Resolver el siguiente sistema de ecuaciones: 3x + 7y = 17 2x + 5y = 12
Understand the Problem
La pregunta presenta un sistema de dos ecuaciones lineales con dos variables (x e y) que necesita ser resuelto. El objetivo es encontrar los valores de 'x' e 'y' que satisfacen ambas ecuaciones simultáneamente. Este problema se resuelve comúnmente utilizando métodos como sustitución, igualación o eliminación.
Answer
$x = 1$, $y = 2$
Answer for screen readers
$x = 1$, $y = 2$
Steps to Solve
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Multiply equations to eliminate a variable We can eliminate $x$. Multiply the first equation by 2 and the second equation by -3. $$2(3x + 7y) = 2(17) \implies 6x + 14y = 34$$ $$-3(2x + 5y) = -3(12) \implies -6x - 15y = -36$$
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Add the equations to eliminate $x$
Add the two resulting equations: $$(6x + 14y) + (-6x - 15y) = 34 + (-36)$$ $$6x - 6x + 14y - 15y = 34 - 36$$ $$-y = -2$$
- Solve for $y$
$$-y = -2 \implies y = 2$$
- Substitute $y$ back into one of the original equations to solve for $x$
Using the first original equation: $$3x + 7y = 17$$ $$3x + 7(2) = 17$$ $$3x + 14 = 17$$ $$3x = 17 - 14$$ $$3x = 3$$ $$x = 1$$
$x = 1$, $y = 2$
More Information
The solution to the system of equations is $x = 1$ and $y = 2$. This means that the point (1, 2) is the intersection of the two lines represented by the equations.
Tips
A common mistake is not distributing the multiplication correctly across all terms in the equation. For example, multiplying only the $x$ term but not the $y$ term or the constant. Another common mistake is making errors with the signs, especially when multiplying by a negative number.
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