Question

Find an equation of the line that passes through the points, and sketch the line. $$ \left(\frac{7}{8}, \frac{3}{4}\right),\left(\frac{5}{4},-\frac{1}{4}\right) $$

Short answer

The equation of the line which passes through the given points is \(y = -2x + \frac{11}{4}\).

Step-by-step solution

Find the Slope

The slope, m, between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. We apply this formula to find the slope between the given points \[\left(\frac{7}{8}, \frac{3}{4}\right)\] and \[\left(\frac{5}{4},-\frac{1}{4}\right)\]. It gives us \[m = \frac{-\frac{1}{4} - \frac{3}{4}}{\frac{5}{4} - \frac{7}{8}} = -2\].

Write the Line Equation

Now that we have the slope, we can use one of the given points to write the line equation in point-slope form \(y - y_1 = m(x - x_1)\). Using the point \(\left(\frac{7}{8}, \frac{3}{4}\right)\) and slope -2, we get the equation \(y - \frac{3}{4} = -2(x - \frac{7}{8})\). Simplifying, the final equation is \(y = -2x + \frac{11}{4}\).

Sketch the Line

To sketch the line, first plot the points \(\left(\frac{7}{8}, \frac{3}{4}\right)\) and \(\left(\frac{5}{4},-\frac{1}{4}\right)\) on the graph. Then draw a line that passes through these points. The line will have a negative slope, which means it falls from left to right.