Problem 42 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line satisfying the given conditions. Passing through \((1,-2)\) with slope \(\frac{-3}{2}\)

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{-3}{2}x - \frac{1}{2}\). Plot the y-intercept (\(0, -\frac{1}{2}\)) and another point based on the slope.

Step 1: Understand the Slope-Intercept Form

The equation of a line can be expressed in slope-intercept form, which is \[y = mx + b\] where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Substitute the Given Slope

We know the slope (\(m\)) is \(\frac{-3}{2}\). Substituting this into the slope-intercept form gives: \[y = \frac{-3}{2}x + b\]

Step 3: Substitute the Given Point

The line passes through the point \((1, -2)\). Substitute \(x = 1\) and \(y = -2\) into the equation to find \(b\): \(-2 = \frac{-3}{2} \times 1 + b\)

Step 4: Solve for the Y-Intercept

Solve the equation for \(b\): \[-2 = \frac{-3}{2} + b\] Rearrange to find \(b\): \[b = -2 + \frac{3}{2} = -\frac{4}{2} + \frac{3}{2} = -\frac{1}{2}\]

Step 5: Write the Equation of the Line

Now that we know the values for \(m\) and \(b\), the equation of the line is: \[y = \frac{-3}{2}x - \frac{1}{2}\]

Step 6: Sketch the Graph

To sketch the graph, start by plotting the y-intercept (\(0, -\frac{1}{2}\)). Then, use the slope (\(\frac{-3}{2}\)), which means from the intercept, move up 3 units and right 2 units to plot another point. Draw a straight line through these points.

Key Concepts

Understanding the Slope-Intercept Form

To graph a linear equation, we often use the slope-intercept form: \(y = mx + b\). This form is very useful because it gives you the slope (\(m\) and the y-intercept (\(b\) directly. The slope (\