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bthemesandtricks
Samuel's Formula for solving Simultaneous equations From the general equation ax + by = c ----------------- (1) dx + ey = f -----------------(2) Where a is the coefficient of x and b is the coefficient of y; d is the coefficient of x and e is the coefficient of f. c and f are the constants.
Using the elimination method of solving Simultaneous Equation ax + by = c -------- (1) multiply with d dx + ey = f -------- (2) multiply with a adx + bdy =cd ------------ (3) adx + aey = af ------------- (4) subtract equation 4 from equation 3 adx - adx + bdy - (+aey) = cd -af bdy - aey = cd - af factorise the Left Hand Side y(bd - ae) = cd - af y = cd- af/bd-ae
Using the elimination method of solving Simultaneous Equation ax + by = c ------------- (1) multiply with e dx + ey = f ------------- (2) multiply with b aex + bey = ce --------- (3) bdx + bey = bf --------- (4) Subtract equation 4 from equation 3 aex - bdx + (bey - bey) = ce - bf aex - bdx = ce - bf Factorise the Left Hand Side x(ae - bd) = ce - bf x = ce - bf/ae - bd
How does It work For example, 2x + 3y = 5 x + 4y = 5 Using Samuel's Formula a = 2 b =3 c = 5 d =1 e =4 f = 5
x = ce-bf/ae - bd x = 5 (4) - 3 (5)/2 (4) - 3 (1) x = 20 - 15 / 8 - 3 x = 5/5 x=1 y = cd - af/ bd - ae y = 5 (1) - 2 (5)/ 3 (1) - 2 (4) y= 5 -10/3 - 8 y= (-5)/(-5) y=1 Therefore, x =1 and y =1 You can try other questions too
Thanks everyone please drop your comments
@bthemesandtricks would like to know if you have seen this formula before in another guise and if you will proof it for errors. Please respond. Thanks.
Nah I've not seen it before. I derived the formula while in high school.
@bthemesandtricks With these two simultaneous equations, will you use your formula results to post the coordinates of the common solution, if it exists? System of Equations: ------------------- (2x - 5y = 7) and (3x + 11y = 19)
Perhaps. You could try it out
I could. I had in mind to compare my traditional solution technique answers to those of your formula.
yeah try it out and you'll seee it works
In general, I'd use regular substitution, or if I was feeling fancy an augmented matrix or even Cramer's Rule.
It looks like what you posted is the same as Cramer's Rule.
It's a specialized case of using determinants of matrices to solve equations, but it amounts to regular substitution. For two linear equations in standard form with two unknowns, the general substitution method of solving follows the same steps every time, so it lends itself well to a single solution equation (or set of equations to be more precise) that works every time. I'd say it's analogous to how the quadratic formula is a general solution form of completing the square.