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Chapter 7: Problem 77

Graph each line passing through the given point and having the given slope $$ (-1,-2) ; m=3 $$

Short Answer

Expert verified
The equation of the line is \(y = 3x + 1\).

Step by step solution

01

Identify the point and slope

The line passes through the point \((-1, -2)\) with a slope \(m = 3\).
02

Use the point-slope form of the equation

Use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\). Here, \(x_1 = -1\), \(y_1 = -2\), and \(m = 3\). Substituting these into the equation we get: \(y - (-2) = 3(x - (-1))\).
03

Simplify the equation

Simplify the equation from the previous step: \(y + 2 = 3(x + 1)\).
04

Distribute and isolate y

Distribute the 3 to get: \(y + 2 = 3x + 3\). Then, isolate \(y\) by subtracting 2 from both sides: \(y = 3x + 1\). This is the equation of the line.
05

Graph the line

Plot the point \((-1, -2)\) on a coordinate plane. Using the slope \(m = 3\) (which means rise over run), from the point move up 3 units and 1 unit to the right to locate another point. Draw a line through these points to graph the line.

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Key Concepts

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

Point-Slope Form of a Line
The point-slope form of a linear equation is a handy way to write the equation of a line when you know one point on the line and the slope. The equation is \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) is the point, and \( m \) is the slope. Using this form helps to quickly write down the equation without needing to rearrange or solve for y first. For instance, if a line passes through the point \( (-1, -2) \) and has a slope of 3, substitute these values into the point-slope form: \( y - (-2) = 3(x - (-1)) \).
Understanding Slope
The slope of a line measures its steepness and direction. It is calculated as the 'rise' over the 'run', which means how much y increases for a one-unit increase in x. Mathematically, \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In our exercise, the slope given is 3, meaning for every unit you move to the right (increase in x), you move up 3 units (increase in y). This positive slope indicates an upward slanting line from left to right. Always remember that:
  • A positive slope means the line goes up as you move right.
  • A negative slope means the line goes down as you move right.
  • A slope of zero means the line is horizontal.
  • An undefined slope indicates a vertical line.
Plotting Points on a Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph equations. It has a horizontal x-axis and a vertical y-axis. Each point on the plane is defined by a pair of coordinates \( (x, y) \). To plot the line from our exercise:
  • Start by plotting the given point \( (-1, -2) \).
  • Using the slope 3, move up (rise) 3 units and 1 unit to the right (run) to find the next point.
  • Draw a straight line through these points to represent the equation \( y = 3x + 1 \).
Practice plotting different points and ensuring they align with the slope to build accuracy in graphing. This visual representation helps in understanding the relationship between the algebraic equation and its graph.

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Most popular questions from this chapter

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