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Chapter 11: Problem 263

In the following exercises, graph the line given a point and the slope. $$ (-3,3) ; m=2 $$

Short Answer

Expert verified
The equation of the line is \( y = 2x + 9 \) and it passes through points such as \((-3, 3)\) and \((-2, 5)\).

Step by step solution

01

Identify the General Form of the Line Equation

Start with the point-slope form of the equation of a line, which is: \( y - y_1 = m(x - x_1) \). Here, \(x_1, y_1\) is the point \((-3, 3)\) and \( m \) is the slope 2.
02

Substitute the Point and Slope into the Equation

Using the point \((-3, 3)\) and the slope \(m = 2\), substitute these values into the point-slope form equation: \( y - 3 = 2(x + 3) \).
03

Simplify the Equation

Distribute the slope value 2 on the right-hand side: \( y - 3 = 2x + 6 \). Then, add 3 to both sides to solve for \( y \): \( y = 2x + 9 \).
04

Graph the Equation

Plot the given point \((-3, 3)\) on the graph. Next, use the slope to find another point. From \((-3, 3)\), move up 2 units and right 1 unit to reach the point \((-2, 5)\). Draw a line through these two points to graph the line with the given slope and point.

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Key Concepts

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

point-slope form
The point-slope form is a powerful way to write the equation of a line. It is expressed as: \( y - y_1 = m(x - x_1) \).
This form is useful when you know a specific point \(x_1, y_1\) on the line and the slope \(m\).
In your exercise, you have the point \((-3, 3)\) and the slope \(m = 2\).
By substituting these values into the equation, it becomes: \( y - 3 = 2(x + 3) \).
This form is very flexible and lets you immediately see a point on the line and its slope.
slope-intercept form
The slope-intercept form is another common way to write the equation of a line. It is in the format: \( y = mx + b \).
Here, \(m\) represents the slope and \(b\) is the y-intercept, which is where the line crosses the y-axis.
To convert from point-slope form to slope-intercept form, you simply simplify the equation.
From the exercise, starting with \( y - 3 = 2(x + 3) \), distribute the slope on the right side to get: \( y - 3 = 2x + 6 \).
Add 3 to both sides: \( y = 2x + 9 \). Now, the equation is in slope-intercept form.
This form is often easier to graph because you can quickly identify the slope and where the line starts (\(b\)).
plotting points
Plotting points is essential for graphing linear equations.
Here is how you do it:
  • Start with the point given in the problem. In this case, \((-3, 3)\).
  • Place a dot on the graph at that coordinate.
  • Next, use the slope to find another point. Slope is rise over run, so with \(m = 2\), move up 2 units and right 1 unit.
  • From \((-3, 3)\), moving up 2 units and right 1 unit lands you at \((-2, 5)\).
  • Plot this second point (\(-2, 5)\) on the graph.
  • Draw a straight line through your two points. This line is the graph of the equation.
This way of plotting points and using slopes ensures accurate and clear graphing of linear equations.

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