Are the lines y = 3x + 1 and 3y - 9x = 7 parallel? Algebra LINEAR EQUATIONS and SLOPE

[@panlomito]

y = 3x + 1 so the slope is +3/1 or one step to the right and 3 steps up. 3y = 9x + 7 or y = 3x + 7/3 so again the slope is +3/1 and so these lines are parallel ! The formula for a line is y = ax + b or better with a = dy / dx y = (dy/dx) . x + b For example the slope of the line (b is not important for the slope): y = x is y = (+1/1) . x so 1 step right and 1 step up y = -x is y = (-1/1) . x so 1 step right and 1 step down y = 2x is y = (+2/1) . x so 1 step right and 2 steps up y = -¾x is y = (-3/4) . x so 4 steps right and 3 steps down b is the crossing with the y-axis while the formula of the y-axis is x = 0 so y = a . 0 + b = b and the crossing of the line with the y-axis at ( 0 , b ). So the first line y1 = 3x + 1 has a crossing with the y-axis at b = 1 or at ( 0 , 1 ) And the second line y2 = 3x + 7/3 has a crossing with the y-axis at b = 7/3 or at ( 0 , 7/3 ) 7/3 ~ 2.333 so the second line y = 3x + 7/3 is about 1.333 above (vertically) the first line. Since the 2 lines are parallel there will be no crossing of the 2 lines that could be found with y1 = y2 : 3x + 1 = 3x + 7/3 so 3x - 3x = 7/3 - 1 or 0 = 4/3 so no common x1 = x2 found (no crossing). For example: y1 = 2x - 6 and y2 = - 4x + 2 so first line is crossing the y-axis at y1 = -6 ( 0 , -6) and the second line is crossing the y-axis at y2 = 2 ( 0 , 2 ) Slope first line a1 = 2 = (+2/1) Slope second line a2 = -4 = (-4/1) Crossing of the 2 lines at y1 = y2 or 2x - 6 = -4x + 2 so 2x + 4x = 6 + 2 or 6x = 8 so x1 = x2 = 4/3 and the y1 = 2 . (4/3) - 6 = 8/3 - 18/3 = - 10/3 or y2 = -4 . (4/3) + 2 = - 16/3 + 6/3 = - 10/3 so the crossing point of x1 = x2 or y1 = y2 at ( 4/3 , - 10/3 )

[@roundtwo3321]

Yes, they are parallel to each other because they both have a slope of 3, m = 3 in the equation y = mx + b.

[@PrithwirajSen-nj6qq]

It is not a sufficient reason.,I think. The second reason is as under The lines have different y intercepts. When gradients are same and y intercepts are same, the the two Iines will be a single line. Ex 3x + 4y =10 9x + 12y=30 These lines have same gradients and same y intercepts. Hence they may not be parallel. They are not different lines. They are a single line. Algebraic approach to this problem will not give any solution to the equations if the lines are parallel. So we may say that when we do not obtain sol, then the lines are parallel. And will give infinite numbers of solutions of the lines are same. So when we obtain sol /sols, then lines are not parallel.

[@Stylux-z1p]

Using System of equations: to proof Elimination Method y = 3x + 1 --> rearrange/flip the equation 3y - 9x = 7 --> rearrange/flip the equation [3x - y = -1] [3] --> multiply to eliminate 9x - 3y = 7 -- --- --- -- --- 9x - 3y = -3 -9x + 3y = 7 + ------------------- 0 = 4 False ✅ No Solution Inconsistent System Means it represents Parallel Lines that never intersect

[@chrisdissanayake6979]

Answer: yes, the two lines are parallel. The two equations that represent the 2 lines, have the same slope of 3. ----------- For two lines to be parallel, they have to have the same slope. In the equation, Y=mX+b, m is the slope. Therefore, in the equation, Y=3X+1, m=3. Slope =3 In the equation, 3Y-9X=7, 3Y=9X+7 Y=3X+7/3 In the above equation, m=3 . Slope =3 Therefore, the lines that are represented by the two equations, both having a slope of 3, are parallel.

[@stevendebettencourt7651]

Interesting application of functions here. As always, we assume flat Euclidean space unless otherwise specified. y = 3x + 1 & 3y - 9x = 7 Two conditions have to be met for these two lines to be parallel. The first is that the two lines have the same slope, as defined as follows: y = mx + b, where m is the slope and b is the y value when x=0. The first equation is already in the above form, so we see it’s slope is 3. The second equation is not in the form of y=mx+b, so let’s solve for y in terms of x: 3y - 9x = 7 3y = 9x + 7 y = (9x + 7) / 3 y = 9x / 3 + 7 / 3 y = 3x + 7/3 So the second equation has the same slope as the first. However, before we declare them parallel lines, there’s a second condition: The two lines must have a different y-intercept. That is, when x=0, the two equations will give a different value. If they give the same value, the lines are not parallel because they’d be the same line. Since we have both equations in what is known as “slope-intercept form,” we just have to compare their b values: y = 3x + 1 & y = 3x + 7/3 1 =/= 7/3 Since the two lines have the same slope and different y-intercepts, they are parallel. Questions?

[@greorbowlfinder7078]

Yes.

[@alancragg1839]

Yes

[@erdemmemisyazici3950]

I remember seeing this in Geometry class a lot. Didn't know it was considered Algebra.

[@tomtke7351]

yes

[@PrithwirajSen-nj6qq]

3x - y +1 =0 --(1) 9x -3y +7=0 > 3x -y + 7/3 =0 ---(2) The two equations differ only in constant values Hence these are parallel lines

[@russelllomando8460]

i remembered Y = MX + B just couldn't remember what the B was. thanks for the fun.

[@raya.pawley3563]

Yes. Y=3x+b

[@davidmurphy563]

{{3, -1}, {-9, 3}} Determinant is zero. It's singular. A line in R2.

[@tomtke7351]

y = mx + b

[@Rev03FFL]

I can't find the 3 dot symbol on my keyboard; is it accessible? Otherwise I'll have to use QED, or maybe, Bob's your uncle.

[@discobear5752]

Bob's not my uncle. Is he yours? I'd never heard of this expression before tonight. Wackypedia says it's a British expression. Are you in or from Britain? The three dots symbol isn't on my keyboard either. I had to use the web to find it, ∴ you should too .

[@davidmurphy563]

Bob's your uncle is a common enough expression this side of the pond. Start using it as if it's in common use and look at people strangely when they don't know it. In a few years I'm sure you can kick start it. 👍

[@josephlaura7387]

Yes

[@SS-gp5kf]

yes

[@mikiokabe6689]

yes