To easily find the slope of the line, you can rearrange the equation to the \(\displaystyle y=mx+b\) form. To do this, isolate the \(\displaystyle y\) by moving the \(\displaystyle x\) to the other side of the equation. This gives you \(\displaystyle 3y=9x+15\). Then, divide both sides by 3 to isolate the \(\displaystyle y\), which leaves you with \(\displaystyle y=3x+5\). Now, the equation is in the \(\displaystyle y=mx+b\) form, and you can easily see that the slope is 3.

The equation is written in standard from: \(\displaystyle \small Ax+By=C\). In this format, the slope is \(\displaystyle \small -\frac{A}{B}\).

\(\displaystyle 6x+3y=15\)

In our equation, \(\displaystyle \small A=6\) and \(\displaystyle \small B=3\).

\(\displaystyle \small m=-\frac{A}{B}=-\frac{6}{3}\)

\(\displaystyle \small -\frac{6}{3}=-2\)

\(\displaystyle 4x - 6y = 15\)

\(\displaystyle -4x + 4x - 6y = -4x + 15\)

\(\displaystyle - 6y = -4x + 15\)

\(\displaystyle - 6y \div (-6)=\left ( -4x + 15 \right )\div (-6)\)

\(\displaystyle y =\frac{2}{3 }x- \frac{5}{2}\)

The slope is the coefficient of \(\displaystyle x\): \(\displaystyle m = \frac{2}{3}\)

Use the slope formula: 

\(\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{-2-(-4)}{8-5} = \frac{2}{3}\)

Use the slope formula: 

\(\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{-2-4}{8-5} = \frac{-6}{3} = -2\)

To calculate the slope of a line, we pick two points, and calculate the change in \(\displaystyle y\) divided by the change in \(\displaystyle x\). To calculate the change in value, we use subtraction. So, 

slope = \(\displaystyle \frac{\text{Change in Y}}{\text{Change in X}} = \frac{y_2-y_1}{x_2-x_1}\) 

So we can pick any two points and plug them into this formula. Let's choose (0,3) and (4,6). We get 

\(\displaystyle \frac{6-3}{4-0}= \frac{3}{4}\)

The general formula of a straight line is \(\displaystyle y=mx+b\), where \(\displaystyle \small m\) is the slope and \(\displaystyle \small b\) is the y-intercept.

To evaluate our original equation, we can compare it to this formula.

\(\displaystyle y=mx+b\)

\(\displaystyle y=-2x-3\)

The slope is \(\displaystyle \small -2\) and the y-intercept is at \(\displaystyle \small -3\). Since the y-intercept is a point on the line, it is written as \(\displaystyle \small (0,-3)\).

Rewrite in slope-intercept form:

\(\displaystyle 2x + 6y = 15\)

\(\displaystyle -2x + 2x + 6y = -2x +15\)

\(\displaystyle 6y = -2x +15\)

\(\displaystyle 6y \div 6= \left (-2x +15 \right )\div 6\)

\(\displaystyle y= - \frac{1}{3}x +\frac{5}{2}\)

The slope is the coefficient of \(\displaystyle x\):

 \(\displaystyle m = - \frac{1}{3}\)

This function describes a straight line. We can compare the given equation with the formula for a straight line in slope-intercept form.

\(\displaystyle y=-\frac{1}{3}x-2\)

\(\displaystyle y=mx+b\)

In the formula, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept. Looking at our given equation, we can see that \(\displaystyle m=-\frac{1}{3}\) and \(\displaystyle b=-2\).

Since the y-intercept is a point, we want to write it in point notation: \(\displaystyle (0,-2)\)

\(\displaystyle 1-10x+5y=2\)

You see that this equation is not written explicitly in terms of \(\displaystyle x\). Before we can determine the slope and y-intercept, we need to write the equation explicitly in terms of \(\displaystyle x\) by solving for \(\displaystyle y\).

First, add \(\displaystyle 10x\) to both sides.

\(\displaystyle 1-10x+10x+5y=10x+2\)

\(\displaystyle 1+5y=10x+2\)

Then subtract \(\displaystyle 1\) to both sides.

\(\displaystyle 1-1+5y=10x+2-1\)

\(\displaystyle 5y=10x+1\)

Finally, divide by \(\displaystyle 5\).

\(\displaystyle \frac{5y}{5}=\frac{10x+1}{5}\)

\(\displaystyle y=2x+\frac{1}{5}\)

Now that the equation is explicitly in terms of \(\displaystyle x\), we can compare it to the general formula of a straight line in slope-intercept form.

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is equal to the slope and \(\displaystyle b\) is equal to the y-intercept. Comparing our given equation to the formula, we can see that \(\displaystyle m=2\) and \(\displaystyle b=\frac{1}{5}\).

Since the y-intercept is a point, we will want to write it in point notation: \(\displaystyle (0,\frac{1}{5})\).