Problem 25 - Elementary Algebra

Question

You are given one point on a line and the slope of the line. Find the coordinates of three other points on the line. $$(3,2), m=\frac{2}{3}$$

Step-by-Step Solution

Answer

The points are (6, 4), (9, 6), and (0, 0).

Step 1: Understand the Equation of a Line

The equation of a line in point-slope form is given as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Here, \(x_1 = 3\), \(y_1 = 2\), and \(m = \frac{2}{3}\).

Step 2: Compute the General Equation of the Line

Substitute \(x_1\), \(y_1\), and \(m\) into the point-slope equation: \(y - 2 = \frac{2}{3}(x - 3)\). Simplify this to get the line equation in slope-intercept form by distributing the slope and solving for \(y\).

Step 3: Expand and Simplify the Equation

Distribute \(\frac{2}{3}\) in the equation: \(y - 2 = \frac{2}{3}x - 2\). Then add 2 to both sides to get \(y = \frac{2}{3}x\). This is the simplified equation of the line.

Step 4: Choose Values for \(x\) to Find Corresponding \(y\)

Pick different values for \(x\). Let's choose \(x = 6, 9, 0\). Substitute each \(x\) into the equation \(y = \frac{2}{3}x\) to find \(y\) values.

Step 5: Compute Coordinates for Chosen \(x\) Values

- For \(x = 6\): \(y = \frac{2}{3} \times 6 = 4\), so point is \((6, 4)\).- For \(x = 9\): \(y = \frac{2}{3} \times 9 = 6\), so point is \((9, 6)\).- For \(x = 0\): \(y = \frac{2}{3} \times 0 = 0\), so point is \((0, 0)\).

Key Concepts

Slope

The slope is a fundamental concept in algebra when dealing with lines on a graph. Essentially, the slope measures the steepness or incline of a line. More formally, it shows how much the line rises or falls for each unit of run. In our given exercise, the slope is \( m = \frac{2}{3} \). This means that for every 3 units we move horizontally (to the right on a graph), the line moves 2 units vertically (upwards).
Understanding this ratio helps in visualizing how quickly a line rises or drops as you move along it.

• Rise: The vertical change between two points on the line (in this case, 2).
• Run: The horizontal change between those same two points (in this case, 3).

This concept is not only essential for graphing lines but also for solving problems involving linear equations and predicting future points on the line.

Point-Slope Form

The point-slope form is a convenient way to write the equation of a line when you know one point on the line and the slope. The general formula is \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) represents a known point on the line, and \( m \) is the slope we talked about earlier.
In the exercise, the given point is \((3, 2)\), and the slope \( m = \frac{2}{3} \).

• Step 1: Substitute \( x_1 = 3 \), \( y_1 = 2 \), and \( m = \frac{2}{3} \) into the formula to create the equation of our line.
• Step 2: This gives us \( y - 2 = \frac{2}{3}(x - 3) \).

Using the point-slope form, you can quickly determine other points on the line by selecting different \( x \) values and solving for \( y \). This method ensures that all calculated points will lie on the same straight line.

Equation of a Line

Once we have a point-slope equation, it's possible to simplify it to the slope-intercept form \( y = mx + b \) for ease of further computation or graphing. For our exercise, the step-by-step solution guides us to simplify by distributing and combining like terms.

• Expand: Distribute \( \frac{2}{3} \) across \( (x - 3) \) to simplify the right side of \( y - 2 = \frac{2}{3}(x - 3) \).
• Simplify: Solve for \( y \) by adding 2 to both sides, yielding \( y = \frac{2}{3}x \).

This is the slope-intercept form, which makes it straightforward to choose values for \( x \) to find corresponding \( y \). By plugging in different \( x \) values, such as 6, 9, or 0, you can glean additional points like \((6, 4)\), \((9, 6)\), and \((0, 0)\) all accurately lying on the line represented by our equation. This form is especially useful for quickly determining the y-value of any point on the line once an x-value is given.