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Chapter 3: Problem 73

Use the point-slope formula to find the equation of the line passing through the two points. $$ (-3,-1),(3,3) $$

Short Answer

Expert verified
The equation of the line is \(y = \frac{2}{3}x + 1\).

Step by step solution

01

Identify the points

We are given the points \((-3, -1)\) and \((3, 3)\) through which the line passes.
02

Calculate the slope

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) to calculate the slope, where \((x_1, y_1) = (-3, -1)\) and \((x_2, y_2) = (3, 3)\). So, \(m = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3}\).
03

Use the point-slope form

The point-slope form of a line is given as \(y - y_1 = m(x - x_1)\). We can use either of the given points, let's use \((-3, -1)\). Substitute \(m = \frac{2}{3}\), \(x_1 = -3\), and \(y_1 = -1\) into the formula: \[y - (-1) = \frac{2}{3}(x - (-3))\].
04

Simplify the equation

Simplify the equation \[y + 1 = \frac{2}{3}(x + 3)\] to \[y + 1 = \frac{2}{3}x + 2\].Subtract 1 from both sides to get:\[y = \frac{2}{3}x + 2 - 1\].Thus, the equation simplifies to \[y = \frac{2}{3}x + 1\].
05

Write the equation

The final equation of the line passing through the given points is \[y = \frac{2}{3}x + 1\].

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Key Concepts

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

Understanding the Equation of a Line
Finding the equation of a line is an essential skill in algebra, helping you to describe relationships between different quantities. An equation of a line mathematically represents how every y-coordinate relates to x-coordinates in a linear manner. For example, the generic form of a linear equation in two variables is given by \(y = mx + b\), where:
  • \(m\) represents the slope of the line.
  • \(b\) is the y-intercept, which is where the line crosses the y-axis.

By using specific forms, like the point-slope formula, we can quickly write equations for lines that pass through certain points, making them ideal for modeling real-world situations. This principle is exactly what we use when finding equations for lines, ensuring we accurately describe the path the line takes across a graph.
Mastering Slope Calculation
The slope of a line is a measure of its steepness and direction. It tells us how much the y-coordinate changes for a specific change in the x-coordinate between two different points. In mathematical terms, the slope \(m\) is calculated as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula requires two sets of points: \((x_1, y_1)\) and \((x_2, y_2)\). By subtracting the y-values and the x-values, you determine how vertical and horizontal differences between the points compare. A positive slope means the line rises as it moves from left to right, whereas a negative slope means it falls. Understanding slope calculation is fundamental for creating and interpreting linear relationships.
Exploring Linear Equations
Linear equations form the backbone of algebra because they provide a straightforward way to represent relationships between variables. When simplified, the equation becomes explicit, showing a constant rate of change. Whether formulated as standard form \(Ax + By = C\) or slope-intercept form \(y = mx + b\), each version serves a unique purpose:
  • The point-slope form \(y - y_1 = m(x - x_1)\) is perfect for constructing the equation when a point and slope are given.
  • The slope-intercept form is great for quickly identifying the slope and y-intercept directly from the equation.

By manipulating these forms, you can transform real-life situations, such as constant velocity or linear growth, into solvable mathematical problems. Linear equations are versatile, making them an essential tool in both academics and beyond.

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Most popular questions from this chapter

Find the equation of a line, given the slope and a point on the line. $$ m=1 / 2 ;(-4,8) $$

Find at least five ordered pair solutions and graph. $$ y=3 $$

Graph the solution set. $$ y \geq-2 x+1 $$

Calculate the midpoint between the given points. (-6,0) and (0,0)

Graph the linear function and state the domain and range. $$ g(x)=-4 x+6 $$
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