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Chapter 3: Problem 6

Determine the slope and \(y\) -intercept. $$ -7 x+2 y=3 $$

Short Answer

Expert verified
Slope = \(\frac{7}{2}\), y-intercept = \(\frac{3}{2}\).

Step by step solution

01

Understand the Equation Format

The equation given is in standard form, which is \(-Ax + By = C\). To identify the slope and y-intercept, we need to transform it into slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
02

Isolate the y-term

Start by moving the \(x\) term to the other side of the equation: \(2y = 7x + 3\).
03

Solve for y

Divide every term by the coefficient of \(y\) to solve for \(y\): \(y = \frac{7}{2}x + \frac{3}{2}\).
04

Identify the Slope and y-intercept

Now that the equation \(y = \frac{7}{2}x + \frac{3}{2}\) is in slope-intercept form, we can identify the slope \(m = \frac{7}{2}\) and the y-intercept \(b = \frac{3}{2}\).

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Key Concepts

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

Understanding Standard Form
The standard form of a linear equation is represented as \(-Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. This format is particularly useful for algebraic operations and solving systems of equations. The advantages of using the standard form include:
  • Clear representation of integer coefficients
  • Ease in finding intercepts
  • Simplicity in performing addition or subtraction with other equations
However, to analyze the slope and y-intercept, we often transform it into the slope-intercept form. This is because the standard form does not explicitly reveal these values. For the given equation, \(-7x + 2y = 3\), rearranging it into slope-intercept form will help us identify both the slope and y-intercept easily.
Grasping the Slope of a Line
The slope of a line, denoted by \(m\) in the slope-intercept form, describes its steepness and direction. It is defined as the change in the vertical direction (\(\Delta y\)) over the change in the horizontal direction (\(\Delta x\)). In simpler terms, the slope tells us how much \(y\) increases or decreases for each unit increase in \(x\). For the given transformed equation \(y = \frac{7}{2}x + \frac{3}{2}\), the slope \(m\) is \(\frac{7}{2}\):
  • A positive slope means the line ascends from left to right, like \(\frac{7}{2}\).
  • A larger slope value indicates a steeper line.
  • If it were negative, the line would descend from left to right.
Recognizing the slope helps us understand the line's behavior without having to graph it first.
Determining the Y-Intercept
The y-intercept of a line is where it crosses the y-axis on a graph. This is represented by \(b\) in the slope-intercept form \(y = mx + b\). It tells us the value of \(y\) when \(x\) equals zero. In our transformed equation \(y = \frac{7}{2}x + \frac{3}{2}\), the y-intercept \(b\) is \(\frac{3}{2}\). You can think of it as where the line starts when graphed:
  • Every linear equation will cross the y-axis once, giving it a unique y-intercept.
  • The y-intercept can be positive, negative, or zero.
Knowing the y-intercept is useful particularly when you need to quickly estimate a line's position on a graph by looking at or calculating just its equation.

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Most popular questions from this chapter

The monthly income in dollars of a commissioned car salesman is given by the function \(I(n)=550 n+1,250,\) where \(n\) represents the number of cars sold in the month. Use the function to determine the salesman's monthly income if he sells 3 cars this month. What is his income if he does not sell any cars in a month?

Graph the linear function and state the domain and range. $$ h(t)=-34 t+3 $$

Find the equation of the line given two points on the line. (7,-6) and (3,-6)

Express in slope-intercept form and identify the slope and y-intercept. $$ 3 x-6 y=24 $$

Find the equation of the line given two points on the line. (10,1) and (10,-3)
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