Derive the equation of the line through the origin
1. y = 3x
2. y = 6x
3. x + y = 3
1. To find the equation of the line through the origin for y = 3x, we notice that the line passes through the origin (0, 0) and has a slope of 3.
The general equation of a line passing through the origin with slope m is y = mx.
So, for y = 3x, the equation of the line passing through the origin is y = 3x.
2. For y = 6x, similarly, the equation of the line passing through the origin is y = 6x.
3. To find the equation of the line passing through the origin for x + y = 3, we need to rewrite the equation in the slope-intercept form (y = mx + b).
x + y = 3
y = -x + 3
Since the line passes through the origin, the y-intercept is 0. Therefore, the equation of the line passing through the origin is y = -x.
wrong
Apologies for the mistake in the previous response. Let's correct them:
1. For the equation y = 3x, since the line passes through the origin, the y-intercept is 0. The slope of the line is 3. Therefore, the equation of the line passing through the origin is y = 3x.
2. For the equation y = 6x, the line passes through the origin and has a slope of 6. Therefore, the equation of the line passing through the origin is y = 6x.
3. For the equation x + y = 3, we rearrange the equation to the slope-intercept form:
y = -x + 3
Since the line passes through the origin, the y-intercept is 0. Therefore, the equation of the line passing through the origin is y = -x.
still wrong
Let's correct the derivation:
1. Given the equation y = 3x, the line passes through the origin (0,0) because when x = 0, y is also 0. The slope of the line is 3. Any line that passes through the origin with slope m has the equation y = mx. Therefore, the equation of the line passing through the origin for y = 3x is y = 3x.
2. For the equation y = 6x, it also passes through the origin (0,0) because when x = 0, y is also 0. The slope of the line is 6. Using the same logic as above, the equation of the line passing through the origin for y = 6x is y = 6x.
3. To find the equation of the line passing through the origin for x + y = 3, we notice that when x = 0, y = 3 and when y = 0, x = 3. This line also passes through the origin. The slope-intercept form of the equation is y = -x + 3. Since the line passes through the origin, the equation of the line passing through the origin for x + y = 3 is y = -x.