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Chapter 2: Problem 47

Write each equation in slope鈥搃ntercept form. Then find the slope and the y-intercept of the line determined by the equation. $$ 3 x-2 y=8 $$

Short Answer

Expert verified
The equation in slope-intercept form is \(y = \frac{3}{2}x - 4\) with a slope of \(\frac{3}{2}\) and a y-intercept of \(-4\).

Step by step solution

01

Identify the Given Equation

The equation given is a linear equation: \(3x - 2y = 8\). Our task is to write this equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Solve for y

To convert the equation into slope-intercept form, we need to solve for \(y\). First, subtract \(3x\) from both sides to get \(-2y = -3x + 8\).
03

Divide by the Coefficient of y

To isolate \(y\), divide every term in the equation \(-2y = -3x + 8\) by \(-2\). This gives us \(y = \frac{3}{2}x - 4\).
04

Identify the Slope and y-intercept

Now that the equation is in the form \(y = mx + b\), we can identify the slope \(m = \frac{3}{2}\) and the y-intercept \(b = -4\).

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Key Concepts

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

Understanding Linear Equations
A linear equation is a fundamental concept in algebra. It's an equation involving two variables that generates a straight line when graphed on a coordinate plane. Linear equations can be written in different forms including standard form, like the given equation \(3x - 2y = 8\), and slope-intercept form \(y = mx + b\).When we talk about linear equations, we refer to:
  • Two variables, usually \(x\) and \(y\)
  • Coefficients (numbers in front of the variables) that define the relationship between the variables
  • A constant that moves the line up or down in a graph
Linear equations are widely used because they model lots of real-world relationships, like calculating the cost of fruits when you know the price per unit and the total quantity.
Explaining Slope
The slope of a line is a measure of its steepness and direction. It's a crucial component in the slope-intercept form of a linear equation, \(y = mx + b\). The slope is represented by \(m\).Here鈥檚 what slope tells us:
  • The number of units the line rises or falls vertically for each unit it moves horizontally to the right
  • A positive slope means the line goes upwards from left to right, while a negative slope means the line goes downwards
  • In the example \(y = \frac{3}{2}x - 4\), the slope \(m = \frac{3}{2}\) tells us the line rises 3 units for every 2 units it moves to the right
Knowing the slope allows you to predict the behavior of the line and understand the rate of change between \(x\) and \(y\).
Understanding the Y-Intercept
The y-intercept is where the line crosses the y-axis on a graph, and it's represented by \(b\) in the slope-intercept equation \(y = mx + b\). This point is found by setting \(x = 0\) and solving for \(y\).When you look at the y-intercept, you can determine:
  • The starting value of \(y\) when \(x = 0\)
  • How the line interacts with the y-axis, which can be crucial in graphing the equation
  • For the equation \(y = \frac{3}{2}x - 4\), the y-intercept \(b = -4\). This shows the line crosses the y-axis at the point (0, -4)
Understanding the y-intercept helps provide a clear starting point on the graph of a linear equation.

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