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Chapter 2: Problem 36

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. Through \(A(-3,1)\) and \(B(2,7)\)

Short Answer

Expert verified
The slope-intercept form is \(y = \frac{6}{5}x + \frac{23}{5}\).

Step by step solution

01

Identify the formula for slope

The slope of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
02

Substitute points into slope formula

Substitute the coordinates from points \(A(-3, 1)\) and \(B(2, 7)\) into the slope formula: \[ m = \frac{7 - 1}{2 - (-3)} = \frac{6}{5} \] The slope \(m\) is \(\frac{6}{5}\).
03

Recall the slope-intercept form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.
04

Use a point to solve for y-intercept

Using point \(A(-3, 1)\) and the slope \(m = \frac{6}{5}\), substitute into the slope-intercept form to find \(b\): \[ 1 = \frac{6}{5}(-3) + b \] \[ 1 = -\frac{18}{5} + b \] \[ b = 1 + \frac{18}{5} = \frac{5}{5} + \frac{18}{5} = \frac{23}{5} \]
05

Write the slope-intercept form

Substitute the slope \(m = \frac{6}{5}\) and y-intercept \(b = \frac{23}{5}\) back into the slope-intercept form: \[ y = \frac{6}{5}x + \frac{23}{5} \]

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Key Concepts

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

Slope Formula
In coordinate geometry, the slope represents how inclined a line is with respect to the horizontal. To find the slope of a line when given two points, we use the slope formula, which plays a crucial role in forming and understanding linear equations. The slope formula is given by:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
  • \( m \) stands for the slope found between two points \((x_1, y_1)\) and \((x_2, y_2)\)
For example, in the given exercise, to determine the slope of the line passing through points \(A(-3,1)\) and \(B(2,7)\), you plug their coordinates into the slope formula and get \( m = \frac{7-1}{2-(-3)} = \frac{6}{5} \). This tells us for every 5 units you move horizontally, the line moves up 6 units.
Y-Intercept
In the slope-intercept form of a linear equation \( y = mx + b \), the y-intercept is denoted by \( b \). It is the point where the line crosses the y-axis. This means that at the y-intercept, the value of \( x \) is zero.
To solve for the y-intercept using a given point and the slope, you just plug them into the slope-intercept form equation.
  • For example, using point \( A(-3, 1) \) and slope \( m = \frac{6}{5} \), do the following:
  • Substitute into the formula: \( 1 = \frac{6}{5}(-3) + b \)
  • Solve for \( b \), which results in \( b = \frac{23}{5} \)
Finding the y-intercept allows us to complete the equation of the line in its slope-intercept form.
Coordinate Geometry
Coordinate Geometry, or analytic geometry, is the study of geometry using a coordinate system. In this system, you can describe geometric figures, such as lines, using equations.
It's a bridge between algebra and geometry, allowing algebraic equations to be represented visually and vice versa.
  • The coordinates \( (x, y) \) are key in mapping points on a plane.
  • Lines can be described with linear equations.
  • The slope tells us the direction and steepness of a line.
  • The y-intercept pinpoints where a line will cross the y-axis.
Coordinate Geometry is essential in practical applications like computer graphics, engineering, and physics, simplifying complex geometric problems into algebraic equations.
Linear Equations
Linear equations graph as straight lines and have a general form. The slope-intercept form of a linear equation is \( y = mx + b \).
  • Here, \( m \) is the slope calculating how steep the line is.
  • \( b \) is the y-intercept showing where the line hits the y-axis.
This form of equation is particularly useful because it gives an instant image of how the line behaves, allowing for quick graphing and easy calculations.
For example, from the problem's step-by-step solution, you find the slope as \( \frac{6}{5} \) and the y-intercept as \( \frac{23}{5} \). These numbers help us construct the equation \( y = \frac{6}{5}x + \frac{23}{5} \), clearly defining the line on the coordinate plane.

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Most popular questions from this chapter

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