# What is the equation of the line with slope  m= -7/3  that passes through  (-17/15,-5/24) ?

Jun 16, 2017

$y = - \frac{7}{3} x - \frac{977}{120}$

or

$7 x + 3 y = - \frac{977}{40}$

or

$280 x + 120 y = - 977$

#### Explanation:

We are finding a line, so it needs to follow the linear form. The easiest way to find the equation in this instance is using the gradient-intercept formula. This is:

$y = m x + c$

Where $m$ is the gradient and $c$ is the $y$-intercept.

We already know what $m$ is, so we can substitute it into the equation:

$m = - \frac{7}{3}$

$\implies y = - \frac{7}{3} x + c$

So now we need to find c. To do this, we can sub in the values of the point we have $\left(- \frac{17}{15} , - \frac{5}{24}\right)$ and solve for $c$.

$x = - \frac{17}{15}$

$y = - \frac{5}{24}$

$\implies y = - \frac{7}{3} x + c$

Substitute the values in:
$\implies - \frac{5}{24} = - \frac{7}{3} \left(- \frac{17}{15}\right) + c$

Apply the multiplication
$\implies - \frac{5}{24} = \frac{- 7 \cdot - 17}{3 \cdot 5} + c$

$\implies - \frac{5}{24} = \frac{119}{15} + c$

Isolate the unknown constant, so bring all the numbers to one side of the by subtracting $- \frac{119}{15}$

$\implies - \frac{5}{24} - \frac{119}{15} = \cancel{\frac{119}{15}} + c - \cancel{\frac{119}{15}}$

$\implies - \frac{5}{24} - \frac{119}{15} = c$

Multiply numerator and denominator by a number to get a common denominator in both fractions to apply the subtraction

$\implies \frac{- 5 \cdot 5}{24 \cdot 5} - \frac{119 \cdot 8}{15 \cdot 8} = c$

$\implies - \frac{25}{120} - \frac{952}{120} = c$

$\implies \frac{- 25 - 952}{120} = c$

$\implies - \frac{977}{120} = c$

So now we can also substitute c into the equation:

$y = - \frac{7}{3} x + c$

$\implies y = - \frac{7}{3} x - \frac{977}{120}$

We can also put this into the general form, which looks like:

$a x + b y = c$

To do this, we can rearrange the gradient intercept formula into the general formula using the steps shown below:

$\implies y = - \frac{7}{3} x - \frac{977}{120}$

We need to get rid of all the fractions first. So we multiply everything with a denominator (using the smaller one will make it easier in my opinion), and it should get rid of the fractions:

$\implies 3 \left(y\right) = 3 \left(- \frac{7}{3} x - \frac{977}{120}\right)$

$\implies 3 y = 3 \cdot - \frac{7}{3} x - 3 \cdot \frac{977}{120}$

$\implies 3 y = \frac{\cancel{3} \cdot - 7}{\cancel{3}} x - \frac{3 \cdot 977}{120}$

$\implies 3 y = - 7 x - \frac{2931}{120}$

$\implies 3 y = - 7 x - \frac{977}{40}$

Then bring the $x$ value to the other side by adding $- 7 x$ to both sides

$\implies 3 y + 7 x = \cancel{- 7 x} - \frac{977}{40} + \cancel{7 x}$

$\implies 7 x + 3 y = - \frac{977}{40}$

If you want you can get rid of the fraction by multiplying both sides by 40:

$\implies 40 \left(7 x + 3 y\right) = 40 \left(- \frac{977}{40}\right)$

$\implies 40 \cdot 7 x + 40 \cdot 3 y = \frac{\cancel{40} - 977}{\cancel{40}}$

$\implies 280 x + 120 y = - 977$