Tell whether the lines for each pair of equations are parallel perpendicular of neither

y=-3x+7
-2x+6y=3
Tell whether the lines for each pair of equations are parallel perpendicular of neither
y=(-1/5)x=6
-2x+10y=5

To determine whether the lines for each pair of equations are parallel, perpendicular, or neither, we can compare the slopes of the lines.

1) y = -3x + 7
-2x + 6y = 3

To find the slope of the first equation, we can see that the coefficient of x is -3. The slope-intercept form of the equation (y = mx + b) tells us that the slope (m) is -3.

To rewrite the second equation in slope-intercept form, we isolate y:
-2x + 6y = 3
6y = 2x + 3
y = (2/6)x + 1/2
y = (1/3)x + 1/2

Comparing the slopes, we see that the slope of the first equation is -3, while the slope of the second equation is 1/3. Since the slopes are not equal and not negative reciprocals of each other, the lines are neither parallel nor perpendicular.

2) y = (-1/5)x + 6
-2x + 10y = 5

Again, the first equation is already in slope-intercept form, so we can see that the slope (m) is -1/5.

To rewrite the second equation in slope-intercept form, we isolate y:
-2x + 10y = 5
10y = 2x + 5
y = (2/10)x + 1/2
y = (1/5)x + 1/2

Comparing the slopes, we see that the slope of the first equation is -1/5, while the slope of the second equation is 1/5. Since the slopes are negative reciprocals of each other, the lines are perpendicular.

Summary:
1) The lines represented by y = -3x + 7 and -2x + 6y = 3 are neither parallel nor perpendicular.
2) The lines represented by y = (-1/5)x + 6 and -2x + 10y = 5 are perpendicular.

To determine whether two lines are parallel, perpendicular, or neither, we need to compare their slopes.

For the first pair of equations:

1. The equation y = -3x + 7 is in slope-intercept form (y = mx + b), where m is the slope. The slope of this line is -3.

2. The equation -2x + 6y = 3 can be rewritten in slope-intercept form by isolating y: 6y = 2x + 3 -> y = (2/6)x + 1/2. The slope of this line is 2/6 or 1/3.

Since the slopes of the two lines are not equal (-3 ≠ 1/3), these lines are not parallel.

To determine if they are perpendicular, we can check whether the product of their slopes will equal -1.

-3 * (1/3) = -1, which means the lines are perpendicular.

For the second pair of equations:

1. The equation y = (-1/5)x + 6 is again in slope-intercept form. The slope of this line is -1/5.

2. The equation -2x + 10y = 5 can be rewritten in slope-intercept form by isolating y: 10y = 2x + 5 -> y = (2/10)x + 1/2. The slope of this line is 2/10 or 1/5.

Since the slopes of the two lines are equal (-1/5 = 1/5), these lines are parallel.

The lines in the second pair of equations are parallel, not perpendicular or neither.

parallel if same slope

perp. if slopes multiply to -1.