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Chapter 4: Problem 99

Find an equation of the line with slope \(-\frac{11}{6}\) that passes through \((2,-6) .\) Write the equation in slope-intercept form.

Short Answer

Expert verified
The equation is \(y = -\frac{11}{6}x - \frac{25}{6}\).

Step by step solution

01

Identify Given Values

We are given the slope \(m = -\frac{11}{6}\) and the point \((x_1, y_1) = (2, -6)\). Our goal is to find the equation of the line using these values.
02

Substitute Values into Point-Slope Formula

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\). Substituting the given values gives us: \[y + 6 = -\frac{11}{6}(x - 2)\]
03

Distribute the Slope

Distribute the slope \(-\frac{11}{6}\) across the terms in the parenthesis:\[y + 6 = -\frac{11}{6}x + \frac{11}{3}\]
04

Solve for y to Get Slope-Intercept Form

Subtract 6 from both sides of the equation to solve for \(y\):\[y = -\frac{11}{6}x + \frac{11}{3} - 6\]Convert \(-6\) to a fraction with the same denominator to combine:\[y = -\frac{11}{6}x + \frac{11}{3} - \frac{36}{6}\]Combine the constants:\[y = -\frac{11}{6}x - \frac{25}{6}\]
05

State the Final Equation

The equation of the line in slope-intercept form is:\[y = -\frac{11}{6}x - \frac{25}{6}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Line
The equation of a line is a mathematical representation that describes all the points along that line. It's a fundamental concept in algebra and geometry, enabling us to understand the relationship between variables. This equation can take many forms, each suitable for different problem-solving scenarios.
In basic linear algebra, the two most common forms are the **slope-intercept form** and the **point-slope form**.
  • **Slope-Intercept Form:** This is expressed as \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept (the point where the line crosses the y-axis). It's straightforward and often used for quickly graphing lines.
  • **Point-Slope Form:** This is written as \(y - y_1 = m(x - x_1)\). It is particularly useful when you know a point on the line and the slope. From this, you can easily derive the slope-intercept form by solving for \(y\).
Point-Slope Form
The point-slope form is a convenient way to write the equation of a line when you know exactly one point through which the line passes and the slope of the line. This form is written as:
\(y - y_1 = m(x - x_1)\)
where:
  • \( (x_1, y_1) \) are the coordinates of the given point on the line
  • \( m \) represents the slope of the line
This method is powerful because you can quickly transition from a single point and a slope to the full linear equation. Here's an example:
Imagine you have a line with a slope of \(-\frac{11}{6}\) that passes through the point \((2, -6)\). By substituting into the formula, you get:
\[y + 6 = -\frac{11}{6}(x - 2)\]
This equation expresses how every point \((x, y)\) on the line relates to your known point \((2, -6)\) in terms of the slope.
Slope
The slope is a measure of the steepness or incline of a line, expressed as a ratio between the vertical change and the horizontal change between two points on the line. It's often denoted by \(m\) and calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
The slope tells us how much the \(y\)-value (vertical position) changes for a given change in the \(x\)-value (horizontal position). A few things to note about slope:
  • **Positive Slope:** The line rises as you move from left to right.
  • **Negative Slope:** The line falls as you move from left to right, just like in our exercise where the slope \(-\frac{11}{6}\) indicates a fall.
  • **Zero Slope:** The line is horizontal, indicating no change in \(y\) as \(x\) changes.
  • **Undefined Slope:** Occurs in vertical lines where there is no horizontal change and division by zero happens.
By understanding the slope, you can tell a lot about the line and its behavior just from its equation, even without seeing it graphed!

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Most popular questions from this chapter

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {x=3} \\ {3 y=6-2 x} \end{array}\right. $$

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {x+2 y=-4} \\ {x-\frac{1}{2} y=6} \end{array}\right. $$

When using the substitution method, how can you tell whether a. a system of linear equations has no solution? b. a system of linear equations has infinitely many solutions?

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {3 x-6 y=18} \\ {x=2 y+3} \end{array}\right. $$

Without graphing, how can you tell that the graphs of \(y=2 x+1\) and \(y=3 x+2\) intersect?
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