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Chapter 3: Problem 9

Write an equation of the line passing through the given point and having the given slope. Give the final answer in slope-intercept form. \((6,-3), m=1\)

Short Answer

Expert verified
The equation of the line is \( y = x - 9 \).

Step by step solution

01

Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is given by: \[ y = mx + b \], where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the Slope

We are given the slope \( m = 1 \). Substitute this value into the slope-intercept form equation: \[ y = 1x + b \].
03

Substitute the Given Point

Substitute the given point \((6, -3)\) into the equation. Replace \( x \) with 6 and \( y \) with -3: \[ -3 = 1(6) + b \].
04

Solve for the Y-Intercept

Solve the equation for \( b \) (the y-intercept): \[ -3 = 6 + b \] \[ b = -3 - 6 \] \[ b = -9 \].
05

Write the Final Equation

Substitute \( b = -9 \) back into the slope-intercept form equation: \[ y = 1x - 9 \]. Thus, the equation of the line is: \[ y = x - 9 \].

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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 forms a straight line when it is graphed on a coordinate plane. The general form of a linear equation in two dimensions is \(\text{ax + by + c = 0}\), but there's a more intuitive form called the slope-intercept form. This is written as \(\text{y = mx + b}\). Linear equations are fundamental in algebra and have various applications, such as predicting trends and modeling real-world situations.

Key components in a linear equation include:
  • Slope (m): Determines the steepness or incline of the line
  • Y-intercept (b): The point where the line crosses the y-axis
Linear equations are straightforward and immensely useful in understanding relationships between variables.
Slope
The slope of a line, often denoted as \(m\), represents how steep the line is. It measures the change in the vertical direction (y-axis) per unit change in the horizontal direction (x-axis). The slope is determined by the formula: \[ m = \frac{{\Delta y}}{{\Delta x}} \] This formula indicates the ratio of the rise (change in y) to the run (change in x).

Slopes can be:
  • Positive: The line rises as it moves from left to right
  • Negative: The line falls as it moves from left to right
  • Zero: The line is horizontal
  • Undefined: The line is vertical
In our exercise, a slope of 1 means that for every unit increase in x, y increases by the same amount.
Y-Intercept
The y-intercept of a line is where it crosses the y-axis. It's the value of \(y\) when \(x = 0\). In the slope-intercept form \(y = mx + b\), \(b\) represents the y-intercept.

To find the y-intercept, you can substitute the x and y values from a known point on the line into the equation and solve for \(b\). For instance, in the given exercise, we have a point (6, -3) and a slope (1). By solving the equation for \(b\), we found that \(b = -9\). Hence, the y-intercept is -9, indicating that our line crosses the y-axis at \y = -9\.
Algebra
Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It's a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions.

In the context of our exercise with linear equations, algebra helps to:
  • Manipulate equations to find unknown values
  • Understand relationships between different quantities
  • Solve problems systematically
Using algebraic techniques, we substituted the given point and slope into the slope-intercept form to find the y-intercept and ultimately form the equation of the line.

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Most popular questions from this chapter

$$ y=-\frac{1}{3} x+4 $$

Write an equation of the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (-1,-7) and (-8,-2)

Graph each linear equation. $$ y=3 $$

Solve each problem. Suppose that it costs a flat fee of \(\$ 20\) plus \(\$ 15\) per day to rent a pressure washer. Therefore, the cost \(y\) in dollars to rent the pressure washer for \(x\) days is given by the linear equation $$ y=15 x+20 $$ Express each of the following as an ordered pair. (a) When the washer is rented for 5 days, the cost is \(\$ 95\). (b) When the cost is \(\$ 110\), the washer is rented for 6 days.

Write an equation of the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (-1,-7) and (-8,-2)
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