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Chapter 8: Problem 32

Find an equation of the line passing through the given points. Use function notation to write the equation. $$ (-9,-2),(-3,10) $$

Short Answer

Expert verified
The equation of the line is \(f(x) = 2x + 16\).

Step by step solution

01

Identify the Slope Formula

The first step is to use the formula for the slope of a line connecting two points \((x_1, y_1)\) and \((x_2, y_2)\), which is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
02

Substitute Points into the Slope Formula

Substitute the given points \((-9, -2)\) and \((-3, 10)\) into the slope formula: \[ m = \frac{10 - (-2)}{-3 - (-9)} = \frac{12}{6} = 2 \]. Thus, the slope \(m\) of the line is 2.
03

Write the Point-Slope Form Equation

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Substitute the slope \(m = 2\) and one of the points, say \((-9, -2)\), into this equation: \[ y - (-2) = 2(x - (-9)) \].
04

Simplify the Equation to Function Notation

Simplify the equation from the previous step: \(y + 2 = 2(x + 9)\). Distribute and simplify to get \(y + 2 = 2x + 18\), then subtract 2 from both sides to get the function \(y = 2x + 16\).
05

Write the Final Equation in Function Notation

The equation of the line in function notation is \(f(x) = 2x + 16\), representing the line passing through the given points.

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Key Concepts

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

Slope Formula
The slope formula is a key tool in understanding linear equations. It is used to determine the steepness or incline of a line by calculating the ratio of the vertical change (rise) to the horizontal change (run) between two points. This is expressed mathematically as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For the points given in our exercise, \((-9, -2)\) and \((-3, 10)\), the slope \(m\) is found by:
  • Subtracting the \(y\) values: \(10 - (-2) = 12\).
  • Subtracting the \(x\) values: \(-3 - (-9) = 6\).
  • Dividing the differences: \(m = \frac{12}{6} = 2\).
This tells us that for every unit increase in \(x\), \(y\) increases by 2 units.
Point-Slope Form
Once the slope is known, the point-slope form of a linear equation comes into play. This is a convenient form for writing the equation of a line when a point on the line and the slope are known. The point-slope form is expressed as \(y - y_1 = m(x - x_1)\). With the slope \(m = 2\) and the point \((-9, -2)\), we substitute into the equation:
  • The equation becomes \(y - (-2) = 2(x - (-9))\).
  • This simplifies to \(y + 2 = 2(x + 9)\).
This equation is instrumental in linear algebra as it represents the line through a specific point with the calculated slope.
Function Notation
Function notation is a way of expressing the relationship between \(x\) and \(y\), where \(y\) is a function of \(x\). In mathematical terms, this is written as \(f(x)\). Once you've simplified the point-slope expression, you can rewrite it in function notation. Our earlier point-slope form \(y + 2 = 2(x + 9)\) simplifies further to:
  • Distribute \(2\) to both terms: \(y + 2 = 2x + 18\).
  • Isolate \(y\) by subtracting 2: \(y = 2x + 16\).
Finally, rewriting this in function notation, \(f(x) = 2x + 16\), makes it clear that \(y\) depends on \(x\), reflecting the linear relationship between them.
Algebraic Methods
Algebraic methods are the systematic procedures used to find the solutions in math problems like ours. Here's a step-by-step recap using algebraic methods when dealing with linear equations:
  • First, calculate the slope using the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  • Next, apply the point-slope form: \(y - y_1 = m(x - x_1)\).
  • Simplify the equation, making it easier to interpret and use.
  • Use algebraic manipulation to express the equation in function notation: \(f(x) = mx + b\).
Algebraic methods ensure accuracy and clarity, making it possible to communicate the problem and its solution effectively. By understanding these methods, you advance your skills not only in solving equations but in analyzing and interpreting mathematical relationships.

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Most popular questions from this chapter

Complete the following table for the inverse variation \(y=\frac{k}{x}\) over each given value of \(k .\) Plot the points on a rectangular coordinate system. $$ \begin{array}{c|c|c|c|c|c|} \hline x & {\frac{1}{4}} & {\frac{1}{2}} & {1} & {2} & {4} \\ \hline y {=\frac{k}{x}} & {} & {} & {} & {} \\ \hline \end{array} $$ $$ k=3 $$

Find an equation of the perpendicular bisector of the line segment whose endpoints are given. See the previous example. $$ (-6,8):(-4,-2) $$

Simplify. $$ \left(\frac{2}{3}\right)^{3} $$

Simplify. $$ -|7| $$

For each statement, find the constant of variation and the variation equation. See Examples 5 and 6 \(y\) varies jointly as \(x\) and the square of \(z ; y=360\) when \(x=4\) and \(z=3\)
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