linear equation of 2x - 3y + 6 = 0 and 6x + y + 8 = 0

Understand the Problem

The question is asking to find the linear equations represented by the two given equations: 2x - 3y + 6 = 0 and 6x + y + 8 = 0. This involves rearranging the equations into slope-intercept form or another standard form. Additionally, it may require finding the point of intersection of the two lines if desired.

Answer

The equations are $y = \frac{2}{3}x + 2$ and $y = -6x - 8$, with intersection at $\left(-\frac{3}{2}, 1\right)$.

Steps to Solve

Rewrite the first equation in slope-intercept form

To convert the first equation $2x - 3y + 6 = 0$ into slope-intercept form ($y = mx + b$), we isolate $y$:

Subtract $2x$ and $6$ from both sides: $$ -3y = -2x - 6 $$

Now, divide by $-3$: $$ y = \frac{2}{3}x + 2 $$

Rewrite the second equation in slope-intercept form

Next, we rewrite the second equation $6x + y + 8 = 0$ into slope-intercept form:

Subtract $6x$ and $8$ from both sides: $$ y = -6x - 8 $$

Identify slopes and y-intercepts

Now we can identify the slope ($m$) and the y-intercept ($b$) from both equations:

For the first equation $y = \frac{2}{3}x + 2$, the slope is $\frac{2}{3}$ and the y-intercept is $2$.

For the second equation $y = -6x - 8$, the slope is $-6$ and the y-intercept is $-8$.

Find the intersection point (optional)

To find the point of intersection, we can set the two equations equal to each other:

$$ \frac{2}{3}x + 2 = -6x - 8 $$

Multiply through by 3 to eliminate the fraction: $$ 2x + 6 = -18x - 24 $$

Combine like terms: $$ 20x = -30 $$

So: $$ x = -\frac{3}{2} $$

Now substitute $x$ back into one of the original equations to find $y$.

Using the first equation: $$ y = \frac{2}{3}(-\frac{3}{2}) + 2 = -1 + 2 = 1 $$

Thus, the point of intersection is $\left(-\frac{3}{2}, 1\right)$.

The linear equations are:

$y = \frac{2}{3}x + 2$
$y = -6x - 8$

The point of intersection is $\left(-\frac{3}{2}, 1\right)$.

More Information

The equations represent two lines where one has a positive slope, indicating it rises as $x$ increases, while the other has a negative slope, indicating it falls. The point of intersection represents the solution to the system of equations and is the point where both lines cross.

Tips

One common mistake is forgetting to change sign when isolating $y$ from the original equations. Another mistake could be incorrectly finding the intersection point by confusing the steps to solve for $x$ or $y$.

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