Problem 39 - Algebra and Trigonometry

Question

(a) Sketch the line with slope \(\frac{3}{2}\) that passes through the point \((-2,1)\) . (b) Find an equation for this line.

Step-by-Step Solution

Answer

The equation is \( y = \frac{3}{2}x + 4 \).

Step 1: Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line. In this problem, the slope \( m \) is given as \( \frac{3}{2} \). We need to find \( b \) using the point \((-2,1)\).

Step 2: Substitute Slope and Coordinates

Using the given point \((-2, 1)\), substitute these values into the slope-intercept formula to find \( b \): \( 1 = \frac{3}{2}(-2) + b \).

Step 3: Simplify to Find Intercept

Simplify the equation: \( 1 = -3 + b \). Solve for \( b \) by adding \( 3 \) to both sides: \( b = 4 \).

Step 4: Write Equation of the Line

Now that we have \( b = 4 \), substitute \( m = \frac{3}{2} \) and \( b = 4 \) back into the slope-intercept form: \( y = \frac{3}{2}x + 4 \).

Step 5: Sketching the Line

To sketch the line: start at the point \((-2, 1)\). Use the slope \( \frac{3}{2} \) to find another point: for each step to the right by 2 on the x-axis, move 3 steps up on the y-axis. Draw a straight line through these points extending in both directions.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express a linear equation. It is written as \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For the problem at hand, we've been given a slope of \( \frac{3}{2} \). To fully express the line in slope-intercept form, we needed to determine the y-intercept \( b \) using the given point \((-2, 1)\). By substitution into the formula, as demonstrated in the original solution, the intercept can be found, completing our linear equation.

Equation of a Line

The equation of a line such as our example can be derived using a point and a slope. Given the point \((-2,1)\) and a slope of \(\frac{3}{2}\), we substitute these into the slope-intercept formula:

• Substituting \(x = -2\) and \(y = 1\) gives us \(1 = \frac{3}{2}(-2) + b\).

After solving for \(b\), which involves simplifying and rearranging the equation, we find that \(b = 4\). Thus, the final equation of the line becomes \(y = \frac{3}{2}x + 4\). This equation captures all necessary information about the line, allowing you to identify both the slope and where it crosses the y-axis quickly. It is the complete relationship between \(x\) and \(y\) on this particular line.

Graphing Lines

Graphing a line can bring an equation to life, providing a visual representation of the relationship between \(x\) and \(y\). Using the slope-intercept form \(y = \frac{3}{2}x + 4\), graphing becomes easier:

• Start at the y-intercept (\(b = 4\)), which is the point \((0,4)\) on the graph.
• Use the slope \(\frac{3}{2}\) to find additional points: for every 2 units you move to the right on the x-axis, move 3 units up on the y-axis.

Once you have plotted the intercept and a few additional points, a ruler can help draw a straight line through these, extending it in both directions to show the line infinitely. Graphing does not just solve the equation but translates it into a diagram that can also be used for further analysis or solving-related graph-based problems.