Use The Equation Of A Line To Identify A Point The Line Passes Through The Slope Of The Line Y-6=2(x-9)

Use The Equation Of A Line To Identify A Point The Line Passes Through The Slope Of The Line Y-6=2(x-9)

Understanding the equation of a line is fundamental in algebra and coordinate geometry. Whether you're solving for unknown points, graphing lines, or analyzing geometric relationships, knowing how to interpret and manipulate line equations is essential. In this article, we will explore how to use the equation of a line to identify a point that the line passes through, with a specific focus on the given line: Y - 6 = 2(x - 9). This process involves understanding the structure of the line's equation, converting it into different forms, and applying algebraic techniques to find points that satisfy the equation.

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Understanding the Equation of a Line

Before delving into the specifics of the provided line equation, it's important to understand the general forms of a line's equation and what each form represents.

Common Forms of a Line's Equation

There are several standard ways to express the equation of a line:

• Slope-Intercept Form: y = mx + b
• Point-Slope Form: y - y₁ = m(x - x₁)
• Standard Form: Ax + By = C
• Two-Point Form: y - y₁ = ((y₂ - y₁)/(x₂ - x₁))(x - x₁)

Each form is useful in different contexts. The given equation, Y - 6 = 2(x - 9), is in point-slope form, which directly relates to a specific point on the line and its slope.

Understanding Point-Slope Form

The point-slope form is written as:

Y - y₁ = m(x - x₁)

Where:

• (x₁, y₁) is a known point on the line.


• m is the slope of the line.

• Y - 6 = 2(x - 9)

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Analyzing the Given Equation: Y - 6 = 2(x - 9)

The primary goal is to find a point that the line passes through, other than the known point (9, 6). To do this, we need to understand the structure of the line's equation fully.

Rewriting the Equation in Slope-Intercept Form

Converting the point-slope form to the slope-intercept form makes it easier to analyze and identify points:

Starting with:

Y - 6 = 2(x - 9)

Distribute the 2:

Y - 6 = 2x - 18

Add 6 to both sides:

Y = 2x - 18 + 6

Y = 2x - 12

Now, the slope-intercept form is:

Y = 2x - 12

This form clearly shows:

• The slope m = 2


• The y-intercept b = -12

Finding Points on the Line

To identify a point that the line passes through, select a value for x and compute Y:

| x | Y = 2x - 12 | Point (x, Y) |
|---|--------------|--------------|
| 0 | 2(0) - 12 = -12 | (0, -12) |
| 5 | 2(5) - 12 = 10 - 12 = -2 | (5, -2) |
| 10 | 2(10) - 12 = 20 - 12 = 8 | (10, 8) |

Thus, the line passes through the points:

• (0, -12)


• (5, -2)


• (10, 8)

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Using the Equation to Find Specific Points

Suppose the goal is to find a point other than the initial known point (9,6) that the line passes through. Here’s a step-by-step process:

Step 1: Choose a value for x

Select any value for x that is different from 9 to find a new point.

For example, choose x = 4.

Step 2: Calculate y using the slope-intercept form

Y = 2(4) - 12 = 8 - 12 = -4

Therefore, the point is (4, -4).

Step 3: Verify that the point satisfies the original equation

Substitute x = 4 into the original equation:

Y - 6 = 2(x - 9)

Y - 6 = 2(4 - 9) = 2(-5) = -10

Add 6 to both sides:

Y = -10 + 6 = -4

This confirms the point (4, -4) lies on the line.

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General Approach to Using the Equation of a Line to Find Points

The process of identifying points on a line from its equation involves:

• Rewriting the equation in a convenient form (typically slope-intercept or standard form).
• Choosing values for x (or y in some cases).
• Calculating corresponding y (or x) values using the equation.
• Verifying the points satisfy the original equation.

This method is useful in many contexts, including graphing, solving geometric problems, and understanding the relationships between variables.

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Application: Finding a Point on a Line Given in Point-Slope Form

Let's look at a practical example of how to find a point on the line given in point-slope form, such as Y - 6 = 2(x - 9).

Example Problem:

Find an additional point on the line other than the known point (9, 6).

Solution Steps:

• Choose an arbitrary value for x; for simplicity, select x = 7.
1. Calculate Y using the equation:

   Y - 6 = 2(7 - 9)

   Y - 6 = 2(-2) = -4

   Y = -4 + 6 = 2





• Thus, the point (7, 2) is on the line.


2. Verify by plugging into the original equation:

   Y - 6 = 2(x - 9)

   2 - 6 = 2(7 - 9)

   -4 = 2(-2) = -4

   Confirmed!

The point (7, 2) lies on the line.

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Understanding the Slope and Its Significance

The slope of the line, m = 2, indicates the steepness and direction of the line.

Interpreting the Slope

• Positive slope (like 2) means the line rises as x increases.
• Magnitude of the slope indicates steepness; larger absolute values mean steeper lines.
• In this case, for every 1 unit increase in x, Y increases by 2 units.

Using the Slope to Find Points

Knowing the slope, you can also find points relative to a known point:

• Starting at (9, 6), moving 1 unit right (increase x by 1):

Point: (10, 8), which matches our earlier calculation.

• Moving 1 unit left (decrease x by 1):

Point: (8, 4)

This demonstrates how to generate points along the line based on the slope and a known point.

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Graphing the Line Using Its Equation

Graphing is an effective way to visualize the line and verify points.

Steps to Graph the Line:

• Plot the known point (9, 6).
1. Use the slope 2 to find additional points:
   • From (9,6), move 1 unit right to x=10, y=8.
   • From (9,6), move 1 unit left to x=8, y=4.
2. Draw a straight