Answer :
To solve this problem, let's examine the slopes of each line. Lines are parallel if and only if they have the same slope.
1. Line 1: [tex]\(y = -3x + 10\)[/tex]
- This equation is already in slope-intercept form [tex]\(y = mx + b\)[/tex], where the slope [tex]\(m\)[/tex] is -3.
2. Line 2: [tex]\(y = \frac{2}{5}x - 8\)[/tex]
- This equation is also in slope-intercept form with the slope [tex]\(m\)[/tex] being [tex]\(\frac{2}{5}\)[/tex].
3. Line 3: [tex]\(-5x + 2y = -16\)[/tex]
- We need to convert this to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 2y = 5x - 16 \][/tex]
[tex]\[ y = \frac{5}{2}x - 8 \][/tex]
- Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
4. Line 4: [tex]\(2x - 5y = 30\)[/tex]
- Convert this to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ -5y = -2x + 30 \][/tex]
[tex]\[ y = \frac{2}{5}x - 6 \][/tex]
- Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{5}\)[/tex].
5. Line 5: [tex]\(6x + 2y = -10\)[/tex]
- Convert this to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 2y = -6x - 10 \][/tex]
[tex]\[ y = -3x - 5 \][/tex]
- Here, the slope [tex]\(m\)[/tex] is -3.
Now let's list our slopes:
- Line 1: [tex]\(-3\)[/tex]
- Line 2: [tex]\(\frac{2}{5}\)[/tex]
- Line 3: [tex]\(\frac{5}{2}\)[/tex]
- Line 4: [tex]\(\frac{2}{5}\)[/tex]
- Line 5: [tex]\(-3\)[/tex]
From this, we observe:
- Line 1 and Line 5 both have the slope [tex]\(-3\)[/tex], making them parallel.
- Line 2 and Line 4 both have the slope [tex]\(\frac{2}{5}\)[/tex], making them parallel.
Therefore, the lines that are parallel to each other are:
- Line 1 and Line 5
- Line 2 and Line 4
1. Line 1: [tex]\(y = -3x + 10\)[/tex]
- This equation is already in slope-intercept form [tex]\(y = mx + b\)[/tex], where the slope [tex]\(m\)[/tex] is -3.
2. Line 2: [tex]\(y = \frac{2}{5}x - 8\)[/tex]
- This equation is also in slope-intercept form with the slope [tex]\(m\)[/tex] being [tex]\(\frac{2}{5}\)[/tex].
3. Line 3: [tex]\(-5x + 2y = -16\)[/tex]
- We need to convert this to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 2y = 5x - 16 \][/tex]
[tex]\[ y = \frac{5}{2}x - 8 \][/tex]
- Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
4. Line 4: [tex]\(2x - 5y = 30\)[/tex]
- Convert this to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ -5y = -2x + 30 \][/tex]
[tex]\[ y = \frac{2}{5}x - 6 \][/tex]
- Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{5}\)[/tex].
5. Line 5: [tex]\(6x + 2y = -10\)[/tex]
- Convert this to slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 2y = -6x - 10 \][/tex]
[tex]\[ y = -3x - 5 \][/tex]
- Here, the slope [tex]\(m\)[/tex] is -3.
Now let's list our slopes:
- Line 1: [tex]\(-3\)[/tex]
- Line 2: [tex]\(\frac{2}{5}\)[/tex]
- Line 3: [tex]\(\frac{5}{2}\)[/tex]
- Line 4: [tex]\(\frac{2}{5}\)[/tex]
- Line 5: [tex]\(-3\)[/tex]
From this, we observe:
- Line 1 and Line 5 both have the slope [tex]\(-3\)[/tex], making them parallel.
- Line 2 and Line 4 both have the slope [tex]\(\frac{2}{5}\)[/tex], making them parallel.
Therefore, the lines that are parallel to each other are:
- Line 1 and Line 5
- Line 2 and Line 4
