/*! 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 38 Find the slope of each line. \... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 38

Find the slope of each line. \(-5 x+y=10\)

Short Answer

Expert verified
The slope of the line is 5.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in the standard form: \(-5x + y = 10\). The standard form of a linear equation is \(Ax + By = C\). Our goal is to find the slope of the line represented by this equation.
02

Convert to Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope. Start by isolating \(y\) in the equation \(-5x + y = 10\):Add \(5x\) to both sides: \[y = 5x + 10\]
03

Identify the Slope

Now, the equation is in the slope-intercept form \(y = mx + b\). Here, \(m = 5\) which represents the slope of the line.

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Key Concepts

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

Linear Equation
A linear equation is an equation that creates a straight line when graphed on a coordinate plane. It's one of the simplest forms of equations you'll encounter in algebra. In a two-dimensional space, a linear equation can be represented as:
  • \(Ax + By = C\), known as the standard form.
  • \(y = mx + b\), known as the slope-intercept form.
Linear equations are characterized by their constant rate of change or slope and can have one, infinitely many, or no solutions at all, depending on their configuration. Understanding linear equations is foundational for solving more complex mathematical problems. Think of them as the most basic expressions of change over a defined set of points, which forms the path of a line on the graph.
Standard Form
The standard form of a linear equation is written as \(Ax + By = C\), where:
  • \(A\), \(B\), and \(C\) are constants.
  • \(x\) and \(y\) are the variables.
This form is particularly useful for quickly identifying the relationship between \(x\) and \(y\) without needing to draw the line first. In our example, the given equation is \(-5x + y = 10\), which fits perfectly into the standard form. Transforming the equation into another form can help identify specific characteristics like the slope or y-intercept more easily. This process usually involves rearranging the terms to solve for \(y\). Though it might seem complex at first, it's a straightforward conversion that aids in understanding the geometric representation of the equation.
Slope-Intercept Form
The slope-intercept form of a linear equation is \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept鈥攚here the line crosses the y-axis. For the equation \(y = 5x + 10\), it's clear that the slope \(m\) is 5.
  • The slope, \(m\), indicates the steepness of the line and the direction in which the line moves. A positive slope means the line rises as it moves from left to right.
  • The y-intercept, \(b\), shows the point at which the line crosses the y-axis.
Reformatting to the slope-intercept form makes it easier to understand the dynamics of a line because you can quickly see the impact of changes in \(x\) on \(y\). This form is particularly useful for graphing and analyzing linear relationships, giving an immediate visual sense of how input values affect output results.

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Most popular questions from this chapter

Write the equations of three lines parallel to \(10 x-5 y=-7\) (See the third Concept Check.)

Write an equation in standard form of the line that contains the point \((4,0)\) and is a. parallel to the line \(y=-2 x+3\) b. perpendicular to the line \(y=-2 x+3\)

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 value of \(x^{2}-3 x+1\) for each given value of \(x .\) See Section 1.7. $$ 2 $$

Solve each equation for \(y .\) See Section 2.5 \(5 x+2 y=7\)
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