Problem 27 - 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. $$(-2,-4), m=\frac{1}{2}$$

Step-by-Step Solution

Answer

Three points are (0, -3), (2, -2), and (4, -1).

Step 1: Understand the Slope-Point Formula

The slope-point form of a line is given by the formula: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. Substitute \((-2, -4)\) and \(m=\frac{1}{2}\) into the formula.

Step 2: Write the Equation of the Line

Substitute the values into the formula: \( y + 4 = \frac{1}{2}(x + 2) \). Simplify to express \(y\) in terms of \(x\). Distribute and solve: \(y + 4 = \frac{1}{2}x + 1\) then \(y = \frac{1}{2}x - 3\).

Step 3: Choose Values for x

To find other points, choose convenient values for \(x\). For example, try \(x = 0, 2,\) and \(4\).

Step 4: Calculate y for x = 0

When \(x = 0\), plug into the equation: \(y = \frac{1}{2}(0) - 3 = -3\). The point is \((0, -3)\).

Step 5: Calculate y for x = 2

When \(x = 2\), plug into the equation: \(y = \frac{1}{2}(2) - 3 = 1 - 3 = -2\). The point is \((2, -2)\).

Step 6: Calculate y for x = 4

When \(x = 4\), plug into the equation: \(y = \frac{1}{2}(4) - 3 = 2 - 3 = -1\). The point is \((4, -1)\).

Key Concepts

Slope-Point Formula

One of the key tools in coordinate geometry is the slope-point formula. This formula helps to create the equation of a line from a given point and a slope. It is written as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope of the line.
The slope describes the line's steepness, indicating how much the line rises or falls as it moves horizontally. By substituting the values into the formula, we can start constructing the line's equation.
For example, with the point \((-2,-4)\) and slope \(m = \frac{1}{2}\), you substitute them into the formula as follows:

• \( y + 4 = \frac{1}{2}(x + 2) \)

This step lays the foundation for finding the line's equation, which we'll explore in the next section.

Equation of a Line

Once you have substituted your values into the slope-point formula, the next step is to solve it to find the equation of the line in the slope-intercept form \(y = mx + b\).
Let's continue with our example from before. We started with:

• \( y + 4 = \frac{1}{2}(x + 2) \)

The goal is to express \(y\) solely in terms of \(x\). Begin by distributing the slope across \(x + 2\):

• \( y + 4 = \frac{1}{2}x + 1 \)

Next, isolate \(y\) by subtracting 4 from both sides:

• \( y = \frac{1}{2}x - 3 \)

This is the equation of the line where \(\frac{1}{2}\) is the slope and \(-3\) is the y-intercept, the value of \(y\) when \(x=0\). This equation describes a line that extends infinitely in both directions on the graph.

Finding Points on a Line

With the equation of a line in hand, finding additional points on that line involves selecting different \(x\) values and computing the corresponding \(y\) values. This process allows us to identify specific coordinate points that satisfy the line's equation.
Using our line equation \(y = \frac{1}{2}x - 3\), let's choose some \(x\) values:

• When \(x = 0\), \(y = \frac{1}{2}(0) - 3 = -3\). Thus, the point is \((0, -3)\).
• When \(x = 2\), \(y = \frac{1}{2}(2) - 3 = -2\). Thus, the point is \((2, -2)\).
• When \(x = 4\), \(y = \frac{1}{2}(4) - 3 = -1\). Thus, the point is \((4, -1)\).

You can choose any \(x\) value, and this method will produce a corresponding \(y\) value, contributing to a clear understanding of how the line behaves on the Cartesian plane. By repeating this process, you can plot numerous points, effectively mapping out the line.