Question
4x+22=8x+10 3x+18=5x+8 consider the two equations that you underlined, which forms system with one solution solve the equations.
Answers
GPT-4o mini
To solve the system of equations you provided, we'll tackle them one at a time.
The first equation is:
1. **Equation 1:** \(4x + 22 = 8x + 10\)
Let's rearrange it to isolate \(x\):
1. Subtract \(4x\) from both sides:
\[
22 = 8x - 4x + 10
\]
This simplifies to:
\[
22 = 4x + 10
\]
2. Next, subtract \(10\) from both sides:
\[
22 - 10 = 4x
\]
This gives:
\[
12 = 4x
\]
3. Finally, divide both sides by \(4\):
\[
x = 3
\]
Now, let's solve the second equation:
2. **Equation 2:** \(3x + 18 = 5x + 8\)
Again, we'll rearrange it:
1. Subtract \(3x\) from both sides:
\[
18 = 5x - 3x + 8
\]
This simplifies to:
\[
18 = 2x + 8
\]
2. Next, subtract \(8\) from both sides:
\[
18 - 8 = 2x
\]
This gives:
\[
10 = 2x
\]
3. Finally, divide both sides by \(2\):
\[
x = 5
\]
### Summary of Solutions
- From the first equation, we found \(x = 3\).
- From the second equation, we found \(x = 5\).
Since \(x\) has different values from the two equations, the system does not have a solution where both equations are satisfied simultaneously. Therefore, this system of equations has no intersection point, meaning it has no solutions where both conditions hold true.
If you are asking about combining both findings, then **the system does not have one solution**, but rather two different independent solutions from the respective equations.
The first equation is:
1. **Equation 1:** \(4x + 22 = 8x + 10\)
Let's rearrange it to isolate \(x\):
1. Subtract \(4x\) from both sides:
\[
22 = 8x - 4x + 10
\]
This simplifies to:
\[
22 = 4x + 10
\]
2. Next, subtract \(10\) from both sides:
\[
22 - 10 = 4x
\]
This gives:
\[
12 = 4x
\]
3. Finally, divide both sides by \(4\):
\[
x = 3
\]
Now, let's solve the second equation:
2. **Equation 2:** \(3x + 18 = 5x + 8\)
Again, we'll rearrange it:
1. Subtract \(3x\) from both sides:
\[
18 = 5x - 3x + 8
\]
This simplifies to:
\[
18 = 2x + 8
\]
2. Next, subtract \(8\) from both sides:
\[
18 - 8 = 2x
\]
This gives:
\[
10 = 2x
\]
3. Finally, divide both sides by \(2\):
\[
x = 5
\]
### Summary of Solutions
- From the first equation, we found \(x = 3\).
- From the second equation, we found \(x = 5\).
Since \(x\) has different values from the two equations, the system does not have a solution where both equations are satisfied simultaneously. Therefore, this system of equations has no intersection point, meaning it has no solutions where both conditions hold true.
If you are asking about combining both findings, then **the system does not have one solution**, but rather two different independent solutions from the respective equations.