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Chapter 13: Problem 20

Find the slope and \(y\) -intercept of each line. $$2 x-y=5$$

Short Answer

Expert verified
The slope of the line is \(2\) and the y-intercept is \(-5\).

Step by step solution

01

Rearrange the equation

Begin by rearranging the equation \(2x - y = 5\) to slope-intercept form \(y = mx + b\). To do this, you isolate \(y\) by adding \(y\) to both sides then subtracting \(5\) from both sides. This obtains: \(y = 2x - 5\).
02

Identify the Slope

Once the equation is in slope-intercept form, it's easy to identify the slope, which is simply the coefficient of \(x\). In this instance, the slope \(m\) is \(2\).
03

Identify the y-intercept

From the slope-intercept form of the equation, the y-intercept \(b\) is the number on its own, without \(x\). In this case, the y-intercept \(b\) is \(-5\).

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Key Concepts

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

Equation of a Line
The equation of a line is a mathematical expression that describes all the points along a straight line. One of the most common ways to represent this equation is by using the slope-intercept form: \(y = mx + b\). Here, \(y\) is the dependent variable, \(m\) is the slope of the line, \(x\) is the independent variable, and \(b\) is the \(y\)-intercept.

The slope-intercept form is incredibly useful because it gives you a clear understanding of the line's characteristics just by looking at the equation. You can immediately determine two crucial aspects:
  • The slope \(m\), which tells you how steep the line is.
  • The \(y\)-intercept \(b\), which indicates where the line crosses the \(y\)-axis.
In summary, the slope-intercept form helps you visualize and understand the line's position and direction on a graph.
Slope
The slope of a line, denoted as \(m\) in the slope-intercept form \(y = mx + b\), is a measure of the steepness or inclination of the line. It essentially tells you how much \(y\) changes for a given change in \(x\).

In the equation \(y = 2x - 5\), the slope \(m\) is \(2\). This means that for every unit increase in \(x\), \(y\) increases by \(2\).

The concept of slope can be easily understood through:
  • Positive slope: The line rises as it moves from left to right.
  • Negative slope: The line falls as it moves from left to right.
  • Zero slope: The line is horizontal and \(y\) is constant.
  • Undefined slope: The line is vertical and \(x\) is constant.
In this case, the slope is positive, so the line moves upward as you go from left to right on the graph.
Y-Intercept
The \(y\)-intercept is the point where the line crosses the \(y\)-axis. This is denoted by \(b\) in the slope-intercept form \(y = mx + b\). It is the value of \(y\) when \(x\) equals zero.

In the equation \(y = 2x - 5\), the \(y\)-intercept \(b\) is \(-5\). This means that when you plot this line on a graph, it will intersect the \(y\)-axis at \(-5\).

The \(y\)-intercept provides useful information about where the line is positioned relative to the origin. It helps in:
  • Quickly graphing the line by starting from the \(y\)-axis.
  • Understanding how the line shifts up or down depending on the value of \(b\).
  • Determining initial conditions in real-world applications where \(x\) represents time or another starting parameter.
Overall, knowing the \(y\)-intercept allows for faster and more accurate plotting of linear equations.

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Most popular questions from this chapter

A line passes through points \((-2,-1)\) and \((4,3) .\) Where does the line intersect the \(x\) -axis? the \(y\) -axis?

A line through \(H(3,1)\) and \(J(5, a)\) has positive slope and makes a \(60^{\circ}\) angle measured counterclockwise with the positive \(x\)-axis. Find the value of \(a\).

Find the midpoints of the legs, then the length of the median of the trapezoid with vertices \(C(-4,-3), D(-1,4), E(4,4),\) and \(F(7,-3)\).

Given points \(A(1,1), B(13,9),\) and \(C(3,7) . D\) is the midpoint of \(\overline{A B},\) and \(E\) is the midpoint of \(\overline{A C}\). a. Find the coordinates of \(D\) and \(E\). b. Use slopes to show that \(\overline{D E} \| \overline{B C}\). c. Use the distance formula to show that \(D E=\frac{1}{2} B C\).

Find the center and the radius of each circle. $$(x+3)^{2}+y^{2}=49$$
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