Resolver el siguiente sistema de ecuaciones: 3x + 7y = 17 2x + 5y = 12
Understand the Problem
La pregunta presenta un sistema de ecuaciones lineales con dos variables (x e y). El objetivo es encontrar los valores de x e y que satisfacen ambas ecuaciones simultáneamente.
Answer
$x = 1, y = 2$
Answer for screen readers
$x = 1, y = 2$
Steps to Solve
-
Multiply the equations by constants to set up elimination Multiply the first equation by 2 and the second equation by -3. This will allow us to eliminate $x$ when we add the equations together. $$2(3x + 7y) = 2(17) \implies 6x + 14y = 34$$ $$-3(2x + 5y) = -3(12) \implies -6x - 15y = -36$$
-
Add the two equations together Adding the modified equations will eliminate the $x$ term, leaving us with an equation in terms of $y$ alone. $$(6x + 14y) + (-6x - 15y) = 34 + (-36)$$ $$6x - 6x + 14y - 15y = 34 - 36$$ $$-y = -2$$
-
Solve for $y$ Multiply both sides of the equation by $-1$ to find the value of $y$. $$(-1)(-y) = (-1)(-2)$$ $$y = 2$$
-
Substitute $y$ into one of the original equations Substitute the value of $y$ into the first equation to solve for $x$. $$3x + 7(2) = 17$$ $$3x + 14 = 17$$
-
Solve for $x$ Subtract 14 from both sides of the equation. $$3x + 14 - 14 = 17 - 14$$ $$3x = 3$$ Divide both sides by 3. $$\frac{3x}{3} = \frac{3}{3}$$ $$x = 1$$
$x = 1, y = 2$
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
- Arithmetic errors when multiplying the equations by constants or when adding the equations together. Double-check each step to ensure accuracy.
- Incorrectly substituting the value of $y$ when solving for $x$. Make sure to substitute into one of the original equations and perform the calculations carefully.
- Forgetting to solve for both $x$ and $y$. It's important to find both variables to fully solve the system of equations.
AI-generated content may contain errors. Please verify critical information
