Problem 36 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

The equation of the line is \( y = 2x - 4 \).

Step 1 - Identify the slope-intercept form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. The given slope is 2.

Step 2 - Use the point to find the y-intercept

Substitute the given point \((3, 2)\) and the slope \( m = 2 \) into the slope-intercept form equation to find \( b \).Starting with \[ y = mx + b \]Substitute \( y = 2 \), \( x = 3 \), and \( m = 2 \):\[ 2 = 2(3) + b \]Simplify:\[ 2 = 6 + b \]Solve for \( b \):\[ b = 2 - 6 \]\[ b = -4 \]Therefore, the y-intercept \( b \) is -4.

Step 3 - Write the equation of the line

With the slope \( m = 2 \) and the y-intercept \( b = -4 \), the equation of the line is:\[ y = 2x - 4 \]

Step 4 - Plot the line

To sketch the graph, start by plotting the y-intercept \( (0, -4) \).Next, use the slope to find another point. The slope \( 2 \) means a rise of 2 units for every run of 1 unit. From \( (0, -4) \), move up 2 units and right 1 unit to plot \( (1, -2) \).Finally, draw a straight line through these points.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a fundamental concept in graphing lines. The formula is expressed as: \[ y = mx + b \] Where:

• \(m\): This is the slope of the line. It indicates how steep the line is. The slope tells you how much \(y\) changes for a change in \(x\).
• \(b\): This represents the y-intercept of the line. It is the point where the line crosses the y-axis.

Understanding the slope-intercept form helps simplify the process of plotting linear equations. For this exercise, the given slope \(m\) is \(2\). This means that for every unit increase in \(x\), \(y\) will increase by 2 units.

The slope-intercept form is particularly useful because:

• It makes it easy to graph the line just by knowing the slope and y-intercept.
• It allows for quick identification of key features of the graph.

finding y-intercept

Finding the y-intercept \(b\) is critical as it gives you one of the key points to start your graph. You can find \(b\) by substituting a known point on the line and the slope \(m\) into the slope-intercept form equation.

For the given problem, we substitute \((3, 2)\) and the slope \(m = 2\) using the equation below:

\[ y = mx + b \]

\[ 2 = 2(3) + b \]
\[ 2 = 6 + b \]
The next step is to isolate \(b\) by subtracting 6 from both sides:
\[ b = 2 - 6 \]
\[ b = -4 \]

Thus, the y-intercept \(b\) is -4. This is the point where the line crosses the y-axis, and it helps in plotting the graph accurately.

plotting points

Plotting points is the final step to graphing a linear equation. Start with the y-intercept you found earlier, which in this case is \( (0, -4) \). From here, use the slope to plot additional points.

For the slope \( 2 \), it indicates a rise of 2 units for every run of 1 unit. So, starting from \( (0, -4) \):

• Move up 2 units
• Move right 1 unit to land at the next point \( (1, -2) \)

Continue this process to find more points if needed. For example: Starting at \( (1, -2) \), move:

• Up 2 units to get \( -2 + 2 = 0 \)
• Right 1 unit to get \( 1 + 1 = 2 \)

So your point will be \( (2, 0) \). After plotting these points, draw a straight line through them. This line represents your linear equation \( y = 2x - 4 \).