Answer:
(x, y) = (2, 1)
Explanation:
You want the solution to the system of equations ...
- 148x +231y = 527
- 231x +148y = 610
Solution
When the coefficients are not nicely related, graphical methods or matrix methods are preferred, as elimination or substitution can result in messy arithmetic with large numbers.
The first attachment shows a calculator's solution using row-reduction of the augmented matrix of coefficients. It tells us the solution is ...
(x, y) = (2, 1)
Cross Multiplication method
A solution similar to the use of Cramer's rule can be found using the cross-multiplication method. For this, we rewrite the equations to general form, and make a list of the coefficients with the x-coefficients repeated at the end of the list:
- 148x +231y -527 = 0 ⇒ 148, 231, -527, 148
- 231x +148y -610 = 0 ⇒ 231, 148, -610, 231
Now, we form the differences of the cross products in each pair of columns:
∆ = (148)(148) -(231)(231) = -31,457
∆x = (231)(-610) -(148)(-527) = -62,914
∆y = (-527)(231) -(-610)(148) = -31,457
And the solution is ...
x = ∆x/∆ = -62,914/-31,457 = 2
y = ∆y/∆ = -31,457/-31,457 = 1
The solution is (x, y) = (2, 1).
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Additional comment
You might consider this to be "messy arithmetic with large numbers." Yes, it looks that way, but there are actually fewer operations required here than when using the elimination method.
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