Hey guys, this is my first post and im hoping someone can help me out these kind of problems have been giving me all sorts of trouble, it would be great if somebody could explain step by step.
Solve the system of two equations in 2 variables
1) x+9y=-63
-5x+10y=-70
2) -6x+7y=-21
-3x+3y=-9
3) -7x-21=-7y
-3x-5y=-63
These three problems have been giving trouble if anyone can explain that would be awesome
There is no need to move this to the proper forum, which is algebra. We are not that fussy about people posting to the right forum on their first visit.
There are several ways to solve linear equations in multiple variables.
Basic fact: if you have n linear equations with n unknowns, there are three possibilities:
(1) there is no solution (which means that at least two of the equations are inconsistent and logically contradict each other);
(2) there are an infinite number of solutions (which means that at least two of the equations are not independent and logically repeat each other); or
(3) there is a unique solution.
If the number of equations is large, solving the system the way I am about to show you can be very tedious, but it is easy with just two equations.
\(\displaystyle [\alpha]\ 4x + 2y = 30.\)
\(\displaystyle [\beta]\ -2x + 3y = 13.\) Two equations in two unknowns.
We use one of the equations to find one of the unknowns in terms of the other. We can use either equation and find either unknown, but let's use equation alpha and solve for y. We get:
\(\displaystyle 4x + 2y = 30 \implies 2y = 30 - 4x \implies\)
\(\displaystyle [\gamma]\ y = \dfrac{30 - 4x}{2}.\) With me so far?
We now use equation gamma to eliminate y from the so far unused original equation (equation beta in this case) and get this:
\(\displaystyle -2x + 3\left(\dfrac{30 - 4x}{2}\right) = 13 \implies 2\left\{-2x + 3\left(\dfrac{30 - 4x}{2}\right)\right\} = 2 * 13 \implies 26 = - 4x + 90 - 12x = \implies 16x = 64 \implies\)
\(\displaystyle [\delta]\ x = 4.\) Did you follow that bit of algebraic manipulation? If so, you can see that we have found one of the two variables.
We use equation gamma to find the other: \(\displaystyle [\epsilon]\ y = \dfrac{30 - 4 * 4}{2} = \dfrac{30 - 16}{2} = 15 - 8 = 7.\)
Now we check our work by seeing whether the original equations alpha and beta are true with these values of x and y.
\(\displaystyle 4(4) + 2(7) = 16 + 14 = 30.\ CHECKS.\)
\(\displaystyle (- 2)(4) + 3(7) = - 8 + 21 = 13.\ CHECKS.\)
Try that approach on your problems.
