/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Find the slope of each line. \... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 43

Find the slope of each line. \(x=2 y\)

Short Answer

Expert verified
The slope is \(\frac{1}{2}\).

Step by step solution

01

Rearrange the Equation

The given equation is written as \(x = 2y\). To find the slope, we should rearrange it in slope-intercept form \(y = mx + b\). Let's isolate \(y\) by dividing both sides of the equation by \(2\): \[y = \frac{x}{2}\].
02

Identify the Slope

Once the equation is in the form \(y = mx + b\), the slope \(m\) is the coefficient of \(x\). In \(y = \frac{1}{2}x\), the slope \(m\) is \(\frac{1}{2}\).

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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 is a way of expressing linear equations. It makes it simple to identify the slope of a line and its y-intercept. The standard format for slope-intercept form is \( y = mx + b \). In this equation:
  • \(y\): The dependent variable
  • \(m\): The slope of the line
  • \(x\): The independent variable
  • \(b\): The y-intercept, where the line crosses the y-axis
The magic of the slope-intercept form is its ability to reveal the slope \( m \) directly as the coefficient of \( x \). This makes it an efficient tool for quickly understanding how steep a line is and which direction it tilts on a graph. Anytime you need to find the slope, rearranging a linear equation into this form simplifies your task.
Linear Equations
Linear equations are equations of the first degree, which means they have no exponents greater than one. They graph as straight lines on a coordinate plane. Their general form is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
To better comprehend how linear equations behave, let's discuss some of their key features:
  • Linearity: As the name suggests, linear equations produce straight lines. This property arises because the change between any two points on the line is constant.
  • Slope: The slope is a measure of the line's steepness. It indicates how much \( y \) changes for a change in \( x \). For positive slopes, the line inclines, whereas negative slopes lead to a decline.
  • Intersections: Linear equations generally intersect the x-axis and the y-axis at specific points, determined by setting each variable to zero.
Understanding these aspects helps in visualizing and interpreting data points on a graph.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for unknown variables or to convert equations into a useful form. In the context of the original exercise, algebraic manipulation is crucial for finding the slope of the line.

Here's how algebraic manipulation is typically carried out:
  • Rearranging Terms: Move terms around in the equation to isolate the variable you are solving for. Prioritize getting all terms with \( y \) (or any desired variable) on one side of the equation.
  • Division or Multiplication: Use multiplication or division to simplify the equation. In the example \( x = 2y \), dividing both sides by two isolates \( y \).
  • Identifying Coefficients: Once in the form \( y = mx + b \), the slope \( m \) can be read directly.
Algebraic manipulation is a fundamental skill in mathematics, allowing us to transform equations and unveil useful properties such as the slope in linear equations.

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