Points on a Line and Solving for Unknowns

TL;DR

When a point is on a line, its coordinates make the line's equation true. You can use this fact to check if a point is on a line or to find an unknown coordinate. Substitute the known values and solve the resulting equation for the missing piece.

1. The Mental Model

Imagine a line as a path, and a point as a specific spot. If the spot is on the path, its location perfectly fits the path's description (its equation). If it's not on the path, its location won't fit.

2. The Core Material

Lines in algebra are usually described by equations like y = mx + b (slope-intercept form) or Ax + By = C (standard form). Every point on that line has an x coordinate and a y coordinate that, when plugged into the equation, make the equation true.

Checking if a Point is on a Line

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To check if a given point (x, y) lies on a line, you just substitute its x and y values into the line's equation. If both sides of the equation are equal, the point is on the line. If they're not equal, it's not.

Example: Is the point (2, 7) on the line y = 3x + 1?
Substitute x = 2 and y = 7:
7 = 3(2) + 1
7 = 6 + 1
7 = 7
Since 7 = 7 is true, the point (2, 7) is on the line.

Solving for an Unknown Coordinate

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Often, you'll be given a point with one coordinate missing (e.g., (3, y)) and told it's on a specific line. Your job is to find that missing coordinate. The process is similar: substitute the known coordinate into the equation, then solve for the unknown.

Let's use a diagram to illustrate the process of solving for an unknown:

graph TD
    A["Start: Given a point (x, y) with one unknown and a line equation"] --> B["Substitute the known x or y value into the line's equation"]
    B --> C["Simplify the equation"]
    C --> D["Isolate the variable (the unknown coordinate)"]
    D --> E["Solve for the unknown value"]
    E --> F["End: The point with the found coordinate"]

Example: The point (-1, y) is on the line y = -2x + 5. What's the value of y?
Here, you know x = -1.
Substitute x = -1 into the equation:
y = -2(-1) + 5
y = 2 + 5
y = 7
So, the full point is (-1, 7).

Example: The point (x, 10) is on the line 3x + y = 16. What's the value of x?
Here, you know y = 10.
Substitute y = 10 into the equation:
3x + 10 = 16
Subtract 10 from both sides:
3x = 16 - 10
3x = 6
Divide by 3:
x = 6 / 3
x = 2
So, the full point is (2, 10).

3. Worked Example

A construction crew is laying a pipeline. The path of the pipeline can be described by the equation y = (1/2)x - 3, where x and y are distances in meters from a reference point. A supply dump is planned at coordinates (10, k). If the supply dump must be exactly on the pipeline, what must k be?

Here, x = 10 and y = k. We need to find k.

1. Substitute the known values into the equation:
   k = (1/2)(10) - 3

2. Simplify the equation:
   k = 5 - 3

3. Solve for k:
   k = 2

So, the supply dump must be located at (10, 2) to be on the pipeline.

4. Key Takeaways

• Every point on a line satisfies that line's equation.
• To check if a point is on a line, substitute its coordinates into the equation and verify if it holds true.
• If a point has an unknown coordinate and is on a line, substitute the known coordinate and solve the equation for the unknown.
• The x coordinate is always the first number in the (x, y) pair, and y is the second.
• Simplifying expressions and solving linear equations are crucial skills for this topic.

Common Mistakes to Avoid:
- Swapping x and y when substituting them into the equation.
- Not following the order of operations (PEMDAS/BODMAS) when simplifying.
- Making arithmetic errors when solving the resulting equation.
- Forgetting to find the value of the unknown, rather than just plugging things in.

5. Now Try It

The line 4x - 2y = 8 passes through the point (p, -4). Find the value of p.

What to do: Substitute the known y value into the equation and solve for p.
What success looks like: You should find a single numerical value for p.