Question
Use the substitution method to solve the system 2x+3y=4 , −4x−6y=−8 .(1 point)
Responses
no solution
no solution
(2−1)
left parenthesis 2 comma negative 1 right parenthesis
(−1,2)
left parenthesis negative 1 comma 2 right parenthesis
infinite solutions
Responses
no solution
no solution
(2−1)
left parenthesis 2 comma negative 1 right parenthesis
(−1,2)
left parenthesis negative 1 comma 2 right parenthesis
infinite solutions
Answers
Answer
are you sure ?
Answer
Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point)
Responses
no solution
no solution
infinite solutions
infinite solutions
(−313,23)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis
(23,−313)
Responses
no solution
no solution
infinite solutions
infinite solutions
(−313,23)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis
(23,−313)
Answer
Use the substitution method to solve the system 2.5x+y=−2 , 3x+2y=0 .(1 point)
Responses
no solution
no solution
(3,−2)
left parenthesis 3 comma negative 2 right parenthesis
(−2,3)
left parenthesis negative 2 comma 3 right parenthesis
infinite solutions
Responses
no solution
no solution
(3,−2)
left parenthesis 3 comma negative 2 right parenthesis
(−2,3)
left parenthesis negative 2 comma 3 right parenthesis
infinite solutions
Answer
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
(4,8) and (3,5)
left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis
(−4,−16) and (−3,−13)
left parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis
(3,5)
left parenthesis 3 comma 5 right parenthesis
Responses
(4,8)
left parenthesis 4 comma 8 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,−13)
left parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis
(3,5)
left parenthesis 3 comma 5 right parenthesis
Answered by
GPT-4o mini
To solve the system of equations using the substitution method, we first look at the given equations:
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)
Let's simplify Equation 2. Notice that if we multiply Equation 1 by 2, we can see if it is equivalent to Equation 2:
\[
2(2x + 3y) = 2(4) \implies 4x + 6y = 8
\]
Now we rewrite Equation 2:
\[
-4x - 6y = -8 \implies 4x + 6y = 8
\]
Now, we can see that Equation 2 is just the negative of our modified Equation 1:
\[
4x + 6y = 8 \quad \text{and} \quad - (4x + 6y) = -8
\]
This shows that the two equations represent the same line, meaning they have infinite solutions that satisfy both equations.
Thus, the correct response is:
**infinite solutions**
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)
Let's simplify Equation 2. Notice that if we multiply Equation 1 by 2, we can see if it is equivalent to Equation 2:
\[
2(2x + 3y) = 2(4) \implies 4x + 6y = 8
\]
Now we rewrite Equation 2:
\[
-4x - 6y = -8 \implies 4x + 6y = 8
\]
Now, we can see that Equation 2 is just the negative of our modified Equation 1:
\[
4x + 6y = 8 \quad \text{and} \quad - (4x + 6y) = -8
\]
This shows that the two equations represent the same line, meaning they have infinite solutions that satisfy both equations.
Thus, the correct response is:
**infinite solutions**
Answered by
GPT-4o mini
Let's verify the steps to ensure that the solution is accurate.
We have the two equations:
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)
First, we can simplify Equation 2 by dividing through by -2:
\[
-4x - 6y = -8 \Rightarrow 2x + 3y = 4
\]
This shows that Equation 2 is actually the same as Equation 1:
\[
2x + 3y = 4
\]
Since both equations represent the same line, it implies that there are infinite solutions to the system.
To summarize:
Both equations describe the same line, which means every point on this line is a solution to the system of equations. Therefore, the system has **infinite solutions**.
So the answer is indeed:
**infinite solutions**
We have the two equations:
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)
First, we can simplify Equation 2 by dividing through by -2:
\[
-4x - 6y = -8 \Rightarrow 2x + 3y = 4
\]
This shows that Equation 2 is actually the same as Equation 1:
\[
2x + 3y = 4
\]
Since both equations represent the same line, it implies that there are infinite solutions to the system.
To summarize:
Both equations describe the same line, which means every point on this line is a solution to the system of equations. Therefore, the system has **infinite solutions**.
So the answer is indeed:
**infinite solutions**
Answered by
GPT-4o mini
To solve the system of equations using the substitution method, we start with the given equations:
1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)
We can substitute Equation 1 into Equation 2.
Substituting \( x \) from Equation 1 into Equation 2:
\[
(y - 4) + 8y = 2
\]
Now combine like terms:
\[
y - 4 + 8y = 2 \\
9y - 4 = 2
\]
Now, add 4 to both sides:
\[
9y = 6
\]
Next, divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]
Now that we have the value for \( y \), we can substitute it back into Equation 1 to find \( x \):
\[
x = y - 4 \\
x = \frac{2}{3} - 4
\]
To express 4 as a fraction with the same denominator:
\[
4 = \frac{12}{3} \\
x = \frac{2}{3} - \frac{12}{3} = \frac{-10}{3}
\]
So the solution to the system of equations is:
\[
\left( x, y \right) = \left( -\frac{10}{3}, \frac{2}{3} \right)
\]
Looking at the answer choices, it looks like the closest answer matches is:
\[
(−\frac{10}{3}, \frac{2}{3})
\]
Thus, the correct response is:
**(−\frac{10}{3}, \frac{2}{3})**
1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)
We can substitute Equation 1 into Equation 2.
Substituting \( x \) from Equation 1 into Equation 2:
\[
(y - 4) + 8y = 2
\]
Now combine like terms:
\[
y - 4 + 8y = 2 \\
9y - 4 = 2
\]
Now, add 4 to both sides:
\[
9y = 6
\]
Next, divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]
Now that we have the value for \( y \), we can substitute it back into Equation 1 to find \( x \):
\[
x = y - 4 \\
x = \frac{2}{3} - 4
\]
To express 4 as a fraction with the same denominator:
\[
4 = \frac{12}{3} \\
x = \frac{2}{3} - \frac{12}{3} = \frac{-10}{3}
\]
So the solution to the system of equations is:
\[
\left( x, y \right) = \left( -\frac{10}{3}, \frac{2}{3} \right)
\]
Looking at the answer choices, it looks like the closest answer matches is:
\[
(−\frac{10}{3}, \frac{2}{3})
\]
Thus, the correct response is:
**(−\frac{10}{3}, \frac{2}{3})**
Answered by
GPT-4o mini
To solve the system of equations using substitution, we first have the two equations:
1. \( 2.5x + y = -2 \) (Equation 1)
2. \( 3x + 2y = 0 \) (Equation 2)
### Step 1: Solve for \( y \) in terms of \( x \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[
y = -2 - 2.5x
\]
### Step 2: Substitute into Equation 2
Now substitute \( y \) from Equation 1 into Equation 2:
\[
3x + 2(-2 - 2.5x) = 0
\]
Distributing the \( 2 \):
\[
3x - 4 - 5x = 0
\]
Combine like terms:
\[
-2x - 4 = 0
\]
### Step 3: Solve for \( x \)
Add \( 4 \) to both sides:
\[
-2x = 4
\]
Now divide by \( -2 \):
\[
x = -2
\]
### Step 4: Substitute back to find \( y \)
Now substitute \( x = -2 \) back into the equation we derived for \( y \):
\[
y = -2 - 2.5(-2)
\]
This simplifies to:
\[
y = -2 + 5 = 3
\]
### Final Solution
Thus, the solution to the system of equations is:
\[
(x, y) = (-2, 3)
\]
Now, let's check the provided options:
1. **no solution**
2. **(3,−2)**
3. **(−2, 3)** ✔️
4. **infinite solutions**
The correct response is:
**(−2, 3)**
1. \( 2.5x + y = -2 \) (Equation 1)
2. \( 3x + 2y = 0 \) (Equation 2)
### Step 1: Solve for \( y \) in terms of \( x \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[
y = -2 - 2.5x
\]
### Step 2: Substitute into Equation 2
Now substitute \( y \) from Equation 1 into Equation 2:
\[
3x + 2(-2 - 2.5x) = 0
\]
Distributing the \( 2 \):
\[
3x - 4 - 5x = 0
\]
Combine like terms:
\[
-2x - 4 = 0
\]
### Step 3: Solve for \( x \)
Add \( 4 \) to both sides:
\[
-2x = 4
\]
Now divide by \( -2 \):
\[
x = -2
\]
### Step 4: Substitute back to find \( y \)
Now substitute \( x = -2 \) back into the equation we derived for \( y \):
\[
y = -2 - 2.5(-2)
\]
This simplifies to:
\[
y = -2 + 5 = 3
\]
### Final Solution
Thus, the solution to the system of equations is:
\[
(x, y) = (-2, 3)
\]
Now, let's check the provided options:
1. **no solution**
2. **(3,−2)**
3. **(−2, 3)** ✔️
4. **infinite solutions**
The correct response is:
**(−2, 3)**
Answered by
GPT-4o mini
To solve the linear-quadratic system using the substitution method, we have the following equations:
1. \( y = x^2 - 4x + 8 \) (Quadratic Equation)
2. \( y = 3x - 4 \) (Linear Equation)
### Step 1: Set the equations equal to each other
Since both equations are equal to \( y \), we can set them equal to each other:
\[
x^2 - 4x + 8 = 3x - 4
\]
### Step 2: Rearrange the equation
Rearranging gives us:
\[
x^2 - 4x - 3x + 8 + 4 = 0
\]
Combine like terms:
\[
x^2 - 7x + 12 = 0
\]
### Step 3: Factor the quadratic equation
We can factor this quadratic equation:
\[
(x - 3)(x - 4) = 0
\]
This gives us two solutions for \( x \):
1. \( x = 3 \)
2. \( x = 4 \)
### Step 4: Find corresponding \( y \) values
Now we can find the \( y \) values by substituting \( x \) back into the linear equation \( y = 3x - 4 \):
1. For \( x = 3 \):
\[
y = 3(3) - 4 = 9 - 4 = 5
\]
Thus, we have the point \( (3, 5) \).
2. For \( x = 4 \):
\[
y = 3(4) - 4 = 12 - 4 = 8
\]
Thus, we have the point \( (4, 8) \).
### Final Solutions
The solutions to the system are:
1. \( (3, 5) \)
2. \( (4, 8) \)
### Conclusion
The correct answer from the choices provided is:
**(4, 8) and (3, 5)**.
1. \( y = x^2 - 4x + 8 \) (Quadratic Equation)
2. \( y = 3x - 4 \) (Linear Equation)
### Step 1: Set the equations equal to each other
Since both equations are equal to \( y \), we can set them equal to each other:
\[
x^2 - 4x + 8 = 3x - 4
\]
### Step 2: Rearrange the equation
Rearranging gives us:
\[
x^2 - 4x - 3x + 8 + 4 = 0
\]
Combine like terms:
\[
x^2 - 7x + 12 = 0
\]
### Step 3: Factor the quadratic equation
We can factor this quadratic equation:
\[
(x - 3)(x - 4) = 0
\]
This gives us two solutions for \( x \):
1. \( x = 3 \)
2. \( x = 4 \)
### Step 4: Find corresponding \( y \) values
Now we can find the \( y \) values by substituting \( x \) back into the linear equation \( y = 3x - 4 \):
1. For \( x = 3 \):
\[
y = 3(3) - 4 = 9 - 4 = 5
\]
Thus, we have the point \( (3, 5) \).
2. For \( x = 4 \):
\[
y = 3(4) - 4 = 12 - 4 = 8
\]
Thus, we have the point \( (4, 8) \).
### Final Solutions
The solutions to the system are:
1. \( (3, 5) \)
2. \( (4, 8) \)
### Conclusion
The correct answer from the choices provided is:
**(4, 8) and (3, 5)**.
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