Problem 65
Question
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,2)\) and parallel to the line whose equation is \(2 x-3 y-7=0\)
Step-by-Step Solution
Verified Answer
The equation of the parallel line passing through (-2,2) in point-slope form is \(y - 2 = 2/3 (x + 2)\) and in slope-intercept form is \(y = 2/3x + 10/3\).
1Step 1: Find the slope of the given line
Slope is coefficient of x divided by coefficient of y with opposite sign. For given line, slope, m = -A/B = -2/(-3) = 2/3
2Step 2: Write down the point-slope form
The point-slope form of a line is \(y - y1 = m(x - x1)\). Substituting given point (-2,2) and calculated slope, 2/3 in point-slope form, we get \(y - 2 = 2/3 (x - (-2)) => y - 2 = 2/3 (x + 2)\)
3Step 3: Convert to slope-intercept form
Solve to the equation from step 2 to get slope-intercept form of line. \(y - 2 = 2/3 x + 4/3 => y = 2/3x + 4/3 + 2 => y = 2/3x + 10/3\).
Key Concepts
Understanding Point-Slope FormExploration of Slope-Intercept FormInvestigating Parallel Lines
Understanding Point-Slope Form
The point-slope form of a line is a reliable way to describe a line using just a point on the line and the slope. If you know both these elements, you can easily write the equation of the line. Here's the basic formula:
In the exercise example, we have a point \((-2, 2)\) and a slope \(\frac{2}{3}\). When these values are substituted into the point-slope form, the equation becomes:
\[ y - 2 = \frac{2}{3}(x + 2) \]
Understanding the point-slope form helps you connect a line's behavior (shown through its slope) to particular locations (coordinates) on the line. It forms the foundation for converting to other line equations like the slope-intercept form.
- Formula: \( y - y_1 = m(x - x_1) \)
- \( (x_1, y_1) \) represents a particular point on the line.
- \( m \) is the slope of the line.
In the exercise example, we have a point \((-2, 2)\) and a slope \(\frac{2}{3}\). When these values are substituted into the point-slope form, the equation becomes:
\[ y - 2 = \frac{2}{3}(x + 2) \]
Understanding the point-slope form helps you connect a line's behavior (shown through its slope) to particular locations (coordinates) on the line. It forms the foundation for converting to other line equations like the slope-intercept form.
Exploration of Slope-Intercept Form
Slope-intercept form is perhaps the most widely used equation format when discussing lines. It's straightforward and clearly shows you the slope and the y-intercept, which tells you where the line crosses the y-axis.
To move from point-slope form to slope-intercept form, solve the equation for \(y\). Let's see how this works using the equation from the previous section:
\[ y - 2 = \frac{2}{3}x + \frac{4}{3} \]
By adding \(2\) to both sides, we convert it to:
\[ y = \frac{2}{3}x + \frac{10}{3} \]
This format quickly tells us:
Knowing how to convert and use the slope-intercept form is crucial for graphing lines and understanding their interactions with other equations.
- Formula: \( y = mx + b \)
- \( m \) represents the slope.
- \( b \) is the y-intercept.
To move from point-slope form to slope-intercept form, solve the equation for \(y\). Let's see how this works using the equation from the previous section:
\[ y - 2 = \frac{2}{3}x + \frac{4}{3} \]
By adding \(2\) to both sides, we convert it to:
\[ y = \frac{2}{3}x + \frac{10}{3} \]
This format quickly tells us:
- The slope \(m = \frac{2}{3}\)
- The y-intercept \(b = \frac{10}{3}\)
Knowing how to convert and use the slope-intercept form is crucial for graphing lines and understanding their interactions with other equations.
Investigating Parallel Lines
Parallel lines have a unique characteristic: they always have the same slope. This means they run alongside each other at a constant distance. Let's explore what that means for equations of parallel lines.
In the original problem, the line of interest is parallel to the given line \(2x - 3y - 7 = 0\). First, solving this equation for \(y\) shows the slope is \(\frac{2}{3}\).
This tells us that any line parallel to this one must also have a slope of \(\frac{2}{3}\).
With this knowledge, you can quickly find equations for lines that stay parallel to a known line by matching their slopes. When working with parallel lines:
Understanding these concepts allows you to easily construct and manipulate parallel lines in various applications.
- Parallel lines never intersect.
- Their slopes are equal, which makes their direction and steepness the same.
In the original problem, the line of interest is parallel to the given line \(2x - 3y - 7 = 0\). First, solving this equation for \(y\) shows the slope is \(\frac{2}{3}\).
This tells us that any line parallel to this one must also have a slope of \(\frac{2}{3}\).
With this knowledge, you can quickly find equations for lines that stay parallel to a known line by matching their slopes. When working with parallel lines:
- Ensure both lines share the same slope.
- Consider different y-intercepts to create unique parallel lines.
Understanding these concepts allows you to easily construct and manipulate parallel lines in various applications.
Other exercises in this chapter
Problem 65
a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1,\) and \(g(x)=f\left(\frac{1}{2} x\right),\) and \(h(x)=f\left(\frac{1}{4} x\right)\
View solution Problem 65
Find the domain of each function. $$ f(x)=\frac{1}{\sqrt{x-3}} $$
View solution Problem 65
If \(f(x)=3 x\) and \(g(x)=x+5,\) find \((f \circ g)^{-1}(x)\) and \(\left(g^{-1} \circ f^{-1}\right)(x)\)
View solution Problem 65
Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.
View solution