The graphs of two equations in a linear system have the same x-intercept. One equation in the system is \frac{1}{5}x + 4y = a. What is the solution of the system?

Steps to Solve

1. Identify the x-intercept of the first equation

To find the x-intercept, set $y = 0$ in the equation $\frac{1}{5}x + 4y = a$.

$$ \frac{1}{5}x + 4(0) = a \implies \frac{1}{5}x = a \implies x = 5a $$

So, the x-intercept of the first equation is $(5a, 0)$.

2. Set up the second equation

Since the two equations share the same x-intercept, we can use the second equation given in the problem:

$$ \frac{1}{3}x + 4y = 0 $$

3. Find the x-intercept of the second equation

To find its x-intercept, set $y = 0$:

$$ \frac{1}{3}x + 4(0) = 0 \implies \frac{1}{3}x = 0 \implies x = 0 $$

So, the x-intercept of the second equation is $(0, 0)$.

4. Set equations equal for a solution

Since both equations must have the same x-intercept, we set them equal:

$$ 5a = 0 \implies a = 0 $$

5. Find the y-intercept for both equations

The y-intercept can be found by setting $x = 0$. For the first equation with $a = 0$:

$$ \frac{1}{5}(0) + 4y = 0 \implies 4y = 0 \implies y = 0 $$

So, the y-intercept for the first equation is $(0, 0)$.

The second equation already has its y-intercept at $(0, 0)$.

6. Conclusion

The solution for the system of equations is the point where both lines intersect:

So, the solution is:

$$(0, 0)$$