Asked by Yes.
Use the table to answer the question.
System A System B
−x+y=1
x+y=11 x+y=11
?
Which of the following shows the equation that creates a true equivalent system?
Responses
−x−y=−1
−2x−2y=−22
2x+2y=12
2y=12
System A System B
−x+y=1
x+y=11 x+y=11
?
Which of the following shows the equation that creates a true equivalent system?
Responses
−x−y=−1
−2x−2y=−22
2x+2y=12
2y=12
Answers
Answered by
Yes.
Use the table to answer the question.
System A System B
7x+4y=−4
5x+8y=28
7x+4y=−4
?
Which of the following terms will make the systems equivalent?
Responses:
−12x−12y=24
−14x−8y=8
12x+12y=24
−14x−8y=−8
System A System B
7x+4y=−4
5x+8y=28
7x+4y=−4
?
Which of the following terms will make the systems equivalent?
Responses:
−12x−12y=24
−14x−8y=8
12x+12y=24
−14x−8y=−8
Answered by
Yes.
Solve the following system of equations using the elimination method.
3x−4y=−4, x+3y=−10
Responses
(-2,-4)
(−16,2)
(−16,−2)
(−4,−2)
3x−4y=−4, x+3y=−10
Responses
(-2,-4)
(−16,2)
(−16,−2)
(−4,−2)
Answered by
Yes.
Use the elimination method to solve the linear-quadratic system y=x^2+10 , y=−7x−2 .(1 point)
Responses
(3,−23) and (4,−30)
(−3,19)
(−4,26)
(−3,19) and (−4,26)
Responses
(3,−23) and (4,−30)
(−3,19)
(−4,26)
(−3,19) and (−4,26)
Answered by
Yes.
Use the elimination method to solve the linear-quadratic system y=x^2−2x−2 , y=2x−6 .(1 point)
Responses
(−2,−2) and (−2,2)
(−2,−10)
(2,−2)
(2,−2) and (−2,2)
Responses
(−2,−2) and (−2,2)
(−2,−10)
(2,−2)
(2,−2) and (−2,2)
Answered by
GPT-4o mini
To determine which option creates a true equivalent system with the original equations, we need to check which option can be derived from the given equations.
We have:
**System A:**
1. \( -x + y = 1 \)
2. \( x + y = 11 \)
First, let's manipulate the second equation in System A for consideration:
From Equation 2:
\[
x + y = 11
\]
We can multiply this entire equation by 2 to find an equivalent equation:
\[
2(x + y) = 2(11) \implies 2x + 2y = 22 \quad (1)
\]
Now let's analyze each option provided:
1. **−x−y=−1**
- This is equivalent to \( x + y = 1 \), which does not relate to our equations.
2. **−2x−2y=−22**
- Dividing the equation by -2 gives us \( x + y = 11 \). This matches one of our original equations.
3. **2x+2y=12**
- This simplifies to \( x + y = 6 \), which does not match our equations.
4. **2y=12**
- Dividing by 2 gives \( y = 6 \). This does not create an equivalent system since it does not match either equation.
The correct answer that transforms into an equivalent system is:
**−2x−2y=−22**
This option \( (-2x - 2y = -22) \) correctly corresponds to the equation \( (x + y = 11) \).
We have:
**System A:**
1. \( -x + y = 1 \)
2. \( x + y = 11 \)
First, let's manipulate the second equation in System A for consideration:
From Equation 2:
\[
x + y = 11
\]
We can multiply this entire equation by 2 to find an equivalent equation:
\[
2(x + y) = 2(11) \implies 2x + 2y = 22 \quad (1)
\]
Now let's analyze each option provided:
1. **−x−y=−1**
- This is equivalent to \( x + y = 1 \), which does not relate to our equations.
2. **−2x−2y=−22**
- Dividing the equation by -2 gives us \( x + y = 11 \). This matches one of our original equations.
3. **2x+2y=12**
- This simplifies to \( x + y = 6 \), which does not match our equations.
4. **2y=12**
- Dividing by 2 gives \( y = 6 \). This does not create an equivalent system since it does not match either equation.
The correct answer that transforms into an equivalent system is:
**−2x−2y=−22**
This option \( (-2x - 2y = -22) \) correctly corresponds to the equation \( (x + y = 11) \).
Answered by
GPT-4o mini
To determine which equation will create an equivalent system with the given equations in System A, we need to look at the existing equations:
**System A:**
1. \( 7x + 4y = -4 \)
2. \( 5x + 8y = 28 \)
For an equivalent system, we can manipulate one of the existing equations to derive a new equation.
### Step 1: Manipulate the first equation
We can multiply the first equation by a constant to find an equivalent equation. Let's multiply \( 7x + 4y = -4 \) by -2:
\[
-2(7x + 4y) = -2(-4)
\]
This gives us:
\[
-14x - 8y = 8 \quad \text{ (1)}
\]
### Step 2: Check the options
Now let's examine each option to see if it matches the result:
1. **−12x−12y=24**
- This does not match our manipulated equation.
2. **−14x−8y=8**
- This matches the derived equation (1).
3. **12x+12y=24**
- This does not match our manipulated equation.
4. **−14x−8y=−8**
- This is not equivalent to our derived equation as it has a different constant on the right side.
### Conclusion
The equation that will create an equivalent system is:
**−14x−8y=8**
This option correctly corresponds to one of the equations in System A after manipulation.
**System A:**
1. \( 7x + 4y = -4 \)
2. \( 5x + 8y = 28 \)
For an equivalent system, we can manipulate one of the existing equations to derive a new equation.
### Step 1: Manipulate the first equation
We can multiply the first equation by a constant to find an equivalent equation. Let's multiply \( 7x + 4y = -4 \) by -2:
\[
-2(7x + 4y) = -2(-4)
\]
This gives us:
\[
-14x - 8y = 8 \quad \text{ (1)}
\]
### Step 2: Check the options
Now let's examine each option to see if it matches the result:
1. **−12x−12y=24**
- This does not match our manipulated equation.
2. **−14x−8y=8**
- This matches the derived equation (1).
3. **12x+12y=24**
- This does not match our manipulated equation.
4. **−14x−8y=−8**
- This is not equivalent to our derived equation as it has a different constant on the right side.
### Conclusion
The equation that will create an equivalent system is:
**−14x−8y=8**
This option correctly corresponds to one of the equations in System A after manipulation.
Answered by
GPT-4o mini
To solve the system of equations using the elimination method, we have the following two equations:
1. \( 3x - 4y = -4 \) (Equation 1)
2. \( x + 3y = -10 \) (Equation 2)
### Step 1: Align the equations
We can either multiply one or both equations to eliminate one variable. Let's try to eliminate \(x\) by making the coefficients of \(x\) in both equations the same.
From Equation 2, let's multiply it by 3 to match the coefficient of \(x\) in Equation 1:
\[
3(x + 3y) = 3(-10)
\]
This gives us:
\[
3x + 9y = -30 \quad (Equation 3)
\]
### Step 2: Subtract the equations
Now we have:
1. \( 3x - 4y = -4 \) (Equation 1)
2. \( 3x + 9y = -30 \) (Equation 3)
Next, we subtract Equation 1 from Equation 3:
\[
(3x + 9y) - (3x - 4y) = -30 - (-4)
\]
This simplifies to:
\[
3x + 9y - 3x + 4y = -30 + 4
\]
Combine the like terms:
\[
13y = -26
\]
### Step 3: Solve for \(y\)
Now, we can solve for \(y\):
\[
y = \frac{-26}{13} = -2
\]
### Step 4: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute this value back into either of the original equations to find \(x\). We'll use Equation 2:
\[
x + 3(-2) = -10
\]
This simplifies to:
\[
x - 6 = -10
\]
Now, solve for \(x\):
\[
x = -10 + 6 = -4
\]
### Step 5: Solution
Therefore, the solution to the system of equations is:
\[
(x, y) = (-4, -2)
\]
### Response
The correct response is:
**(−4,−2)**
1. \( 3x - 4y = -4 \) (Equation 1)
2. \( x + 3y = -10 \) (Equation 2)
### Step 1: Align the equations
We can either multiply one or both equations to eliminate one variable. Let's try to eliminate \(x\) by making the coefficients of \(x\) in both equations the same.
From Equation 2, let's multiply it by 3 to match the coefficient of \(x\) in Equation 1:
\[
3(x + 3y) = 3(-10)
\]
This gives us:
\[
3x + 9y = -30 \quad (Equation 3)
\]
### Step 2: Subtract the equations
Now we have:
1. \( 3x - 4y = -4 \) (Equation 1)
2. \( 3x + 9y = -30 \) (Equation 3)
Next, we subtract Equation 1 from Equation 3:
\[
(3x + 9y) - (3x - 4y) = -30 - (-4)
\]
This simplifies to:
\[
3x + 9y - 3x + 4y = -30 + 4
\]
Combine the like terms:
\[
13y = -26
\]
### Step 3: Solve for \(y\)
Now, we can solve for \(y\):
\[
y = \frac{-26}{13} = -2
\]
### Step 4: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute this value back into either of the original equations to find \(x\). We'll use Equation 2:
\[
x + 3(-2) = -10
\]
This simplifies to:
\[
x - 6 = -10
\]
Now, solve for \(x\):
\[
x = -10 + 6 = -4
\]
### Step 5: Solution
Therefore, the solution to the system of equations is:
\[
(x, y) = (-4, -2)
\]
### Response
The correct response is:
**(−4,−2)**
Answered by
GPT-4o mini
To solve the linear-quadratic system of equations using the elimination method, we have the following two equations:
1. \( y = x^2 + 10 \) (Equation 1)
2. \( y = -7x - 2 \) (Equation 2)
Since both equations are equal to \(y\), we can set them equal to each other:
\[
x^2 + 10 = -7x - 2
\]
### Step 1: Rearranging the equation
Now, let's rearrange the equation to set it to 0:
\[
x^2 + 7x + 12 = 0
\]
### Step 2: Factoring the quadratic equation
Next, we can factor the quadratic:
\[
(x + 3)(x + 4) = 0
\]
### Step 3: Finding the values of \(x\)
Setting each factor equal to zero gives us:
1. \( x + 3 = 0 \) → \( x = -3 \)
2. \( x + 4 = 0 \) → \( x = -4 \)
### Step 4: Finding the corresponding \(y\) values
Now we need to find the corresponding \(y\) values for each \(x\). We can use either original equation; we will use Equation 2:
1. For \( x = -3 \):
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
So one solution is \( (-3, 19) \).
2. For \( x = -4 \):
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
So the second solution is \( (-4, 26) \).
### Step 5: Solutions
Thus, the solutions to the system are:
\[
(-3, 19) \quad \text{and} \quad (-4, 26)
\]
### Response
The correct response is:
**(−3,19) and (−4,26)**
1. \( y = x^2 + 10 \) (Equation 1)
2. \( y = -7x - 2 \) (Equation 2)
Since both equations are equal to \(y\), we can set them equal to each other:
\[
x^2 + 10 = -7x - 2
\]
### Step 1: Rearranging the equation
Now, let's rearrange the equation to set it to 0:
\[
x^2 + 7x + 12 = 0
\]
### Step 2: Factoring the quadratic equation
Next, we can factor the quadratic:
\[
(x + 3)(x + 4) = 0
\]
### Step 3: Finding the values of \(x\)
Setting each factor equal to zero gives us:
1. \( x + 3 = 0 \) → \( x = -3 \)
2. \( x + 4 = 0 \) → \( x = -4 \)
### Step 4: Finding the corresponding \(y\) values
Now we need to find the corresponding \(y\) values for each \(x\). We can use either original equation; we will use Equation 2:
1. For \( x = -3 \):
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
So one solution is \( (-3, 19) \).
2. For \( x = -4 \):
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
So the second solution is \( (-4, 26) \).
### Step 5: Solutions
Thus, the solutions to the system are:
\[
(-3, 19) \quad \text{and} \quad (-4, 26)
\]
### Response
The correct response is:
**(−3,19) and (−4,26)**
Answered by
GPT-4o mini
To solve the linear-quadratic system of equations using the elimination method, we have:
1. \( y = x^2 - 2x - 2 \) (Equation 1)
2. \( y = 2x - 6 \) (Equation 2)
### Step 1: Set the equations equal to each other
Since both equations equal \(y\), we can set them equal to each other:
\[
x^2 - 2x - 2 = 2x - 6
\]
### Step 2: Rearranging the equation
Now, let's rearrange the equation to set it to 0:
\[
x^2 - 2x - 2 - 2x + 6 = 0
\]
\[
x^2 - 4x + 4 = 0
\]
### Step 3: Factoring the quadratic equation
Now we can factor the quadratic:
\[
(x - 2)(x - 2) = 0
\]
### Step 4: Finding the values of \(x\)
Setting the factor equal to zero gives us:
\[
x - 2 = 0 \implies x = 2
\]
### Step 5: Finding the corresponding \(y\) value
Now, we need to find the corresponding \(y\) value for \(x = 2\). We can use either of the original equations; we'll use Equation 2:
\[
y = 2(2) - 6 = 4 - 6 = -2
\]
So, the only solution to the system is \( (2, -2) \).
### Step 6: Conclusion
Thus, the solution to the linear-quadratic system is:
\[
(2, -2)
\]
### Response
The correct response is:
**(2,−2)**.
1. \( y = x^2 - 2x - 2 \) (Equation 1)
2. \( y = 2x - 6 \) (Equation 2)
### Step 1: Set the equations equal to each other
Since both equations equal \(y\), we can set them equal to each other:
\[
x^2 - 2x - 2 = 2x - 6
\]
### Step 2: Rearranging the equation
Now, let's rearrange the equation to set it to 0:
\[
x^2 - 2x - 2 - 2x + 6 = 0
\]
\[
x^2 - 4x + 4 = 0
\]
### Step 3: Factoring the quadratic equation
Now we can factor the quadratic:
\[
(x - 2)(x - 2) = 0
\]
### Step 4: Finding the values of \(x\)
Setting the factor equal to zero gives us:
\[
x - 2 = 0 \implies x = 2
\]
### Step 5: Finding the corresponding \(y\) value
Now, we need to find the corresponding \(y\) value for \(x = 2\). We can use either of the original equations; we'll use Equation 2:
\[
y = 2(2) - 6 = 4 - 6 = -2
\]
So, the only solution to the system is \( (2, -2) \).
### Step 6: Conclusion
Thus, the solution to the linear-quadratic system is:
\[
(2, -2)
\]
### Response
The correct response is:
**(2,−2)**.
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