Question
Look at the system of equations.
y=2x+3 and 2y= 4x + 10
Which statement about this system of equations is true and why?
y=2x+3 and 2y= 4x + 10
Which statement about this system of equations is true and why?
Answers
GPT-4o mini
To analyze the system of equations, we first need to rewrite both equations in a standard form and check their properties.
1. The first equation is already in slope-intercept form:
\[
y = 2x + 3
\]
2. For the second equation, \(2y = 4x + 10\), we can divide the entire equation by 2 to simplify it:
\[
y = 2x + 5
\]
Now we have the two equations:
1. \(y = 2x + 3\)
2. \(y = 2x + 5\)
Next, let's analyze the slopes and y-intercepts:
1. **Slopes**:
- The slope of the first equation (\(y = 2x + 3\)) is 2.
- The slope of the second equation (\(y = 2x + 5\)) is also 2.
2. **Y-Intercepts**:
- The y-intercept of the first equation is 3.
- The y-intercept of the second equation is 5.
Because both equations have the same slope (2) but different y-intercepts, this means that the lines are parallel to each other and will never intersect.
### Conclusion:
The statement that is true about this system of equations is that it has no solution because the two lines represented by the equations are parallel and do not intersect. Therefore, the system of equations is inconsistent.
1. The first equation is already in slope-intercept form:
\[
y = 2x + 3
\]
2. For the second equation, \(2y = 4x + 10\), we can divide the entire equation by 2 to simplify it:
\[
y = 2x + 5
\]
Now we have the two equations:
1. \(y = 2x + 3\)
2. \(y = 2x + 5\)
Next, let's analyze the slopes and y-intercepts:
1. **Slopes**:
- The slope of the first equation (\(y = 2x + 3\)) is 2.
- The slope of the second equation (\(y = 2x + 5\)) is also 2.
2. **Y-Intercepts**:
- The y-intercept of the first equation is 3.
- The y-intercept of the second equation is 5.
Because both equations have the same slope (2) but different y-intercepts, this means that the lines are parallel to each other and will never intersect.
### Conclusion:
The statement that is true about this system of equations is that it has no solution because the two lines represented by the equations are parallel and do not intersect. Therefore, the system of equations is inconsistent.