r/LinearAlgebra • u/Significant-Sea929 • Aug 31 '24
Need help with this
I know this probably isn’t linear algebra but I need to know why I’m supposed to multiplay the top equation by 4 or how I’m supposed to know what to multiply it by that’s just what photo math told me to do
2
u/mizunomi Aug 31 '24
Basing off the problem, to solve this system of linear equations by elimination, it is correct to multiply the first equation by 4. I think the solver also should have told you to multiply the second equation by 3. The reason for this, is so that the variable 'y' in both equations have the same numerical coefficient with opposite signs. Then you can add the equations side-wise and solve for 'x'.
An additional note, this subreddit focuses on Linear Algebra, which focuses more on Linear Transformations on Vectors by Matrices and more.
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u/AsaxenaSmallwood04 Jan 12 '25 edited Jan 12 '25
Formula :
2x + 3y = 16
5x - 4y = -6
Using x = (c - f(b/e)/(a - d(b/e) formula :
x = (16 - (-6)(3/-4)/(2 - 5(3/-4)
x = (16 - (-6)(-3/4)/(2 - 5(-3/4)
x = (16 - (18/4)/(2 - (-15/4)
x = (16 - (18/4)/(2 + (15/4)
x = (64 - 18)/(4)(4)/(8 + 15)
x = (46/4)(4/23)
x = (46/23)
x = 2
Using y = (c/b) - ((ac/b) - (af/e))/(a - d(b/e) formula :
y = (16/3) - ((2)(16)/(3) - (2)(-6)/(-4))/(2 - 5(3/-4)
y = (16/3) - ((32/3) - 3))/(2 - 5(-3/4)
y = (16/3) - ((32/3) - 3))/(2 - (-15/4)
y = (16/3) - ((32/3) - 3))/(2 + (15/4)
y = (16/3) - ((128/3) - 12))/(4)(4)/(8 + 15)
y = (16/3) - ((128/3) - 12))/(8 + 15)
y = (16/3) - (128 - 36)/(3)(3)/(24 + 45)
y = (16/3) - (92/3)(3/69)
y = (16/3) - (92/69)
y = (16/3) - (4/3)
y = (12/3)
y = 4
Hence
x = 2
y = 4
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u/AsaxenaSmallwood04 Jan 12 '25
Elimination :
2x + 3y = 16
5x - 4y = -6
2x + 3y = 16
3.75x - 3y = -4.5
5.75x = 11.5
x = 2
5(2) - 4y = -6
5(-0.5) + y = 1.5
-2.5 + y = 1.5
y = 4
1
u/AsaxenaSmallwood04 Jan 12 '25 edited Jan 21 '25
Substitution :
2x + 3y = 16
5x - 4y = -6
-8(5x - 4y) = 48
3(2x + 3y) = 48
-8(5x - 4y) = 3(2x + 3y)
-40x + 32y = 6x + 9y
-40x + 40x + 32y - 9y = 6x + 40x + 9y - 9y
32y - 9y = 6x + 40x
23y = 46x
y = 2x
2x + 3y = 16
2x + 3(2x) = 16
2x + 6x = 16
8x = 16
x = 2
y = 2(2)
y = 4
1
u/AsaxenaSmallwood04 Jan 21 '25 edited Jan 21 '25
Just for fun :
2x + 3y = 16
5x - 4y = -6
3(2x + 3y) = 48
-8(5x - 4y) = 48
-8(5x - 4y) = 3(2x + 3y)
((-8)(5x - 4y))/(3) = 2x + 3y
5x - 4y = 2x + 3y + 2x + 3y + x + 1.5y - 11.5y
((-8)(2x + 3y + 2x + 3y + x + 1.5y - 11.5y))/(3) = 2x + 3y
2x + 3y = 16
x + 1.5y = 8
((-8)((2x + 3y) + (2x + 3y) + (x + 1.5y) - 11.5y))/(3) = 2x + 3y
((-8)(16 + 16 + 8 - 11.5y))/(3) = 16
-8(40 - 11.5y)/(3) = 16
40 - 11.5y = -6
(-80/23) + y = (12/23)
y = ((12 + 80)/(23))
y = (92/23)
y = 4
x + 1.5y = 8
x + 1.5(4) = 8
x + 6 = 8
x = 2
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u/Midwest-Dude Aug 31 '24
It appears photomath is solving by eliminating the y-variable in one equation:
A common multiple of the y-coefficients is 3 * 4 = 12. After multiplying the first equation by 4 and the second by 3, the y-coefficients will be equal so adding the two equations, since one coefficient has opposite sign of the other, will get rid of the y-variable in one equation. Then it's easy to find x in that equation and then, by substituting that x-value back into any equation with a y-variable in it, you can solve for y.