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Chapter 3: Problem 4

Write an equation of the line with each given slope, \(m,\) and \(y\) -intercept, \((0, b) .\) See Example \(1\). $$ m=2, b=\frac{3}{4} $$

Short Answer

Expert verified
The equation is \( y = 2x + \frac{3}{4} \).

Step by step solution

01

Understanding the Line Equation

The general form of the equation of a line with given slope \( m \) and y-intercept \( b \) is \( y = mx + b \). This equation represents a linear function where \( m \) is the slope and \( b \) is the y-coordinate where the line intersects the y-axis.
02

Substitute the Given Values

We are given the slope \( m = 2 \) and the y-intercept \( b = \frac{3}{4} \). Substitute these values into the line equation to get \( y = 2x + \frac{3}{4} \).
03

Write the Final Equation

After substituting the given values, the equation of the line is written as \( y = 2x + \frac{3}{4} \). This is the equation of the line with slope 2 and y-intercept \( \frac{3}{4} \).

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Key Concepts

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

Slope-Intercept Form
In mathematics, the slope-intercept form is a widely used way to express the equation of a straight line. This form is special because it straightforwardly reveals two essential characteristics of the line: its slope and its y-intercept. The formula is articulated as \( y = mx + b \), where:
  • \( y \) is the dependent variable/output, corresponding to any point along the line.
  • \( x \) is the independent variable/input, which you select along the x-axis.
  • \( m \) is the slope of the line, indicating the angle or steepness of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
Using the slope-intercept form, you can quickly construct the graph of the line and identify key features. It simplifies the task of interpreting and studying linear equations.
Slope of a Line
The slope of a line \( m \) is a fundamental aspect in the study of linear equations. It essentially measures the steepness of a line and determines how much the line rises or falls as you move from one point to another along the line.
  • If the slope \( m \) is positive, the line ascends when moving from left to right.
  • If the slope \( m \) is negative, the line descends.
  • If the slope \( m \) is zero, the line is horizontal, indicating no rise or fall.
  • If the slope \( m \) is undefined, the line is vertical, only moving up or down.
One can calculate the slope given two points, \((x_1, y_1)\) and \((x_2, y_2)\), on the line by using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]The slope is crucial for understanding how fast or slow changes occur between variables.
Y-Intercept
The y-intercept in a line's equation is a key component that determines where the line intersects the y-axis. It is represented by the \( b \) in the slope-intercept form \( y = mx + b \). The y-intercept provides an anchor or starting point from which the rest of the line can be drawn.
  • When \( x \) is zero, \( y \) equals \( b \), making \( (0, b) \) the coordinates of the y-intercept.
  • This point reflects the value of \( y \) when there is no contribution from the \( x \) variable, essentially serving as a baseline.
Understanding the y-intercept is crucial when graphing linear equations, as it helps establish the exact position of the line on a graph. Knowing the y-intercept assists in sketching the line correctly and provides deep insight into the equation's behavior at the starting point.

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Most popular questions from this chapter

Answer each exercise with true or false. Point \((3,0)\) lies on the \(y\) -axis.

Write an equation in standard form of the line that contains the point \((3,-5)\) and is a. parallel to the line \(3 x+2 y=7\) b. perpendicular to the line \(3 x+2 y=7\)

The graph of \(y=-\frac{1}{3} x+2\) has a slope of \(-\frac{1}{3} .\) The graph of \(y=-2 x+2\) has a slope of \(-2 .\) The graph of \(y=-4 x+2\) has a slope of \(-4 .\) Graph all three equations on a single coordinate system. As the absolute value of the slope becomes larger, how does the steepness of the line change?

Find the slope of the line that is (a) parallel and (b) perpendicular to the line through each pair of points. \((-8,-4)\) and \((3,5)\)

Find the slope of each line. \(-3 x-4 y=6\)
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