Question
Solve the system by substitution.
minus, 5, y, minus, 10, equals, x
−5y−10=
x
x, plus, 9, y, equals, minus, 2
x+9y=
−2
minus, 5, y, minus, 10, equals, x
−5y−10=
x
x, plus, 9, y, equals, minus, 2
x+9y=
−2
Answers
Answer
Solve the system by substitution.
minus, 4, x, plus, 6, y, equals, minus, 10
−4x+6y=
−10
2, y, plus, 4, equals, x
2y+4=
x
minus, 4, x, plus, 6, y, equals, minus, 10
−4x+6y=
−10
2, y, plus, 4, equals, x
2y+4=
x
Answer
Solve the following system of equations algebraically:
y, equals, x, squared, minus, x, minus, 65
y=
x
2
−x−65
y, equals, minus, x, minus, 1
y=
−x−1
Answer
Attempt 1 out of 2
Two solutions
and and
y, equals, x, squared, minus, x, minus, 65
y=
x
2
−x−65
y, equals, minus, x, minus, 1
y=
−x−1
Answer
Attempt 1 out of 2
Two solutions
and and
Answer
Solve the following system of equations algebraically:
y, equals, x, squared, minus, 7, x, minus, 7
y=
x
2
−7x−7
y, equals, minus, 3, x, minus, 2
y=
−3x−2
Answer
Attempt 1 out of 2
Two solutions
and and
y, equals, x, squared, minus, 7, x, minus, 7
y=
x
2
−7x−7
y, equals, minus, 3, x, minus, 2
y=
−3x−2
Answer
Attempt 1 out of 2
Two solutions
and and
Answered by
GPT-4o mini
To solve the system of equations by substitution, we start with the given equations:
1. \(-5y - 10 = x\) (we can rearrange this to express \(x\) in terms of \(y\))
2. \(x + 9y = -2\)
From equation 1, we can express \(x\) as:
\[
x = -5y - 10
\]
Now, we can substitute this expression for \(x\) into equation 2:
\[
(-5y - 10) + 9y = -2
\]
Now, simplify the left side:
\[
-5y - 10 + 9y = -2
\]
Combine like terms:
\[
(9y - 5y) - 10 = -2
\]
\[
4y - 10 = -2
\]
Next, add 10 to both sides:
\[
4y = -2 + 10
\]
\[
4y = 8
\]
Now, divide by 4:
\[
y = 2
\]
Now that we have \(y\), we will substitute it back into the equation for \(x\):
\[
x = -5(2) - 10
\]
\[
x = -10 - 10
\]
\[
x = -20
\]
So, the solution to the system of equations is:
\[
(x, y) = (-20, 2)
\]
To verify our solution, we can substitute \(x = -20\) and \(y = 2\) back into the original equations:
1. For the first equation:
\[
-5(2) - 10 = -20 \quad \text{(True)}
\]
2. For the second equation:
\[
-20 + 9(2) = -20 + 18 = -2 \quad \text{(True)}
\]
Both equations are satisfied, confirming that the solution is correct.
Final answer:
\[
(x, y) = (-20, 2)
\]
1. \(-5y - 10 = x\) (we can rearrange this to express \(x\) in terms of \(y\))
2. \(x + 9y = -2\)
From equation 1, we can express \(x\) as:
\[
x = -5y - 10
\]
Now, we can substitute this expression for \(x\) into equation 2:
\[
(-5y - 10) + 9y = -2
\]
Now, simplify the left side:
\[
-5y - 10 + 9y = -2
\]
Combine like terms:
\[
(9y - 5y) - 10 = -2
\]
\[
4y - 10 = -2
\]
Next, add 10 to both sides:
\[
4y = -2 + 10
\]
\[
4y = 8
\]
Now, divide by 4:
\[
y = 2
\]
Now that we have \(y\), we will substitute it back into the equation for \(x\):
\[
x = -5(2) - 10
\]
\[
x = -10 - 10
\]
\[
x = -20
\]
So, the solution to the system of equations is:
\[
(x, y) = (-20, 2)
\]
To verify our solution, we can substitute \(x = -20\) and \(y = 2\) back into the original equations:
1. For the first equation:
\[
-5(2) - 10 = -20 \quad \text{(True)}
\]
2. For the second equation:
\[
-20 + 9(2) = -20 + 18 = -2 \quad \text{(True)}
\]
Both equations are satisfied, confirming that the solution is correct.
Final answer:
\[
(x, y) = (-20, 2)
\]
Answered by
GPT-4o mini
To solve the system of equations by substitution, we start with the given equations:
1. \(-4x + 6y = -10\)
2. \(x = 2y + 4\)
Since the second equation already expresses \(x\) in terms of \(y\), we can substitute this expression for \(x\) into the first equation.
**Substituting \(x\) in the first equation:**
\[
-4(2y + 4) + 6y = -10
\]
**Expanding the equation:**
\[
-8y - 16 + 6y = -10
\]
**Combining like terms:**
\[
-8y + 6y - 16 = -10
\]
\[
-2y - 16 = -10
\]
**Adding 16 to both sides:**
\[
-2y = -10 + 16
\]
\[
-2y = 6
\]
**Dividing both sides by -2:**
\[
y = -3
\]
Now that we have \(y\), we will substitute it back into the equation for \(x\):
\[
x = 2(-3) + 4
\]
\[
x = -6 + 4
\]
\[
x = -2
\]
So, the solution to the system of equations is:
\[
(x, y) = (-2, -3)
\]
**Verification:**
To verify our solution, we can substitute \(x = -2\) and \(y = -3\) back into the original equations:
1. For the first equation:
\[
-4(-2) + 6(-3) = 8 - 18 = -10 \quad \text{(True)}
\]
2. For the second equation:
\[
-2 = 2(-3) + 4 = -6 + 4 = -2 \quad \text{(True)}
\]
Both equations are satisfied, confirming that the solution is correct.
**Final answer:**
\[
(x, y) = (-2, -3)
\]
1. \(-4x + 6y = -10\)
2. \(x = 2y + 4\)
Since the second equation already expresses \(x\) in terms of \(y\), we can substitute this expression for \(x\) into the first equation.
**Substituting \(x\) in the first equation:**
\[
-4(2y + 4) + 6y = -10
\]
**Expanding the equation:**
\[
-8y - 16 + 6y = -10
\]
**Combining like terms:**
\[
-8y + 6y - 16 = -10
\]
\[
-2y - 16 = -10
\]
**Adding 16 to both sides:**
\[
-2y = -10 + 16
\]
\[
-2y = 6
\]
**Dividing both sides by -2:**
\[
y = -3
\]
Now that we have \(y\), we will substitute it back into the equation for \(x\):
\[
x = 2(-3) + 4
\]
\[
x = -6 + 4
\]
\[
x = -2
\]
So, the solution to the system of equations is:
\[
(x, y) = (-2, -3)
\]
**Verification:**
To verify our solution, we can substitute \(x = -2\) and \(y = -3\) back into the original equations:
1. For the first equation:
\[
-4(-2) + 6(-3) = 8 - 18 = -10 \quad \text{(True)}
\]
2. For the second equation:
\[
-2 = 2(-3) + 4 = -6 + 4 = -2 \quad \text{(True)}
\]
Both equations are satisfied, confirming that the solution is correct.
**Final answer:**
\[
(x, y) = (-2, -3)
\]
Answered by
GPT-4o mini
To solve the system of equations:
1. \( y = x^2 - x - 65 \)
2. \( y = -x - 1 \)
we can set the two equations equal to each other since they both equal \(y\):
\[
x^2 - x - 65 = -x - 1
\]
Next, we'll add \(x + 1\) to both sides to eliminate the right-hand side:
\[
x^2 - x + x - 65 + 1 = 0
\]
This simplifies to:
\[
x^2 - 64 = 0
\]
Now, we can factor or solve this quadratic equation. This can be factored as follows:
\[
(x - 8)(x + 8) = 0
\]
Setting each factor equal to zero gives us two solutions for \(x\):
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]
Now that we have the \(x\) values, we can find the corresponding \(y\) values by substituting each \(x\) back into either original equation. We'll use the second equation for simplicity:
For \(x = 8\):
\[
y = -8 - 1 = -9
\]
For \(x = -8\):
\[
y = -(-8) - 1 = 8 - 1 = 7
\]
**Thus, the two solutions are:**
1. \( (8, -9) \)
2. \( (-8, 7) \)
So the final answer includes both solutions:
\[
(8, -9) \quad \text{and} \quad (-8, 7)
\]
1. \( y = x^2 - x - 65 \)
2. \( y = -x - 1 \)
we can set the two equations equal to each other since they both equal \(y\):
\[
x^2 - x - 65 = -x - 1
\]
Next, we'll add \(x + 1\) to both sides to eliminate the right-hand side:
\[
x^2 - x + x - 65 + 1 = 0
\]
This simplifies to:
\[
x^2 - 64 = 0
\]
Now, we can factor or solve this quadratic equation. This can be factored as follows:
\[
(x - 8)(x + 8) = 0
\]
Setting each factor equal to zero gives us two solutions for \(x\):
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]
Now that we have the \(x\) values, we can find the corresponding \(y\) values by substituting each \(x\) back into either original equation. We'll use the second equation for simplicity:
For \(x = 8\):
\[
y = -8 - 1 = -9
\]
For \(x = -8\):
\[
y = -(-8) - 1 = 8 - 1 = 7
\]
**Thus, the two solutions are:**
1. \( (8, -9) \)
2. \( (-8, 7) \)
So the final answer includes both solutions:
\[
(8, -9) \quad \text{and} \quad (-8, 7)
\]
Answered by
GPT-4o mini
To solve the system of equations:
1. \( y = x^2 - 7x - 7 \)
2. \( y = -3x - 2 \)
we can set the two expressions for \(y\) equal to each other since both equations are equal to \(y\):
\[
x^2 - 7x - 7 = -3x - 2
\]
Next, let's move all terms to one side of the equation:
\[
x^2 - 7x + 3x - 7 + 2 = 0
\]
Combine like terms:
\[
x^2 - 4x - 5 = 0
\]
Now we can factor this quadratic equation:
\[
(x - 5)(x + 1) = 0
\]
Setting each factor equal to zero gives us the solutions for \(x\):
1. \( x - 5 = 0 \) → \( x = 5 \)
2. \( x + 1 = 0 \) → \( x = -1 \)
Now, we can find the corresponding \(y\) values by substituting each \(x\) value back into either original equation. We'll use the second equation for simplicity:
For \(x = 5\):
\[
y = -3(5) - 2 = -15 - 2 = -17
\]
For \(x = -1\):
\[
y = -3(-1) - 2 = 3 - 2 = 1
\]
**Thus, the two solutions are:**
1. \( (5, -17) \)
2. \( (-1, 1) \)
So the final answer with both solutions is:
\[
(5, -17) \quad \text{and} \quad (-1, 1)
\]
1. \( y = x^2 - 7x - 7 \)
2. \( y = -3x - 2 \)
we can set the two expressions for \(y\) equal to each other since both equations are equal to \(y\):
\[
x^2 - 7x - 7 = -3x - 2
\]
Next, let's move all terms to one side of the equation:
\[
x^2 - 7x + 3x - 7 + 2 = 0
\]
Combine like terms:
\[
x^2 - 4x - 5 = 0
\]
Now we can factor this quadratic equation:
\[
(x - 5)(x + 1) = 0
\]
Setting each factor equal to zero gives us the solutions for \(x\):
1. \( x - 5 = 0 \) → \( x = 5 \)
2. \( x + 1 = 0 \) → \( x = -1 \)
Now, we can find the corresponding \(y\) values by substituting each \(x\) value back into either original equation. We'll use the second equation for simplicity:
For \(x = 5\):
\[
y = -3(5) - 2 = -15 - 2 = -17
\]
For \(x = -1\):
\[
y = -3(-1) - 2 = 3 - 2 = 1
\]
**Thus, the two solutions are:**
1. \( (5, -17) \)
2. \( (-1, 1) \)
So the final answer with both solutions is:
\[
(5, -17) \quad \text{and} \quad (-1, 1)
\]