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Sagot :
To determine which statements about the given system of linear equations are true, let's analyze the system step by step.
### Step 1: Rewrite the Second Equation
The given equations are:
1. [tex]\( y = \frac{1}{3}x - 4 \)[/tex]
2. [tex]\( 3y - x = -7 \)[/tex]
First, we need to put the second equation into slope-intercept form (i.e., [tex]\( y = mx + b \)[/tex]) to easily compare it with the first equation.
Starting with [tex]\( 3y - x = -7 \)[/tex]:
[tex]\[ 3y = x - 7 \][/tex]
Divide both sides by 3:
[tex]\[ y = \frac{1}{3}x - \frac{7}{3} \][/tex]
Now, the second equation in slope-intercept form is:
[tex]\[ y = \frac{1}{3}x - \frac{7}{3} \][/tex]
### Step 2: Identify Slopes and Intercepts
Next, we identify the slopes and y-intercepts of the two lines:
For the first equation, [tex]\( y = \frac{1}{3}x - 4 \)[/tex]:
- Slope ([tex]\( m \)[/tex]) = [tex]\(\frac{1}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = [tex]\(-4\)[/tex]
For the second equation, [tex]\( y = \frac{1}{3}x - \frac{7}{3} \)[/tex]:
- Slope ([tex]\( m \)[/tex]) = [tex]\(\frac{1}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = [tex]\(-\frac{7}{3}\)[/tex]
### Step 3: Compare the Slopes and Intercepts
Now, compare the slopes and y-intercepts of the two lines.
1. Comparing Slopes: Both lines have the slope [tex]\(\frac{1}{3}\)[/tex]. Therefore, the lines are parallel.
2. Comparing Y-Intercepts: The y-intercepts are different: one is [tex]\(-4\)[/tex], and the other is [tex]\(-\frac{7}{3}\)[/tex]. Different y-intercepts imply that the lines are not the same line.
### Conclusion: True Statements
Based on the analysis, the true statements about the system are:
1. The system consists of parallel lines. This is true because both lines have the same slope but different y-intercepts.
2. Both lines have the same slope. This is true because the slope for both equations is [tex]\( \frac{1}{3} \)[/tex].
The other statements are not true because:
- The system does not have one solution since parallel lines never intersect (they have no common solutions).
- Both lines do not have the same y-intercept as their y-intercepts are [tex]\(-4\)[/tex] and [tex]\(-\frac{7}{3}\)[/tex].
- The equations do not represent the same line since their y-intercepts differ.
Thus, the two correct statements are:
1. The system consists of parallel lines.
2. Both lines have the same slope.
### Step 1: Rewrite the Second Equation
The given equations are:
1. [tex]\( y = \frac{1}{3}x - 4 \)[/tex]
2. [tex]\( 3y - x = -7 \)[/tex]
First, we need to put the second equation into slope-intercept form (i.e., [tex]\( y = mx + b \)[/tex]) to easily compare it with the first equation.
Starting with [tex]\( 3y - x = -7 \)[/tex]:
[tex]\[ 3y = x - 7 \][/tex]
Divide both sides by 3:
[tex]\[ y = \frac{1}{3}x - \frac{7}{3} \][/tex]
Now, the second equation in slope-intercept form is:
[tex]\[ y = \frac{1}{3}x - \frac{7}{3} \][/tex]
### Step 2: Identify Slopes and Intercepts
Next, we identify the slopes and y-intercepts of the two lines:
For the first equation, [tex]\( y = \frac{1}{3}x - 4 \)[/tex]:
- Slope ([tex]\( m \)[/tex]) = [tex]\(\frac{1}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = [tex]\(-4\)[/tex]
For the second equation, [tex]\( y = \frac{1}{3}x - \frac{7}{3} \)[/tex]:
- Slope ([tex]\( m \)[/tex]) = [tex]\(\frac{1}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = [tex]\(-\frac{7}{3}\)[/tex]
### Step 3: Compare the Slopes and Intercepts
Now, compare the slopes and y-intercepts of the two lines.
1. Comparing Slopes: Both lines have the slope [tex]\(\frac{1}{3}\)[/tex]. Therefore, the lines are parallel.
2. Comparing Y-Intercepts: The y-intercepts are different: one is [tex]\(-4\)[/tex], and the other is [tex]\(-\frac{7}{3}\)[/tex]. Different y-intercepts imply that the lines are not the same line.
### Conclusion: True Statements
Based on the analysis, the true statements about the system are:
1. The system consists of parallel lines. This is true because both lines have the same slope but different y-intercepts.
2. Both lines have the same slope. This is true because the slope for both equations is [tex]\( \frac{1}{3} \)[/tex].
The other statements are not true because:
- The system does not have one solution since parallel lines never intersect (they have no common solutions).
- Both lines do not have the same y-intercept as their y-intercepts are [tex]\(-4\)[/tex] and [tex]\(-\frac{7}{3}\)[/tex].
- The equations do not represent the same line since their y-intercepts differ.
Thus, the two correct statements are:
1. The system consists of parallel lines.
2. Both lines have the same slope.
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