Asked by adaleta
Use the substitution method to solve the system 2x+3y=4, −4x−6y=−8.(1 point)
Responses
infinite solutions
infinite solutions
no solution
no solution
(−1,2)
left parenthesis negative 1 comma 2 right parenthesis
(2−1)
Responses
infinite solutions
infinite solutions
no solution
no solution
(−1,2)
left parenthesis negative 1 comma 2 right parenthesis
(2−1)
Answers
Answered by
adaleta
Use the substitution method to solve the system x=y−4, x+8y=2.(1 point)
Responses
infinite solutions
infinite solutions
(23,−313)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
(−313,23)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis
no solution
Responses
infinite solutions
infinite solutions
(23,−313)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
(−313,23)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis
no solution
Answered by
adaleta
yes
Answered by
adaleta
Use the substitution method to solve the system x=y−4, x+8y=2.(1 point)
Responses
infinite solutions
infinite solutions
(2/3,−3/13)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
(−3/13,2/3)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis
no solution
Responses
infinite solutions
infinite solutions
(2/3,−3/13)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
(−3/13,2/3)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis
no solution
Answered by
adaleta
Use the substitution method to solve the system 2.5x+y=−2, 3x+2y=0.(1 point)
Responses
(3,−2)
left parenthesis 3 comma negative 2 right parenthesis
infinite solutions
infinite solutions
(−2,3)
left parenthesis negative 2 comma 3 right parenthesis
no solution
Responses
(3,−2)
left parenthesis 3 comma negative 2 right parenthesis
infinite solutions
infinite solutions
(−2,3)
left parenthesis negative 2 comma 3 right parenthesis
no solution
Answered by
adaleta
Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8, y=3x−4?(1 point)
Responses
(4,8)
left parenthesis 4 comma 8 right parenthesis
(3,5)
left parenthesis 3 comma 5 right parenthesis
(4,8) and (3,5)
left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis
(−4,−16) and -3
Responses
(4,8)
left parenthesis 4 comma 8 right parenthesis
(3,5)
left parenthesis 3 comma 5 right parenthesis
(4,8) and (3,5)
left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis
(−4,−16) and -3
Answered by
GPT-4o mini
To solve the system of equations using the substitution method, we first express one variable in terms of the other.
The given system is:
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)
We can start by solving Equation 1 for \( y \):
\[
3y = 4 - 2x
\]
\[
y = \frac{4 - 2x}{3}
\]
Now, we can substitute this expression for \( y \) into Equation 2. However, first, notice that Equation 2 can be simplified directly by dividing all terms by -2:
\[
2x + 3y = 4
\]
This shows that both equations represent the same line because Equation 1 and the simplified Equation 2 are identical.
Since both equations are the same, there are infinite solutions along the line defined by \( 2x + 3y = 4 \).
Thus, the correct response to the system of equations is:
**infinite solutions**.
The given system is:
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)
We can start by solving Equation 1 for \( y \):
\[
3y = 4 - 2x
\]
\[
y = \frac{4 - 2x}{3}
\]
Now, we can substitute this expression for \( y \) into Equation 2. However, first, notice that Equation 2 can be simplified directly by dividing all terms by -2:
\[
2x + 3y = 4
\]
This shows that both equations represent the same line because Equation 1 and the simplified Equation 2 are identical.
Since both equations are the same, there are infinite solutions along the line defined by \( 2x + 3y = 4 \).
Thus, the correct response to the system of equations is:
**infinite solutions**.
Answered by
GPT-4o mini
To solve the system of equations using the substitution method, we have the following two equations:
1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)
We can start by substituting the expression for \( x \) from Equation 1 into Equation 2.
Substituting \( x = y - 4 \) into Equation 2:
\[
(y - 4) + 8y = 2
\]
Combine like terms:
\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]
Now, add 4 to both sides:
\[
9y = 6
\]
Then, divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[
x = y - 4 = \frac{2}{3} - 4
\]
To express it with a common denominator:
\[
x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]
So the solution to the system is:
\[
\left( x, y \right) = \left( -\frac{10}{3}, \frac{2}{3} \right)
\]
Now, we compare this to the given responses. Note that it seems there are no exact matches with my calculation.
Thus, the answer should be:
\[
\left( -\frac{10}{3}, \frac{2}{3} \right)
\]
Verify whether any of the given options reflect this solution accurately:
- \( (23, -313) \)
- \( (-313, 23) \)
- \( \text{no solution} \)
None of these choices are correct based on the calculations. Please double-check your options or provide the correct choices if there's an error. Would you like to confirm or correct the response options?
1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)
We can start by substituting the expression for \( x \) from Equation 1 into Equation 2.
Substituting \( x = y - 4 \) into Equation 2:
\[
(y - 4) + 8y = 2
\]
Combine like terms:
\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]
Now, add 4 to both sides:
\[
9y = 6
\]
Then, divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[
x = y - 4 = \frac{2}{3} - 4
\]
To express it with a common denominator:
\[
x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]
So the solution to the system is:
\[
\left( x, y \right) = \left( -\frac{10}{3}, \frac{2}{3} \right)
\]
Now, we compare this to the given responses. Note that it seems there are no exact matches with my calculation.
Thus, the answer should be:
\[
\left( -\frac{10}{3}, \frac{2}{3} \right)
\]
Verify whether any of the given options reflect this solution accurately:
- \( (23, -313) \)
- \( (-313, 23) \)
- \( \text{no solution} \)
None of these choices are correct based on the calculations. Please double-check your options or provide the correct choices if there's an error. Would you like to confirm or correct the response options?
Answered by
GPT-4o mini
Great! Please provide the correct options or confirm any modifications to the existing options based on our solution of
\[
\left( -\frac{10}{3}, \frac{2}{3} \right).
\]
If possible, I can help verify the correctness of any new options you provide!
\[
\left( -\frac{10}{3}, \frac{2}{3} \right).
\]
If possible, I can help verify the correctness of any new options you provide!
Answered by
GPT-4o mini
Let's solve the system of equations again using the substitution method to ensure accuracy.
The equations are:
1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)
**Step 1: Substitute \( x \) in Equation 2**
From Equation 1, we know:
\[
x = y - 4
\]
Now substitute \( x \) in Equation 2:
\[
(y - 4) + 8y = 2
\]
**Step 2: Combine like terms**
\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]
**Step 3: Solve for \( y \)**
Add 4 to both sides:
\[
9y = 6
\]
Now divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]
**Step 4: Substitute \( y \) back to find \( x \)**
Now use the value of \( y \) to find \( x \):
\[
x = y - 4
\]
\[
x = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]
Now we have:
\[
\left(x, y\right) = \left(-\frac{10}{3}, \frac{2}{3}\right)
\]
However, it seems there is a need to verify if we had made an error or if the answer relates to those options you provided.
### Check the Possible Responses:
1. **infinite solutions**
2. **\((\frac{2}{3}, -\frac{3}{13})\)**
3. **\((- \frac{3}{13}, \frac{2}{3})\)**
4. **no solution**
#### Solution Misalignment:
The solutions we've found do not match any of the specified options:
- Our calculated solution is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).
**Verification of Given Responses:**
1. **Infinite solutions** would apply were the lines coincident, which isn't the case here.
2. **No solution** would apply if the lines were parallel, which isn't the case here either.
3. Upon checking: Neither of \((\frac{2}{3}, -\frac{3}{13})\) nor \((- \frac{3}{13}, \frac{2}{3})\) match our solution.
### Conclusion:
The calculated solution \( \left( -\frac{10}{3}, \frac{2}{3} \right) \) does not match any of the provided options. There might be either a transcription error with the equations or the solution options.
Would you like to verify the problem statement again or see if the equations were appropriately transcribed?
The equations are:
1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)
**Step 1: Substitute \( x \) in Equation 2**
From Equation 1, we know:
\[
x = y - 4
\]
Now substitute \( x \) in Equation 2:
\[
(y - 4) + 8y = 2
\]
**Step 2: Combine like terms**
\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]
**Step 3: Solve for \( y \)**
Add 4 to both sides:
\[
9y = 6
\]
Now divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]
**Step 4: Substitute \( y \) back to find \( x \)**
Now use the value of \( y \) to find \( x \):
\[
x = y - 4
\]
\[
x = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]
Now we have:
\[
\left(x, y\right) = \left(-\frac{10}{3}, \frac{2}{3}\right)
\]
However, it seems there is a need to verify if we had made an error or if the answer relates to those options you provided.
### Check the Possible Responses:
1. **infinite solutions**
2. **\((\frac{2}{3}, -\frac{3}{13})\)**
3. **\((- \frac{3}{13}, \frac{2}{3})\)**
4. **no solution**
#### Solution Misalignment:
The solutions we've found do not match any of the specified options:
- Our calculated solution is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).
**Verification of Given Responses:**
1. **Infinite solutions** would apply were the lines coincident, which isn't the case here.
2. **No solution** would apply if the lines were parallel, which isn't the case here either.
3. Upon checking: Neither of \((\frac{2}{3}, -\frac{3}{13})\) nor \((- \frac{3}{13}, \frac{2}{3})\) match our solution.
### Conclusion:
The calculated solution \( \left( -\frac{10}{3}, \frac{2}{3} \right) \) does not match any of the provided options. There might be either a transcription error with the equations or the solution options.
Would you like to verify the problem statement again or see if the equations were appropriately transcribed?
Answered by
GPT-4o mini
Let's solve the system of equations using the substitution method.
The given system is:
1. \( 2.5x + y = -2 \) (Equation 1)
2. \( 3x + 2y = 0 \) (Equation 2)
### Step 1: Solve Equation 1 for \( y \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[
y = -2 - 2.5x
\]
### Step 2: Substitute \( y \) in Equation 2
Now substitute \( y \) in Equation 2:
\[
3x + 2(-2 - 2.5x) = 0
\]
Distributing the \( 2 \):
\[
3x - 4 - 5x = 0
\]
### Step 3: Combine like terms
Combine the \( x \) terms:
\[
-2x - 4 = 0
\]
### Step 4: Solve for \( x \)
Add \( 4 \) to both sides:
\[
-2x = 4
\]
Now divide by \(-2\):
\[
x = -2
\]
### Step 5: Substitute \( x \) back to find \( y \)
Now substitute \( x = -2 \) back into the equation for \( y \):
\[
y = -2 - 2.5(-2)
\]
\[
y = -2 + 5 = 3
\]
### Conclusion
The solution to the system of equations is:
\[
(x, y) = (-2, 3)
\]
### Compare with the Given Responses:
1. \( (3, -2) \)
2. **infinite solutions**
3. **(-2, 3)**
4. **no solution**
The solution we found matches option **(-2, 3)**.
Thus, the correct response is:
\[
\boxed{(-2, 3)}
\]
The given system is:
1. \( 2.5x + y = -2 \) (Equation 1)
2. \( 3x + 2y = 0 \) (Equation 2)
### Step 1: Solve Equation 1 for \( y \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[
y = -2 - 2.5x
\]
### Step 2: Substitute \( y \) in Equation 2
Now substitute \( y \) in Equation 2:
\[
3x + 2(-2 - 2.5x) = 0
\]
Distributing the \( 2 \):
\[
3x - 4 - 5x = 0
\]
### Step 3: Combine like terms
Combine the \( x \) terms:
\[
-2x - 4 = 0
\]
### Step 4: Solve for \( x \)
Add \( 4 \) to both sides:
\[
-2x = 4
\]
Now divide by \(-2\):
\[
x = -2
\]
### Step 5: Substitute \( x \) back to find \( y \)
Now substitute \( x = -2 \) back into the equation for \( y \):
\[
y = -2 - 2.5(-2)
\]
\[
y = -2 + 5 = 3
\]
### Conclusion
The solution to the system of equations is:
\[
(x, y) = (-2, 3)
\]
### Compare with the Given Responses:
1. \( (3, -2) \)
2. **infinite solutions**
3. **(-2, 3)**
4. **no solution**
The solution we found matches option **(-2, 3)**.
Thus, the correct response is:
\[
\boxed{(-2, 3)}
\]
Answered by
GPT-4o mini
To solve the linear-quadratic system given by the equations \(y = x^2 - 4x + 8\) and \(y = 3x - 4\) using the substitution method, we can follow these steps:
1. **Set the two equations equal to each other:**
Since both equations are equal to \(y\), we can set them equal to each other:
\[
x^2 - 4x + 8 = 3x - 4
\]
2. **Rearrange the equation:**
Move all terms to one side:
\[
x^2 - 4x - 3x + 8 + 4 = 0
\]
This simplifies to:
\[
x^2 - 7x + 12 = 0
\]
3. **Factor the quadratic equation:**
Now, we can factor the quadratic:
\[
(x - 3)(x - 4) = 0
\]
4. **Solve for \(x\):**
Setting each factor to zero gives us the solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
5. **Find corresponding \(y\) values:**
Now we substitute each \(x\) value back into either original equation to find \(y\).
- For \(x = 3\):
\[
y = 3x - 4 = 3(3) - 4 = 9 - 4 = 5
\]
So, one solution is \((3, 5)\).
- For \(x = 4\):
\[
y = 3x - 4 = 3(4) - 4 = 12 - 4 = 8
\]
So, another solution is \((4, 8)\).
### Final Solutions:
The solutions to the system are:
1. \( (3, 5) \)
2. \( (4, 8) \)
### Compare with Given Responses:
1. \( (4, 8) \)
2. \( (3, 5) \)
3. \( (4, 8) \) and \( (3, 5) \)
4. \( (-4, -16) \) and -3
### Conclusion:
The correct response is:
\[
\boxed{(4, 8) \text{ and } (3, 5)}
\]
This corresponds to option **(4, 8) and (3, 5)**.
1. **Set the two equations equal to each other:**
Since both equations are equal to \(y\), we can set them equal to each other:
\[
x^2 - 4x + 8 = 3x - 4
\]
2. **Rearrange the equation:**
Move all terms to one side:
\[
x^2 - 4x - 3x + 8 + 4 = 0
\]
This simplifies to:
\[
x^2 - 7x + 12 = 0
\]
3. **Factor the quadratic equation:**
Now, we can factor the quadratic:
\[
(x - 3)(x - 4) = 0
\]
4. **Solve for \(x\):**
Setting each factor to zero gives us the solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
5. **Find corresponding \(y\) values:**
Now we substitute each \(x\) value back into either original equation to find \(y\).
- For \(x = 3\):
\[
y = 3x - 4 = 3(3) - 4 = 9 - 4 = 5
\]
So, one solution is \((3, 5)\).
- For \(x = 4\):
\[
y = 3x - 4 = 3(4) - 4 = 12 - 4 = 8
\]
So, another solution is \((4, 8)\).
### Final Solutions:
The solutions to the system are:
1. \( (3, 5) \)
2. \( (4, 8) \)
### Compare with Given Responses:
1. \( (4, 8) \)
2. \( (3, 5) \)
3. \( (4, 8) \) and \( (3, 5) \)
4. \( (-4, -16) \) and -3
### Conclusion:
The correct response is:
\[
\boxed{(4, 8) \text{ and } (3, 5)}
\]
This corresponds to option **(4, 8) and (3, 5)**.
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