# How do you write an equation of a line given slope is 3/7 and the ordered pair is (0, 5/6)?

Jul 15, 2016

$y = \frac{3}{7} x + \frac{5}{6}$

#### Explanation:

$\textcolor{red}{\text{Step 1}}$
The general equation of a straight line is: $y = m x + c$

Where $m$ is the gradient and $c$ is a constant which is also the y-intercept.

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$\textcolor{red}{\text{Step 2}}$

We are told that the gradient (slope) is $\frac{3}{7}$ so by substituting this for $m$ we have

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

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$\textcolor{red}{\text{Step 3}}$

We are given a condition where this equation is true. We are told that it passes through the point $\left(x , y\right) \to \left(\textcolor{b l u e}{0 , \frac{5}{6}}\right)$

So we van substitute for both $x$ and $y$ to determine the value of $c$.

$\textcolor{b r o w n}{y = \frac{3}{7} x + x \text{ "->" } \textcolor{b l u e}{\frac{5}{6}} = \frac{3}{7} \left(\textcolor{b l u e}{0}\right) + c}$

But $\frac{7}{7} \times 0 = 0$ giving:

$\frac{5}{6} = c$

Thus the finished equation is:

$y = \frac{3}{7} x + \frac{5}{6}$

Jul 15, 2016

$y = \frac{3}{7} x + \frac{5}{6}$

#### Explanation:

There is a nifty formula for the equation of a line which applies in just such a case where we are given the slope and one point.

Using this formula requires only ONE step with substitution and some easy simplifying. Works like a dream...

$y - {y}_{1} = m \left(x - {x}_{1}\right) \text{ m = slope and "(x_1,y_1)" is a point}$

$y - \frac{5}{6} = \frac{3}{7} \left(x - 0\right)$

$y = \frac{3}{7} x + \frac{5}{6}$

However, in this particular case, no working was required at all.

The slope, m is given as $\frac{3}{7}$ and the point given happens to be the y-intercept because the $x -$value is 0. So $c = \frac{5}{6}$

$y = m x + c \Rightarrow y = \frac{3}{7} x + \frac{5}{6}$

Jul 15, 2016

Alternative approach

$y = \frac{3}{7} x + \frac{5}{6}$

#### Explanation:

Let the slope (gradient) be $m$

The gradient (slope) the change in up or down for a given change in along.

$m = \left(\text{Change in y")/("Change in x}\right) = \frac{3}{7}$

Let given point be ${P}_{o} \to \left({x}_{o} , {y}_{o}\right) = \left(0 , \frac{5}{6}\right)$

Let any other point be ${P}_{i} \to \left({x}_{i} , {y}_{i}\right)$

Then $m = \left(\text{Change in y")/("Change in x}\right) \to \frac{{y}_{i} - {y}_{o}}{{x}_{i} - {x}_{o}} = \frac{{y}_{i} - \frac{5}{6}}{{x}_{i} - 0} = \frac{3}{7}$

${y}_{i} = \frac{3}{7} {x}_{i} + \frac{5}{6}$
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$y = \frac{3}{7} x + \frac{5}{6}$