How do you write the standard form of a line given (5, 2) and which has a y-intercept of 7?

Feb 20, 2017

$\textcolor{red}{1} x + \textcolor{b l u e}{1} y = \textcolor{g r e e n}{7}$

Explanation:

The y-intercept of $7$ is equal to the point $\left(0 , 7\right)$. Using this point and the point from the problem we can determine the slope. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{7} - \textcolor{b l u e}{2}}{\textcolor{red}{0} - \textcolor{b l u e}{5}} = \frac{5}{-} 5 = - 1$

Next, we can use the point-slope formula to find an equation for the line. The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through. Substituting the slope we calculated and the y-intercept of $\left(0 , 7\right)$ gives:

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- 1} \left(x - \textcolor{red}{0}\right)$

The standard form of a linear equation is: $\textcolor{red}{A} x + \textcolor{b l u e}{B} y = \textcolor{g r e e n}{C}$

Where, if at all possible, $\textcolor{red}{A}$, $\textcolor{b l u e}{B}$, and $\textcolor{g r e e n}{C}$are integers, and A is non-negative, and, A, B, and C have no common factors other than 1.

We can convert the equation above to the standard form as follows:

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- 1} \left(x - \textcolor{red}{0}\right)$

$y - \textcolor{red}{7} = - 1 x - \left(- 1 \times \textcolor{red}{0}\right)$

$y - \textcolor{red}{7} = - 1 x - 0$

$y - \textcolor{red}{7} = - 1 x$

$\textcolor{b l u e}{1 x} + y - \textcolor{red}{7} + 7 = \textcolor{b l u e}{1 x} - 1 x + 7$

$1 x + y - 0 = 0 + 7$

$\textcolor{red}{1} x + \textcolor{b l u e}{1} y = \textcolor{g r e e n}{7}$