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Chapter 1: Problem 37

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(6-2 y=10 x-2\)

Short Answer

Expert verified
The slope is 5, and the y-intercept is -4.

Step by step solution

01

Rearrange the Equation

We start with the equation given: \( 6 - 2y = 10x - 2 \). Our goal is to express it in the slope-intercept form \( y = mx + b \). First, we'll isolate terms involving \( y \). Add \( 2y \) to both sides to get: \( 6 = 10x - 2 + 2y \).
02

Simplify the Equation

Now, we add \( 2 \) to both sides to isolate terms involving \( x \) and constant terms on one side: \( 8 = 10x + 2y \).
03

Solve for y

Reorder the equation as \( 2y = 10x - 8 \) by subtracting \( 8 \) from both sides. Divide every term by 2 to express \( y \): \( y = 5x - 4 \).
04

Identify the Slope and y-intercept

Now that we have the equation \( y = 5x - 4 \), we see that it's in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Thus, the slope \( m \) is \( 5 \), and the \( y \)-intercept \( b \) is \( -4 \).

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Key Concepts

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

Linear Equations
Linear equations are foundational components in algebra. These equations describe straight lines when graphed on a coordinate plane. The standard form of a linear equation is given by \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. However, in many scenarios, it's more useful to transform these equations into the slope-intercept form, which is \( y = mx + b \).
  • \( m \) represents the slope of the line, indicating the steepness and direction.
  • \( b \) is the y-intercept, highlighting where the line crosses the y-axis.
Understanding the transformation from the standard form to the slope-intercept form is crucial. This manipulation reveals critical features of the linear relationship, such as slope and intercept, making it easier to interpret graphically.
Slope Calculation
The slope of a line tells us how sharply a line rises or falls as you move from left to right across the graph. It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate. In our solution, the equation was first adjusted to the form \( y = mx + b \) to easily identify \( m \) (the slope). Here's how we transformed the equation:
  • Initial equation: \( 6 - 2y = 10x - 2 \)
  • Rearranged to isolate \( y \): \( 2y = 10x - 8 \)
  • Simplified to slope-intercept: \( y = 5x - 4 \)
In this situation, the slope \( m = 5 \). This value signifies that for every unit increase in x, the value of y increases by 5 units. Thus, the line rises steeply, moving upwards to the right.
Intercept Identification
The y-intercept of a linear equation in slope-intercept form \( y = mx + b \) is the value \( b \). It specifies the point at which the line crosses the y-axis. This is a critical part because it provides a starting point for graphing the line.In the adjusted equation \( y = 5x - 4 \), the intercept \( b = -4 \) reveals that the line crosses the y-axis at the point (0, -4). This means that when \( x = 0 \), \( y \) will be \(-4\). Knowing the y-intercept helps in quickly sketching the graph of the line since you have a specific point to start from. By clearly identifying the y-intercept and combining it with the slope, graphing becomes straightforward.

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Most popular questions from this chapter

How far does a wheel of radius 2 feet roll along level ground in making 150 revolutions?

in Problems 17-22, find the center and radius of the circle with the given equation. \(x^{2}+y^{2}-10 x+10 y=0\)

A regular polygon of \(n\) sides is inscribed in a circle of radius \(r\). Find formulas for the perimeter, \(P\), and area, \(A\), of the polygon in terms of \(n\) and \(r\).

Suppose that \((a, b)\) is on the circle \(x^{2}+y^{2}=r^{2}\). Show that the line \(a x+b y=r^{2}\) is tangent to the circle at \((a, b)\).

Find the exact values in Problems 27-31. Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$
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