Problem 38 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

\[ y = \frac{2}{3}x - \frac{1}{3} \]

Step 1 Title: 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.

Step 2 Title: Substitute the Slope

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

Step 3 Title: Substitute the Point to Find the Intercept

The line passes through the point \( (2, 1) \). Substitute \( x = 2 \) and \( y = 1 \) into the equation to solve for \( b \): \[ 1 = \frac{2}{3}(2) + b \] Simplify: \[ 1 = \frac{4}{3} + b \] Subtract \( \frac{4}{3} \) from both sides: \[ 1 - \frac{4}{3} = b \] \[ b = -\frac{1}{3} \]

Step 4 Title: Write the Equation of the Line

Now that the y-intercept \( b \) is known, write the complete equation of the line: \[ y = \frac{2}{3}x - \frac{1}{3} \]

Step 5 Title: Plot the Line

To sketch the graph, plot the y-intercept \( (0, -\frac{1}{3}) \) and use the slope \( \frac{2}{3} \) to find another point. From \( (0, -\frac{1}{3}) \), move 2 units up and 3 units to the right to reach the point \( (3, \frac{5}{3}) \). Draw a line through the points \( (0, -\frac{1}{3}) \) and \( (3, \frac{5}{3}) \).

Key Concepts

slope-intercept form

The slope-intercept form is a way to write the equation of a straight line. It is expressed as:

\( y = mx + b \)

Where:

• y is the dependent variable (or the value on the y-axis)
• m is the slope of the line
• x is the independent variable (or the value on the x-axis)
• b is the y-intercept (the value where the line crosses the y-axis)

The form is particularly useful as it directly gives you the slope and the y-intercept of the line. This information is essential for graphing linear equations quickly and easily.

slope

The slope of a line measures its steepness and direction. In the slope-intercept form \( y = mx + b \), the slope is represented by m. The slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, it is written as:

\( m = \frac{Δy}{Δx} = \frac{y_2 - y_1}{x_2 - x_1} \)

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• If the slope is zero, the line is horizontal.
• If the slope is undefined, the line is vertical.

In our exercise, the slope is given as \( \frac{2}{3} \), which means for every 3 units you move to the right, the line rises by 2 units.

y-intercept

The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by b. This value tells us where the line will be when the value of x is zero.

To find the y-intercept, you can substitute x = 0 into your line equation and solve for y. In our worked example, we found that the y-intercept b is -\( \frac{1}{3} \). This means the line crosses the y-axis at (0, -\( \frac{1}{3} \)).

coordinate geometry

Coordinate geometry, also called analytic geometry, is the study of geometry using a coordinate system. It provides a link between algebra and geometry through graphs of lines and curves. In this field, every point is defined by a pair of numerical coordinates:

• The x-coordinate represents the point's horizontal position.
• The y-coordinate represents the point's vertical position.

These coordinates are usually written in the form \( (x, y) \). Through coordinate geometry, we can use equations to describe lines, circles, and other geometric shapes. In the given exercise, you are plotting a line through points that satisfy the slope-intercept equation: \( y = \frac{2}{3}x - \frac{1}{3} \). The concepts of the slope and y-intercept help to place these points accurately on a coordinate plane, allowing you to graph the line effectively.