Solve the following system of equations: y = 2x + 1, 3x + y = 0
Understand the Problem
The question asks us to solve a system of two linear equations for (x) and (y). This can be achieved using substitution or elimination to find the values of (x) and (y) that satisfy both equations.
Answer
$x = -\frac{1}{5}$ $y = \frac{3}{5}$
Answer for screen readers
$x = -\frac{1}{5}$ $y = \frac{3}{5}$
Steps to Solve
- Substitute the first equation into the second
Since $y = 2x + 1$, substitute $2x + 1$ for $y$ in the second equation: $$3x + (2x + 1) = 0$$
- Solve for $x$
Combine like terms: $$5x + 1 = 0$$ Subtract 1 from both sides: $$5x = -1$$ Divide by 5: $$x = -\frac{1}{5}$$
- Solve for $y$
Substitute $x = -\frac{1}{5}$ into the first equation $y = 2x + 1$: $$y = 2(-\frac{1}{5}) + 1$$ $$y = -\frac{2}{5} + 1$$ $$y = -\frac{2}{5} + \frac{5}{5}$$ $$y = \frac{3}{5}$$
$x = -\frac{1}{5}$ $y = \frac{3}{5}$
More Information
The solution to the system of equations is $x = -\frac{1}{5}$ and $y = \frac{3}{5}$. This means that the point $(-\frac{1}{5}, \frac{3}{5})$ is the intersection of the two lines defined by the given equations.
Tips
A common mistake is to incorrectly substitute the expression for $y$ from the first equation into the second equation, or to make errors when simplifying the resulting equation. Another common mistake is to incorrectly solve for $y$ after finding $x$. Ensuring careful substitution and simplification can help avoid these errors.
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