x - 2y = 0; 6x + 20y = 0; compare with

Steps to Solve

1. Rewrite the First Equation in Slope-Intercept Form We start with the first equation $x - 2y = 0$. We want to express it in the form $y = mx + b$, where $m$ is the slope.

Rearranging gives: $$ -2y = -x $$ Dividing by -2, we find: $$ y = \frac{1}{2}x $$

2. Rewrite the Second Equation in Slope-Intercept Form Next, we move to the second equation $6x + 20y = 0$. Again, we want to convert this to slope-intercept form.

Rearranging gives: $$ 20y = -6x $$ Dividing by 20, we find: $$ y = -\frac{3}{10}x $$

3. Compare the Slopes Now, we will analyze the slopes of the two equations we derived.

From the first equation, the slope $m_1$ is $\frac{1}{2}$. From the second equation, the slope $m_2$ is $-\frac{3}{10}$.

Since the slopes are different ($\frac{1}{2} \neq -\frac{3}{10}$), the lines represented by these equations will intersect at one point.

4. Find the Point of Intersection To find the intersection point, we set the two equations equal to each other: $$ \frac{1}{2}x = -\frac{3}{10}x $$

To solve for $x$, add $\frac{3}{10}x$ to both sides: $$ \frac{1}{2}x + \frac{3}{10}x = 0 $$

Converting $\frac{1}{2}$ to have a common denominator of 10: $$ \frac{5}{10}x + \frac{3}{10}x = 0 $$ Combine like terms: $$ \frac{8}{10}x = 0 $$ Thus, $x = 0$.

5. Find Corresponding y-Value Now that we have $x = 0$, we substitute this back into one of the original equations. Using the first equation: $$ 0 - 2y = 0 \Rightarrow -2y = 0 \Rightarrow y = 0 $$

6. Final Result The point of intersection of the two lines is $(0, 0)$.