Question
System A 5x+2y=9 and 3x+3y=9
System B 3x+3y=9 and ?
-8x-5y=6
8x+5y=6
-10x -4y=-6
System B 3x+3y=9 and ?
-8x-5y=6
8x+5y=6
-10x -4y=-6
Answers
GPT-4o mini
To solve the systems you've provided, we first need to analyze System A and see if both equations can be solved.
### System A:
1. \( 5x + 2y = 9 \) (Equation 1)
2. \( 3x + 3y = 9 \) (Equation 2)
Let's solve System A using substitution or elimination.
First, we can simplify Equation 2:
\[
3x + 3y = 9 \implies x + y = 3 \quad (Dividing by 3)
\]
This gives us:
\[
y = 3 - x \quad (Equation 3)
\]
Substitute Equation 3 into Equation 1:
\[
5x + 2(3 - x) = 9
\]
\[
5x + 6 - 2x = 9
\]
\[
3x + 6 = 9
\]
\[
3x = 3 \implies x = 1
\]
Now substitute \( x = 1 \) back into Equation 3:
\[
y = 3 - 1 = 2
\]
Thus, the solution for System A is:
\[
(x, y) = (1, 2)
\]
### System B:
Now let's look at System B, which consists of:
\[
3x + 3y = 9 \quad (Same as Equation 2 from System A)
\]
and one of the equations you provided.
The options given are:
- \(-8x - 5y = 6\)
- \(8x + 5y = 6\)
- \(-10x - 4y = -6\)
Let’s see which of these options is consistent with the first equation, \(3x + 3y = 9\).
**Using the second equation option \(8x + 5y = 6\)**:
Now let's solve \( 3x + 3y = 9 \) together with \( 8x + 5y = 6 \):
- We can express \( y \) from the first equation:
\[
y = 3 - x \quad (from \, Equation \, 3)
\]
Substituting this into \( 8x + 5y = 6 \):
\[
8x + 5(3 - x) = 6
\]
\[
8x + 15 - 5x = 6
\]
\[
3x + 15 = 6
\]
\[
3x = -9 \implies x = -3
\]
Substituting \( x = -3 \) back into the first equation:
\[
y = 3 - (-3) = 6
\]
So, the solution is \( (-3, 6) \).
**Checking the other options**:
For \(-8x - 5y = 6\):
Substituting \( y = 3 - x \):
\[
-8x - 5(3 - x) = 6
\]
\[
-8x - 15 + 5x = 6
\]
\[
-3x - 15 = 6
\]
\[
-3x = 21 \implies x = -7
\]
Then,
\[
y = 3 - (-7) = 10
\]
Solution: \((-7, 10)\).
For \(-10x - 4y = -6\):
Substituting \( y = 3 - x \):
\[
-10x - 4(3 - x) = -6
\]
\[
-10x - 12 + 4x = -6
\]
\[
-6x - 12 = -6
\]
\[
-6x = 6 \implies x = -1
\]
Then,
\[
y = 3 - (-1) = 4
\]
Solution: \((-1, 4)\).
### Conclusion:
- The option \( 8x + 5y = 6 \) is one consistent equation in System B and leads to a solution of \( (-3, 6) \).
- The other equations also yield valid (though different) solutions. Thus, both options can serve as an equation in System B alongside \( 3x + 3y = 9 \).
### System A:
1. \( 5x + 2y = 9 \) (Equation 1)
2. \( 3x + 3y = 9 \) (Equation 2)
Let's solve System A using substitution or elimination.
First, we can simplify Equation 2:
\[
3x + 3y = 9 \implies x + y = 3 \quad (Dividing by 3)
\]
This gives us:
\[
y = 3 - x \quad (Equation 3)
\]
Substitute Equation 3 into Equation 1:
\[
5x + 2(3 - x) = 9
\]
\[
5x + 6 - 2x = 9
\]
\[
3x + 6 = 9
\]
\[
3x = 3 \implies x = 1
\]
Now substitute \( x = 1 \) back into Equation 3:
\[
y = 3 - 1 = 2
\]
Thus, the solution for System A is:
\[
(x, y) = (1, 2)
\]
### System B:
Now let's look at System B, which consists of:
\[
3x + 3y = 9 \quad (Same as Equation 2 from System A)
\]
and one of the equations you provided.
The options given are:
- \(-8x - 5y = 6\)
- \(8x + 5y = 6\)
- \(-10x - 4y = -6\)
Let’s see which of these options is consistent with the first equation, \(3x + 3y = 9\).
**Using the second equation option \(8x + 5y = 6\)**:
Now let's solve \( 3x + 3y = 9 \) together with \( 8x + 5y = 6 \):
- We can express \( y \) from the first equation:
\[
y = 3 - x \quad (from \, Equation \, 3)
\]
Substituting this into \( 8x + 5y = 6 \):
\[
8x + 5(3 - x) = 6
\]
\[
8x + 15 - 5x = 6
\]
\[
3x + 15 = 6
\]
\[
3x = -9 \implies x = -3
\]
Substituting \( x = -3 \) back into the first equation:
\[
y = 3 - (-3) = 6
\]
So, the solution is \( (-3, 6) \).
**Checking the other options**:
For \(-8x - 5y = 6\):
Substituting \( y = 3 - x \):
\[
-8x - 5(3 - x) = 6
\]
\[
-8x - 15 + 5x = 6
\]
\[
-3x - 15 = 6
\]
\[
-3x = 21 \implies x = -7
\]
Then,
\[
y = 3 - (-7) = 10
\]
Solution: \((-7, 10)\).
For \(-10x - 4y = -6\):
Substituting \( y = 3 - x \):
\[
-10x - 4(3 - x) = -6
\]
\[
-10x - 12 + 4x = -6
\]
\[
-6x - 12 = -6
\]
\[
-6x = 6 \implies x = -1
\]
Then,
\[
y = 3 - (-1) = 4
\]
Solution: \((-1, 4)\).
### Conclusion:
- The option \( 8x + 5y = 6 \) is one consistent equation in System B and leads to a solution of \( (-3, 6) \).
- The other equations also yield valid (though different) solutions. Thus, both options can serve as an equation in System B alongside \( 3x + 3y = 9 \).