Question

Write an equation in slope-intercept form of the line that passes through the points. $$ \left(\frac{1}{4}, 2\right),\left(-5, \frac{2}{3}\right) $$

Short answer

The final equation of the line that passes through the points (1/4, 2) and (-5, 2/3) can be obtained by substituting the calculated slope 'm' and y-intercept 'b' into the slope-intercept form \( y = m*x + b \).

Step-by-step solution

Calculate the Slope

The slope 'm' of a line that passes through two points (x1, y1) and (x2, y2) is given by: \( m = \frac{y2 - y1}{x2 - x1} \) Here, (x1, y1) is (1/4, 2) and (x2, y2) is (-5, 2/3). Substituting these values into the formula gives: \( m = \frac{2/3 - 2}{-5 - 1/4} \) Simplifying this expression will give the calculated slope.

Calculate the Y-Intercept

Once the slope 'm' is known, it can be substituted back into the slope-intercept equation, along with the coordinates of one of the points, to find the y-intercept 'b'. The equation is given by: \( b = y1 - m*x1 \) Substitute y1 = 2, x1 = 1/4 and 'm' as the calculated slope into the equation, and simplify the expression to find the value of 'b'.

Write the Equation of the Line

After calculating the slope 'm' and the y-intercept 'b', substitute these values into the slope-intercept form. The final equation of the line will be: \( y = m*x + b \)