to another equation. In context, \(-x + y = 7\) is the equation we analyzed, and it also has a different slope, meaning it is correct that this line will intersect with \(x + y = 2\). Therefore, if we treat \(-x + y = 2\) as an incorrect assumption regarding parallelism, we can suggest that the second statement is also false.**
Since both statements appear to have factual inaccuracies based on the original equations discussed, if you are asking which one to choose considering intersection, **the first statement is correct in identifying that the earlier graph represents lines that do intersect. However, this would be under the assumption that the first equation (\(x + y = 2\)) was misrepresented rather than indicating that \(x + y = 2\) itself is incorrect. So I would select**:
1. **The graph of \(x + y = 2\) is incorrect. The lines should intersect.** (as a descriptor of confirming that a misunderstanding may have occurred regarding intersection).
However, be aware that this reflection is based on the details discussed, and if there was a specific graphical representation in question which clearly depicted \(x + y = 2\) inaccurately, selecting option 1 would align with requiring clarification.
1. **The graph of \(x + y = 2\) is incorrect. The lines should intersect.**
2. **The graph of \(-x + y = 2\) is incorrect. The lines should intersect.**
From our previous conversation, we analyzed the system of equations:
1. \(x + y = 2\)
2. \(-x + y = 7\)
### Summary of Findings:
- The first equation, \(x + y = 2\), can be rewritten as \(y = -x + 2\).
- The second equation, \(-x + y = 7\), can be rewritten as \(y = x + 7\).
- These two lines have different slopes (\(-1\) and \(1\)), which means they are not parallel and will intersect at a point.
### Choosing the Correct Statement:
Considering the analysis above:
- **The graph of \(x + y = 2\) is indeed correct, and the lines do intersect. Therefore, the first statement is false.**
- **The second statement mentions \(-x + y = 2\) which was never part of our original discussion; however, it likely refers to another equation. In context, \(-x + y = 7\) is the equation we analyzed, and it also has a different slope, meaning it is correct that this line will intersect with \(x + y = 2\). Therefore, if we treat \(-x + y = 2\) as an incorrect assumption regarding parallelism, we can suggest that the second statement is also false.**
Since both statements appear to have factual inaccuracies based on the original equations discussed, if you are asking which one to choose considering intersection, **the first statement is correct in identifying that the earlier graph represents lines that do intersect. However, this would be under the assumption that the first equation (\(x + y = 2\)) was misrepresented rather than indicating that \(x + y = 2\) itself is incorrect. So I would select**:
1. **The graph of \(x + y = 2\) is incorrect. The lines should intersect.** (as a descriptor of confirming that a misunderstanding may have occurred regarding intersection).
However, be aware that this reflection is based on the details discussed, and if there was a specific graphical representation in question which clearly depicted \(x + y = 2\) inaccurately, selecting option 1 would align with requiring clarification.
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