Problem 37 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{3}{2}x - 2 \).

Step 1: Identify the slope-intercept form

The slope-intercept form of a line's equation is given by: \[ y = mx + c \] where \( m \) is the slope and \( c \) is the y-intercept.

Step 2: Substitute the slope

Given the slope \( m = \frac{3}{2} \), substitute it into the slope-intercept form equation: \[ y = \frac{3}{2}x + c \]

Step 3: Substitute the given point

The line passes through the point \( (2,1) \). Substitute \( x = 2 \) and \( y = 1 \) into the equation to find the value of \( c \): \[ 1 = \frac{3}{2}(2) + c \]

Step 4: Solve for the y-intercept

Solve the equation from the previous step to find \( c \): \[ 1 = 3 + c \] \[ c = 1 - 3 \] \[ c = -2 \]

Step 5: Write the final equation

Substitute \( c = -2 \) back into the slope-intercept form equation: \[ y = \frac{3}{2}x - 2 \]

Step 6: Sketch the graph

To sketch the line, start by plotting the y-intercept \( (0, -2) \) and then use the slope to find another point. From \( (0, -2) \), move up 3 units and right 2 units to locate another point.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a very efficient way to describe a straight line. It's given by the formula: \[ y = mx + c \] Here, m is the slope of the line, which tells us how steep the line is. The c is the y-intercept, which represents the point where the line crosses the y-axis. Understanding this form is crucial because it simplifies plotting and sketching linear equations. You can quickly identify the steepness and where the line begins on the y-axis. It's the go-to method for representing lines in both mathematical problems and real-world applications.

finding y-intercept

To sketch a line, we often need to find the y-intercept, especially when given a specific point and slope. Let's revisit the problem: we are given the point \((2,1)\) and the slope \(m = \frac{3}{2} \) Start by plugging the slope into the slope-intercept form: \[ y = \frac{3}{2}x + c \] Substitute the given point \( (2,1) \) into the equation: \[ 1 = \frac{3}{2}(2) + c \] This equation allows us to solve for c: \[ 1 = 3 + c \] \[ c = 1 - 3 \] \[ c = -2 \] Now we have the full equation of the line, including the y-intercept: \[ y = \frac{3}{2}x - 2 \] The y-intercept c is \( -2 \) This step is critical as it completes the line equation, allowing us to easily sketch it.

plotting points

With our line equation \[ y = \frac{3}{2}x - 2 \] we can now sketch the line. Start by plotting the y-intercept: \( (0, -2) \). This is the point where the line crosses the y-axis. From there, use the slope, which is \( \frac{3}{2} \) to find more points. Slope is described as 'rise over run,' meaning for every 3 units you move up, you move 2 units to the right. Starting from \( (0, -2) \), move 3 units up and 2 units right, which lands us at \( (2,1) \) From these two points, you can draw a straight line, sketching out the entire graph. You can always plot more points if necessary, following the same slope guideline to ensure accuracy. This method effectively visualizes the linear equation, providing a clear representation of the line.