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Chapter 3: Problem 27

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example \(1 .\) Slope \(-\frac{11}{6},\) passes through \((2,-6)\)

Short Answer

Expert verified
The equation is \( y = -\frac{11}{6}x - \frac{7}{3} \).

Step by step solution

01

Recall the Point-Slope Form Equation

The point-slope form of a line's equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a specific point that the line passes through.
02

Plug in the Given Values into the Point-Slope Equation

Given the slope \(-\frac{11}{6}\) and the point \((2, -6)\), substitute \( m = -\frac{11}{6} \), \( x_1 = 2 \), and \( y_1 = -6 \) into the point-slope form equation. This gives us: \[ y - (-6) = -\frac{11}{6}(x - 2) \] Simplify to: \[ y + 6 = -\frac{11}{6}(x - 2) \]
03

Simplify the Right Side of the Equation

Distribute the slope \(-\frac{11}{6}\) to the terms inside the parenthesis: \[ y + 6 = -\frac{11}{6}x + \frac{11}{3} \] This step involves careful multiplication of fractions.
04

Solve for \(y\) to Convert to Slope-Intercept Form

To convert the equation from point-slope to slope-intercept form, solve for \(y\):\[ y = -\frac{11}{6}x + \frac{11}{3} - 6 \] Convert \(6\) to a fraction with a denominator of 3: \(6 = \frac{18}{3}\). Then, subtract: \[ y = -\frac{11}{6}x - \frac{7}{3} \]
05

Write the Final Equation in Slope-Intercept Form

The equation in slope-intercept form is: \( y = -\frac{11}{6}x - \frac{7}{3} \) where \(-\frac{11}{6}\) is the slope and \(-\frac{7}{3}\) is the y-intercept.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most popular ways to express a straight line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.

Why is this form so important?
  • It directly shows the slope, which indicates how steep the line is.
  • The y-intercept makes it easy to graph the line, as you start your graph where the line intersects the y-axis.
  • It simplifies understanding the relationship between the variables, offering a clear picture of how changes in one variable affect the other.
By converting an equation into slope-intercept form, you gain a straightforward way to visualize and analyze a line. This makes it particularly useful in algebra problem solving. As shown in the solution, converting from point-slope form to slope-intercept form can help you understand the line better.
Equation of a Line
The equation of a line is a mathematical statement representing all the points that lie on a straight line. Essentially, it defines the relationship between the variables \( x \) and \( y \) along that line.

There are several forms in which you can express the equation of a line:
  • **Standard Form:** \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. This form is useful for analyzing the line's position concerning other lines.
  • **Point-Slope Form:** \( y - y_1 = m(x - x_1) \), which is beneficial when you have a known point and the slope.
  • **Slope-Intercept Form:** \( y = mx + b \), as previously discussed, provides ease when graphing and understanding the line's incline and y-interception.
In our specific exercise, we utilized the point-slope form to determine the initial equation considering a specific point on the line, then translated it to slope-intercept form for simplicity in interpretation.
Algebra Problem Solving
Algebra problem solving often involves manipulating equations to uncover hidden patterns and insights. The ability to shift between different types of line equations, like from point-slope to slope-intercept form, is a valuable skill in mathematical problem solving.

Here are steps often used in problem-solving processes:
  • **Identify What You Know:** Understand the given values, like slopes and points.
  • **Choose a Strategy:** Decide the best form or method to use for the equation, such as point-slope or slope-intercept form.
  • **Substitute Values:** Plug in the known values, as shown when substituting the slope \(-\frac{11}{6}\) and the point \((2, -6)\) into the point-slope equation.
  • **Simplify and Convert:** Break down the equation into simpler parts and convert if needed, for instance, going to slope-intercept form to simplify graph plotting.
These steps are essential for tackling algebraic challenges, providing a structured approach to solving complex problems in a systematic manner.

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Most popular questions from this chapter

Find the slope of each line. See Examples 4 and \(5 .\) $$3 x=-12$$

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((8,-3)\) and \((8,-8)\) \((11,3)\) and \((22,3)\)

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &y=-9 x-3\\\ &y=-9 x \end{aligned} $$

Is the graph of \(y \geq 3-x\) a function? Explain. CAN'T COPY THE GRAPH

In the function \(y=-5 x+2,\) why do you think the value of \(x\) is called the input and the corresponding value of \(y\) the output?
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