Question

Write an equation in standard form of the line that passes through the two points. $$(1,4),(5,7)$$

Short answer

The equation of the line in standard form is -3x + 4y = 13.

Step-by-step solution

Calculate the Slope

First thing required for the equation of a line is the slope, which we calculate using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points (1,4) and (5,7) into the formula, we get \(m = \frac{7 - 4}{5 - 1} = \frac{3}{4}\).

Write the Equation in Slope-Intercept Form

With the calculated slope and one of the points (1,4), substitute these values into the slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept. So, we get \(4 = \frac{3}{4} * 1 + b\). After solving for b, we get \(b= \frac{13}{4}\). So, the line in slope-intercept form is \(y = \frac{3}{4}x + \frac{13}{4}\).

Convert to Standard Form

The last step is to convert the equation from slope-intercept form into standard form \(Ax + By = C\). The requirement is that A, B and C should be integers. Multiply the entire equation by 4 to clear the fractions: \(4y = 3x + 13\). Then, rearrange this in standard form as -3x + 4y = 13.