What is the solution to the system of equations?
Understand the Problem
The question is asking for the solution to a system of two linear equations represented in slope-intercept form. The user needs to find the point where the two lines intersect, which involves solving for the values of x and y that satisfy both equations.
Answer
The solution is $(4, -2)$.
Answer for screen readers
The solution to the system of equations is $(4, -2)$.
Steps to Solve
- Set the equations equal to each other
Since both equations are equal to $y$, we can set them equal to each other:
$$ -\frac{1}{4}x - 1 = \frac{1}{4}x - 3 $$
- Eliminate the fractions
To get rid of the fractions, we can multiply both sides of the equation by 4:
$$ -x - 4 = x - 12 $$
- Combine like terms
Next, we can add $x$ to both sides and add 4 to both sides:
$$ -4 + 12 = x + x $$
This simplifies to:
$$ 8 = 2x $$
- Solve for x
Now, divide both sides by 2 to find $x$:
$$ x = 4 $$
- Substitute x back into one of the original equations
Now that we have $x$, we can substitute it back into one of the original equations to find $y$. Let's use the second equation:
$$ y = \frac{1}{4}(4) - 3 $$
- Calculate y
Simplifying that gives us:
$$ y = 1 - 3 = -2 $$
The solution to the system of equations is $(4, -2)$.
More Information
The point $(4, -2)$ is where the two lines intersect. This means that at $x = 4$, both equations yield the same $y$ value of $-2$. Such intersection points can often represent solutions to real-world problems, such as finding equilibrium points.
Tips
- Misinterpreting the equations: Sometimes students forget to correctly set the equations equal to each other, which can lead to incorrect solutions.
- Incorrect arithmetic: Mistakes in adding, subtracting, or multiplying can lead to wrong answers; keeping calculations organized helps avoid this.
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