Answer :
To analyze the given system of linear equations, we need to transform and compare both equations:
Given equations:
1. \( 2y = x + 10 \)
2. \( 3y = 3x + 15 \)
First, we convert each equation into slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For the first equation:
[tex]\[ 2y = x + 10 \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{1}{2} x + 5 \][/tex]
Here, the slope (\( m \)) is \(\frac{1}{2}\) and the y-intercept (\( b \)) is 5.
For the second equation:
[tex]\[ 3y = 3x + 15 \][/tex]
Divide both sides by 3:
[tex]\[ y = x + 5 \][/tex]
Here, the slope (\( m \)) is 1 and the y-intercept (\( b \)) is 5.
Now, compare the slopes and y-intercepts of both lines:
1. The system has one solution.
- To determine whether the system has one solution, we look for the intersection of the two lines. Since the slopes are different (\(\frac{1}{2}\) and 1), the lines are not parallel and thus intersect at exactly one point. So, this statement is true.
2. The system graphs parallel lines.
- Parallel lines have the same slope. Here, the slopes are different (\(\frac{1}{2}\) and 1), so the system does not graph parallel lines. Therefore, this statement is false.
3. Both lines have the same slope.
- From the slopes calculated above (\(\frac{1}{2}\) and 1), itâs clear that the lines do not have the same slope. Thus, this statement is false.
4. Both lines have the same y-intercept.
- Both equations in slope-intercept form have the same y-intercept, \( b = 5 \). Therefore, this statement is true.
5. The equations graph the same line.
- For two lines to be identical, they must have the same slope and y-intercept. Although both lines have the same y-intercept (5), their slopes are different (\(\frac{1}{2}\) and 1), so they are not the same line. Thus, this statement is false.
6. The solution is the intersection of the 2 lines.
- Since the lines have different slopes, they intersect at exactly one point. Therefore, solving the system will give us this point of intersection, validating this statement as true.
Hence, the true statements about the system are:
- The system has one solution.
- Both lines have the same y-intercept.
- The solution is the intersection of the 2 lines.
Given equations:
1. \( 2y = x + 10 \)
2. \( 3y = 3x + 15 \)
First, we convert each equation into slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For the first equation:
[tex]\[ 2y = x + 10 \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{1}{2} x + 5 \][/tex]
Here, the slope (\( m \)) is \(\frac{1}{2}\) and the y-intercept (\( b \)) is 5.
For the second equation:
[tex]\[ 3y = 3x + 15 \][/tex]
Divide both sides by 3:
[tex]\[ y = x + 5 \][/tex]
Here, the slope (\( m \)) is 1 and the y-intercept (\( b \)) is 5.
Now, compare the slopes and y-intercepts of both lines:
1. The system has one solution.
- To determine whether the system has one solution, we look for the intersection of the two lines. Since the slopes are different (\(\frac{1}{2}\) and 1), the lines are not parallel and thus intersect at exactly one point. So, this statement is true.
2. The system graphs parallel lines.
- Parallel lines have the same slope. Here, the slopes are different (\(\frac{1}{2}\) and 1), so the system does not graph parallel lines. Therefore, this statement is false.
3. Both lines have the same slope.
- From the slopes calculated above (\(\frac{1}{2}\) and 1), itâs clear that the lines do not have the same slope. Thus, this statement is false.
4. Both lines have the same y-intercept.
- Both equations in slope-intercept form have the same y-intercept, \( b = 5 \). Therefore, this statement is true.
5. The equations graph the same line.
- For two lines to be identical, they must have the same slope and y-intercept. Although both lines have the same y-intercept (5), their slopes are different (\(\frac{1}{2}\) and 1), so they are not the same line. Thus, this statement is false.
6. The solution is the intersection of the 2 lines.
- Since the lines have different slopes, they intersect at exactly one point. Therefore, solving the system will give us this point of intersection, validating this statement as true.
Hence, the true statements about the system are:
- The system has one solution.
- Both lines have the same y-intercept.
- The solution is the intersection of the 2 lines.
