Question
Use the substitution method to solve the system 2x+3y=4 , −4x−6y=−8 .(1 point)
Responses
(−1,2)
left parenthesis negative 1 comma 2 right parenthesis
no solution
no solution
infinite solutions
infinite solutions
(2−1)
Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point)
Responses
no solution
no solution
(2/3,−3 1/3)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
infinite solutions
infinite solutions
(−3 1/3,2/3)
Which of the following shows a system of equations equivalent to 2x+7y=14 and 4x+5y=10 ?(1 point)
Responses
2x+7y=14 and 2x+5y=5
2 x plus 7 y equals 14 and 2 x plus 5 y equals 5
4x+14y=14 and 4x+5y=10
4 x plus 14 y equals 14 and 4 x plus 5 y equals 10
2x+7y=14 and 8x+10y=20
2 x plus 7 y equals 14 and 8 x plus 10 y equals 20
4x+14y=28 and 8x+10y=10
Why is the system of equations 2x+7y=14 and 4x+5y=10 equivalent to the system of equations 4x+14y=28 and 8x+10y=20 ?(1 point)
Responses
All equations share the point (0,2).
All equations share the point left parenthesis 0 comma 2 right parenthesis .
Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 12 to produce the equivalent system of equations.
Both equations, 2 x plus 7 y equals 14 and 4 x plus 5 y equals 10 , can be multiplied by Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations.
The intercepts of the equations 2x+7y=14 and 4x+14y=28 are (0,2) and (7,0).
The intercepts of the equations 2 x plus 7 y equals 14 and 4 x plus 14 y equals 28 are left parenthesis 0 comma 2 right parenthesis and left parenthesis 7 comma 0 right parenthesis .
The intercepts of both 4x+5y=10 and 8x+10y=20 are (0,2.5) and (2,0).
Solve the following system of equations using the elimination method.
3x−4y=−4 , x+3y=−10
(1 point)
Responses
(−4,−2)
left parenthesis negative 4 comma negative 2 right parenthesis
(−16,2)
left parenthesis negative 16 comma 2 right parenthesis
(−16,−2)
left parenthesis negative 16 comma negative 2 right parenthesis
(−2,−4)
please double check and give me a answer summary
Responses
(−1,2)
left parenthesis negative 1 comma 2 right parenthesis
no solution
no solution
infinite solutions
infinite solutions
(2−1)
Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point)
Responses
no solution
no solution
(2/3,−3 1/3)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
infinite solutions
infinite solutions
(−3 1/3,2/3)
Which of the following shows a system of equations equivalent to 2x+7y=14 and 4x+5y=10 ?(1 point)
Responses
2x+7y=14 and 2x+5y=5
2 x plus 7 y equals 14 and 2 x plus 5 y equals 5
4x+14y=14 and 4x+5y=10
4 x plus 14 y equals 14 and 4 x plus 5 y equals 10
2x+7y=14 and 8x+10y=20
2 x plus 7 y equals 14 and 8 x plus 10 y equals 20
4x+14y=28 and 8x+10y=10
Why is the system of equations 2x+7y=14 and 4x+5y=10 equivalent to the system of equations 4x+14y=28 and 8x+10y=20 ?(1 point)
Responses
All equations share the point (0,2).
All equations share the point left parenthesis 0 comma 2 right parenthesis .
Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 12 to produce the equivalent system of equations.
Both equations, 2 x plus 7 y equals 14 and 4 x plus 5 y equals 10 , can be multiplied by Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations.
The intercepts of the equations 2x+7y=14 and 4x+14y=28 are (0,2) and (7,0).
The intercepts of the equations 2 x plus 7 y equals 14 and 4 x plus 14 y equals 28 are left parenthesis 0 comma 2 right parenthesis and left parenthesis 7 comma 0 right parenthesis .
The intercepts of both 4x+5y=10 and 8x+10y=20 are (0,2.5) and (2,0).
Solve the following system of equations using the elimination method.
3x−4y=−4 , x+3y=−10
(1 point)
Responses
(−4,−2)
left parenthesis negative 4 comma negative 2 right parenthesis
(−16,2)
left parenthesis negative 16 comma 2 right parenthesis
(−16,−2)
left parenthesis negative 16 comma negative 2 right parenthesis
(−2,−4)
please double check and give me a answer summary
Answers
There are no human answers yet.
Answered by
GPT-4o mini
Let's review and summarize the solutions for each of the provided systems of equations.
### 1. System:
\( 2x + 3y = 4 \)
\( -4x - 6y = -8 \)
**Solution:**
First, notice that the second equation can be simplified:
\[
-4x - 6y = -8 \implies 4x + 6y = 8 \implies 2x + 3y = 4
\]
This is the same as the first equation. Since both equations are the same, there are infinitely many solutions.
**Response:** **infinite solutions**
### 2. System:
\( x = y - 4 \)
\( x + 8y = 2 \)
**Substitution Method:**
Substitute \( x \) in the second equation:
\[
(y - 4) + 8y = 2 \implies 9y - 4 = 2 \implies 9y = 6 \implies y = \frac{2}{3}
\]
Now substitute \( y \) back to find \( x \):
\[
x = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3} \quad \left(\text{or } -3\frac{1}{3}\right)
\]
**Response:** **(−3 1/3, 2/3)**
### 3. Equivalent System:
\( 2x + 7y = 14 \)
\( 4x + 5y = 10 \)
**Checking Options for Equivalence:**
To create an equivalent system:
- Multiply the first equation by 2:
\[
4x + 14y = 28
\]
- Multiply the second equation by 2:
\[
8x + 10y = 20
\]
Thus, the equivalent system is:
\[
4x + 14y = 28 \quad \text{and} \quad 8x + 10y = 20
\]
**Response:** **4x + 14y = 28 and 8x + 10y = 20**
### 4. Reason for Equivalence:
**Reasoning:**
The equivalence holds because both equations represent the same geometrical lines. A scaling factor can convert the original equations to the new ones.
**Response:** **Both equations, 2x + 7y = 14 and 4x + 5y = 10, can be multiplied by 1/2 to produce the equivalent system of equations.**
### 5. System:
\( 3x - 4y = -4 \)
\( x + 3y = -10 \)
**Elimination Method:**
Rearrange the second equation:
\[
x = -10 - 3y
\]
Substitute \( x \) in the first equation:
\[
3(-10 - 3y) - 4y = -4
\]
Simplifying:
\[
-30 - 9y - 4y = -4 \implies -13y = 26 \implies y = -2
\]
Substitute \( y \) back to find \( x \):
\[
x = -10 - 3(-2) = -10 + 6 = -4
\]
**Response:** **(−4,−2)**
---
### Summary of Responses:
1. **infinite solutions**
2. **(−3 1/3, 2/3)**
3. **4x + 14y = 28 and 8x + 10y = 20**
4. **Both equations can be multiplied by 1/2**
5. **(−4,−2)**
If you have further questions or need more clarification, feel free to ask!
### 1. System:
\( 2x + 3y = 4 \)
\( -4x - 6y = -8 \)
**Solution:**
First, notice that the second equation can be simplified:
\[
-4x - 6y = -8 \implies 4x + 6y = 8 \implies 2x + 3y = 4
\]
This is the same as the first equation. Since both equations are the same, there are infinitely many solutions.
**Response:** **infinite solutions**
### 2. System:
\( x = y - 4 \)
\( x + 8y = 2 \)
**Substitution Method:**
Substitute \( x \) in the second equation:
\[
(y - 4) + 8y = 2 \implies 9y - 4 = 2 \implies 9y = 6 \implies y = \frac{2}{3}
\]
Now substitute \( y \) back to find \( x \):
\[
x = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3} \quad \left(\text{or } -3\frac{1}{3}\right)
\]
**Response:** **(−3 1/3, 2/3)**
### 3. Equivalent System:
\( 2x + 7y = 14 \)
\( 4x + 5y = 10 \)
**Checking Options for Equivalence:**
To create an equivalent system:
- Multiply the first equation by 2:
\[
4x + 14y = 28
\]
- Multiply the second equation by 2:
\[
8x + 10y = 20
\]
Thus, the equivalent system is:
\[
4x + 14y = 28 \quad \text{and} \quad 8x + 10y = 20
\]
**Response:** **4x + 14y = 28 and 8x + 10y = 20**
### 4. Reason for Equivalence:
**Reasoning:**
The equivalence holds because both equations represent the same geometrical lines. A scaling factor can convert the original equations to the new ones.
**Response:** **Both equations, 2x + 7y = 14 and 4x + 5y = 10, can be multiplied by 1/2 to produce the equivalent system of equations.**
### 5. System:
\( 3x - 4y = -4 \)
\( x + 3y = -10 \)
**Elimination Method:**
Rearrange the second equation:
\[
x = -10 - 3y
\]
Substitute \( x \) in the first equation:
\[
3(-10 - 3y) - 4y = -4
\]
Simplifying:
\[
-30 - 9y - 4y = -4 \implies -13y = 26 \implies y = -2
\]
Substitute \( y \) back to find \( x \):
\[
x = -10 - 3(-2) = -10 + 6 = -4
\]
**Response:** **(−4,−2)**
---
### Summary of Responses:
1. **infinite solutions**
2. **(−3 1/3, 2/3)**
3. **4x + 14y = 28 and 8x + 10y = 20**
4. **Both equations can be multiplied by 1/2**
5. **(−4,−2)**
If you have further questions or need more clarification, feel free to ask!
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