Question
Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions? Where do they overlap?
Answers
Answered by
GPT-4o mini
To determine which graph shows that the linear system \( -2x + y = 6 \) and \( 4x - 2y = -12 \) has an infinite number of solutions, we first need to analyze the given equations.
1. **Rearranging the equations:**
- For the first equation, \( -2x + y = 6 \):
\[
y = 2x + 6
\]
- For the second equation, \( 4x - 2y = -12 \):
Rearranging it gives:
\[
-2y = -4x - 12 \implies 2y = 4x + 12 \implies y = 2x + 6
\]
2. **Identifying the relationship:**
Both equations, after being rearranged, are found to be \( y = 2x + 6 \). This means they represent the same line.
3. **Conclusion:**
This means that the two equations are actually equivalent, thus they have an infinite number of solutions since every point on the line described by \( y = 2x + 6 \) is a solution to both equations.
4. **Graph overlap:**
The overlap of the graphs will be along the line \( y = 2x + 6 \). Any graph that depicts this line twice (or as a single line with two different representations) will illustrate that the system of equations has an infinite number of solutions.
To summarize, the graph that shows the equation \( y = 2x + 6 \) (e.g., a line with a slope of 2 and y-intercept of 6) will illustrate that both equations have an infinite number of solutions because they overlap entirely along that line.
1. **Rearranging the equations:**
- For the first equation, \( -2x + y = 6 \):
\[
y = 2x + 6
\]
- For the second equation, \( 4x - 2y = -12 \):
Rearranging it gives:
\[
-2y = -4x - 12 \implies 2y = 4x + 12 \implies y = 2x + 6
\]
2. **Identifying the relationship:**
Both equations, after being rearranged, are found to be \( y = 2x + 6 \). This means they represent the same line.
3. **Conclusion:**
This means that the two equations are actually equivalent, thus they have an infinite number of solutions since every point on the line described by \( y = 2x + 6 \) is a solution to both equations.
4. **Graph overlap:**
The overlap of the graphs will be along the line \( y = 2x + 6 \). Any graph that depicts this line twice (or as a single line with two different representations) will illustrate that the system of equations has an infinite number of solutions.
To summarize, the graph that shows the equation \( y = 2x + 6 \) (e.g., a line with a slope of 2 and y-intercept of 6) will illustrate that both equations have an infinite number of solutions because they overlap entirely along that line.
Answer
(__, ___)
Answered by
GPT-4o mini
The point where the two equations overlap can be expressed as the equation of the line they both represent, \( y = 2x + 6 \). However, if you are looking for specific points of intersection or a representation of the overlap, we can find some points on this line.
For example, when \( x = 0 \):
\[
y = 2(0) + 6 = 6
\]
So one point is \( (0, 6) \).
When \( x = 1 \):
\[
y = 2(1) + 6 = 8
\]
So another point is \( (1, 8) \).
When \( x = -3 \):
\[
y = 2(-3) + 6 = 0
\]
So another point is \( (-3, 0) \).
You can use any value for \( x \) to find corresponding values of \( y \) to describe points along the line.
Thus, the line represented by both equations can also be described by points such as:
- \( (0, 6) \)
- \( (1, 8) \)
- \( (-3, 0) \)
If you want the general form for the infinite number of solutions, it’s simply the line \( (x, 2x + 6) \) for all \( x \) in the set of real numbers.
In a concise format, if you're specifically asking for a format like:
\[
(x, y) = (x, 2x + 6) \quad \text{for } x \in \mathbb{R}
\]
Or using specific points as mentioned:
\[
(0, 6) \text{ or } (1, 8) \text{ or } (-3, 0)
\] etc.
For example, when \( x = 0 \):
\[
y = 2(0) + 6 = 6
\]
So one point is \( (0, 6) \).
When \( x = 1 \):
\[
y = 2(1) + 6 = 8
\]
So another point is \( (1, 8) \).
When \( x = -3 \):
\[
y = 2(-3) + 6 = 0
\]
So another point is \( (-3, 0) \).
You can use any value for \( x \) to find corresponding values of \( y \) to describe points along the line.
Thus, the line represented by both equations can also be described by points such as:
- \( (0, 6) \)
- \( (1, 8) \)
- \( (-3, 0) \)
If you want the general form for the infinite number of solutions, it’s simply the line \( (x, 2x + 6) \) for all \( x \) in the set of real numbers.
In a concise format, if you're specifically asking for a format like:
\[
(x, y) = (x, 2x + 6) \quad \text{for } x \in \mathbb{R}
\]
Or using specific points as mentioned:
\[
(0, 6) \text{ or } (1, 8) \text{ or } (-3, 0)
\] etc.
Answer
Which of these graphs shows that the linear system y=−x+6 and 3x+3y=18 has an infinite number of solutions?(1 point)
It intersects at (__, __)
It intersects at (__, __)
Answered by
GPT-4o mini
To determine whether the linear system represented by the equations \(y = -x + 6\) and \(3x + 3y = 18\) has an infinite number of solutions, we can first check if the two equations actually represent the same line.
1. **Rearranging the second equation**:
\[
3x + 3y = 18
\]
To convert it into slope-intercept form \(y = mx + b\), we can isolate \(y\):
\[
3y = -3x + 18 \implies y = -x + 6
\]
2. **Identifying the relationship**:
As we can see, both equations are:
- \(y = -x + 6\) (first equation)
- \(y = -x + 6\) (second equation after simplification)
Since both equations are equivalent, they represent the same line. This means that they intersect at every point along the line represented by \(y = -x + 6\), indicating that the system has an infinite number of solutions.
3. **Finding a point of intersection**:
To express this overlap in a specific form, we can check a few points that lie on the line \(y = -x + 6\).
For example:
- When \(x = 0\):
\[
y = -0 + 6 = 6 \quad \text{(point is } (0, 6)\text{)}
\]
- When \(x = 6\):
\[
y = -6 + 6 = 0 \quad \text{(point is } (6, 0)\text{)}
\]
- When \(x = -6\):
\[
y = -(-6) + 6 = 12 \quad \text{(point is } (-6, 12)\text{)}
\]
These points indicate that the two lines overlap entirely.
### Conclusion
The graphs showing that the linear system \(y = -x + 6\) and \(3x + 3y = 18\) has an infinite number of solutions will be the same line (overlapping), confirming they intersect at an infinite number of points.
#### Answer
The intersection can be represented as:
\[
(x, y) = (0, 6) \quad \text{(or any other point on the line, such as (6, 0) or (-6, 12))}
\]
1. **Rearranging the second equation**:
\[
3x + 3y = 18
\]
To convert it into slope-intercept form \(y = mx + b\), we can isolate \(y\):
\[
3y = -3x + 18 \implies y = -x + 6
\]
2. **Identifying the relationship**:
As we can see, both equations are:
- \(y = -x + 6\) (first equation)
- \(y = -x + 6\) (second equation after simplification)
Since both equations are equivalent, they represent the same line. This means that they intersect at every point along the line represented by \(y = -x + 6\), indicating that the system has an infinite number of solutions.
3. **Finding a point of intersection**:
To express this overlap in a specific form, we can check a few points that lie on the line \(y = -x + 6\).
For example:
- When \(x = 0\):
\[
y = -0 + 6 = 6 \quad \text{(point is } (0, 6)\text{)}
\]
- When \(x = 6\):
\[
y = -6 + 6 = 0 \quad \text{(point is } (6, 0)\text{)}
\]
- When \(x = -6\):
\[
y = -(-6) + 6 = 12 \quad \text{(point is } (-6, 12)\text{)}
\]
These points indicate that the two lines overlap entirely.
### Conclusion
The graphs showing that the linear system \(y = -x + 6\) and \(3x + 3y = 18\) has an infinite number of solutions will be the same line (overlapping), confirming they intersect at an infinite number of points.
#### Answer
The intersection can be represented as:
\[
(x, y) = (0, 6) \quad \text{(or any other point on the line, such as (6, 0) or (-6, 12))}
\]
Answer
Solve the given system of linear equations by inspection.
y=3/5x−5
y=−3/5x−5
(1 point)
Responses
(−5,0)
left parenthesis negative 5 comma 0 right parenthesis
(0,−5)
left parenthesis 0 comma negative 5 right parenthesis
no solution
no solution
infinitely many solutions
y=3/5x−5
y=−3/5x−5
(1 point)
Responses
(−5,0)
left parenthesis negative 5 comma 0 right parenthesis
(0,−5)
left parenthesis 0 comma negative 5 right parenthesis
no solution
no solution
infinitely many solutions
Answered by
GPT-4o mini
To solve the given system of linear equations by inspection, we can analyze the two equations:
1. \(y = \frac{3}{5}x - 5\)
2. \(y = -\frac{3}{5}x - 5\)
Here's a step-by-step approach to solving them:
### Step 1: Set the equations equal to each other
Since both equations equal \(y\), we can set them equal to find the intersection point:
\[
\frac{3}{5}x - 5 = -\frac{3}{5}x - 5
\]
### Step 2: Simplify the equation
First, we can add \(5\) to both sides:
\[
\frac{3}{5}x = -\frac{3}{5}x
\]
### Step 3: Combine the terms
Now, add \(\frac{3}{5}x\) to both sides:
\[
\frac{3}{5}x + \frac{3}{5}x = 0
\]
\[
\frac{6}{5}x = 0
\]
### Step 4: Solve for \(x\)
Dividing both sides by \(\frac{6}{5}\):
\[
x = 0
\]
### Step 5: Substitute \(x\) back into one of the original equations
Now we substitute \(x = 0\) into either equation to find \(y\). We'll use the first equation:
\[
y = \frac{3}{5}(0) - 5 = -5
\]
### Conclusion
Thus, we have found the solution to the system of equations:
\[
(x, y) = (0, -5)
\]
### Final Answer
The solution is:
\[
\text{(0, -5)}
\]
So the correct choice from the responses is **(0,−5)**.
1. \(y = \frac{3}{5}x - 5\)
2. \(y = -\frac{3}{5}x - 5\)
Here's a step-by-step approach to solving them:
### Step 1: Set the equations equal to each other
Since both equations equal \(y\), we can set them equal to find the intersection point:
\[
\frac{3}{5}x - 5 = -\frac{3}{5}x - 5
\]
### Step 2: Simplify the equation
First, we can add \(5\) to both sides:
\[
\frac{3}{5}x = -\frac{3}{5}x
\]
### Step 3: Combine the terms
Now, add \(\frac{3}{5}x\) to both sides:
\[
\frac{3}{5}x + \frac{3}{5}x = 0
\]
\[
\frac{6}{5}x = 0
\]
### Step 4: Solve for \(x\)
Dividing both sides by \(\frac{6}{5}\):
\[
x = 0
\]
### Step 5: Substitute \(x\) back into one of the original equations
Now we substitute \(x = 0\) into either equation to find \(y\). We'll use the first equation:
\[
y = \frac{3}{5}(0) - 5 = -5
\]
### Conclusion
Thus, we have found the solution to the system of equations:
\[
(x, y) = (0, -5)
\]
### Final Answer
The solution is:
\[
\text{(0, -5)}
\]
So the correct choice from the responses is **(0,−5)**.