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**Hint:**The equation of line is $y=mx+b$.

This is equation of a line in which is called as slope intercept form where $m$ is the slope and $b$ is the $y$-intercept for finding equations of line first we have to find $m$ slope and then use the slope to find the $y$-intercept. Then you can find the equation of line for finding slope use.

$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ this formula. Then put the values in the equation. You will get the answer.

**Complete step by step solution:**

To point given $\left( 4,3 \right)$ and $\left( -6,3 \right)$

The equation of line is $y=mx+b$ where $m$ is the slope and $b$ is the $y$-intercept for finding equations of line first we have to find $m$ slope and then use the slope to find the $y$-intercept. Then you can find the equation of line for finding slope.

The formula for slope is $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$

$\left( {{x}_{1}}{{y}_{1}} \right)=\left( 4,3 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( -6,3 \right)$

$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\dfrac{-3-3}{-6-4}=\dfrac{-6}{-10}=\dfrac{3}{5}$

$m=\dfrac{3}{5}$

So, the slope of the line passing through the point $\left( 4,3 \right)$ and $\left( -6,-3 \right)$ is $3.$

Now we will use the slope to find the $y$intercept we know the slope of the line is $\dfrac{3}{5}$ we can put the value of slope $m$ in the equation of line in slope intercept from be,

$y=mx+b$

$\Rightarrow m=\dfrac{3}{5}$

$\Rightarrow y=\dfrac{3}{5}x+b$

Next choose one of the two points to put plug in for values of $x,y.$ It does not matter which one of the two points you should get the same answer in either case.

We will take $\left( x,y \right)$ $\left( 4:3 \right)$

Put this value in this equation.

$y=\dfrac{3}{5}x+b$

$\Rightarrow 3=\dfrac{3}{5}.4+b$

$\Rightarrow 3=\dfrac{12}{5}+b$

$\Rightarrow b=3-\dfrac{12}{5}$

$b=\dfrac{3}{5}$

Now, we have slope $m=\dfrac{3}{5}$ and the $y$-intercept $-b=\dfrac{3}{5}$

Put this value in the equation of the line in slope intercept form is

$y=\dfrac{3}{5}x+\dfrac{3}{5}$

**Additional Information:**

Slope intercept equation of vertical and horizontal lines. The equation of vertical lines is $x=b$ Since a vertical goes straight point on a vertical line is the same. Therefore whatever the $x$ value is also the value of $b.$

For instance the red line in the picture below is graph of the $x=1$

The equation of a horizontal line is $0$ is the general formula for the standard equation $y=mx+b$ becomes ${{y}_{0}}x+b$ $y=b$

Also since the line horizontal every point on that line has the same $y$ value. The $y$ value is therefore also the $y$ intercept for instance the red line.

**Note:**While solving this type of problem also slope intercept it is easy to solve.

Use the correct formula for students making mistakes on slope formulas.

It is $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ students write $\dfrac{{{x}_{2}}-{{x}_{1}}}{{{y}_{2}}-{{y}_{1}}}$

Sometimes so write carefully.

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