Problem 41 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

The line passes through \(1, -2\) with a slope of \(\frac{-2}{3}\). Equation: \(y = \frac{-2}{3}x - \frac{4}{3}\).

Step 1: Identify given information

The given point is \(1, -2\), and the slope of the line is \(\frac{-2}{3}\text{\)}.

Step 2: Use point-slope form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \m\ is the slope. Substitute the values: \(y + 2 = \frac{-2}{3}(x - 1)\).

Step 3: Distribute and simplify

Distribute the slope on the right side: \(y + 2 = \frac{-2}{3}x + \frac{2}{3}\).

Step 4: Isolate y

Subtract 2 from both sides to isolate \y\: \(y = \frac{-2}{3}x + \frac{2}{3} - 2\).\text{Convert } -2 \text{ to} \ \frac{-6}{3}\ \text{and simplify the y-intercept term.} \(y = \frac{-2}{3}x - \frac{4}{3}\).This is the slope-intercept form.

Step 5: Plot the given point

Mark the point \(1, -2\) on the graph.

Step 6: Use slope to find another point

From point \(1, -2\), use the slope \(\frac{-2}{3}\text{\)} to plot another point: move 3 units to the right \ and 2 units down to reach \(4, -4\).

Step 7: Draw the line

Draw a straight line passing through the points \((1, -2)\) and \((4, -4)\).

Key Concepts

Point-Slope Form of a Linear Equation

The point-slope form is very useful for writing an equation when you have a point and the slope of the line. It is given by the formula: \( y - y_1 = m(x - x_1) \). Here,

• \( (x_1, y_1) \) are the coordinates of the given point
• \( m \) is the slope of the line

You simply substitute the point and the slope into the formula to get the equation. In the provided exercise, the given point is \( (1, -2) \) and the slope is \( \frac{-2}{3} \). Substituting these values in, we get: \( y + 2 = \frac{-2}{3}(x - 1) \), which is our equation in point-slope form.

Slope-Intercept Form of a Linear Equation

The slope-intercept form is another popular way to write the equation of a line. Its formula is \( y = mx + b \), where

• \( m \) is the slope
• \( b \) is the y-intercept (where the line crosses the y-axis)

After obtaining the point-slope form, you can convert it to the slope-intercept form by solving for \( y \). For example:
Starting from \( y + 2 = \frac{-2}{3}(x - 1) \), distribute the \( \frac{-2}{3} \) on the right side:
\( y + 2 = \frac{-2}{3}x + \frac{2}{3} \)
Subtract 2 from both sides to isolate \( y \):
\( y = \frac{-2}{3}x + \frac{2}{3} - 2 \)
To simplify, convert \( -2 \) to a fraction with the same denominator:
\( -2 = \frac{-6}{3} \)
Simplifying further, you'll get:\( y = \frac{-2}{3}x - \frac{4}{3} \). Now the equation is in slope-intercept form.

Plotting Points on a Graph

Plotting points is a fundamental skill in graphing linear equations. Each point has coordinates \( (x, y) \). For example, the point \( (1, -2) \) means you go 1 unit to the right (along the x-axis) and 2 units down (along the y-axis). To plot this point on a graph:

• Start from the origin \( (0, 0) \)
• Move 1 unit to the right
• Move 2 units down

Mark this point. In the exercise, this is our starting point. After this, we can use the slope to find more points on the line and eventually draw it.

Calculating Slope

The slope of a line tells us how steep the line is. It is calculated as the 'rise' over the 'run' between two points on the line. The formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line

For example, if the points are \( (1, -2) \) and \( (4, -4) \), we can use the formula to find the slope:
\( m = \frac{-4 - (-2)}{4 - 1} = \frac{-4 + 2}{4 - 1} = \frac{-2}{3} \). This tells us that for every 3 units we move to the right, we move 2 units down.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a straight line through them. To graph a line based on its equation:

• Convert the equation to slope-intercept form to identify the slope and y-intercept
• Plot the y-intercept on the y-axis
• Use the slope to find another point starting from the y-intercept
• Draw a straight line through the points

For instance, given \( y = \frac{-2}{3}x - \frac{4}{3} \):

• The y-intercept is \( - \frac{4}{3}\), i.e., mark the point \( (0, - \frac{4}{3}) \)
• From this point, use the slope \( \frac{-2}{3} \) to find another point, i.e., move 3 units right and 2 units down

You can repeat this process to ensure the line is accurate. Then, connect the points with a straight line, and you've successfully graphed the linear equation!