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Magazine Chapter 3: Problem 38
Find an equation of the line passing through each pair of points. Write the equation in the form \(A x+B y=C .\) $$ (0,0) \text { and }\left(-\frac{1}{2}, \frac{1}{3}\right) $$
Short Answer
Expert verified
The equation of the line is \(2x + 3y = 0\).
Step by step solution
01
Identify the Formula for the Equation of a Line
To find the equation of a line passing through two points, we will first determine the slope-intercept form of the line, which is given by \( y = mx + c \). Then we will convert it to the standard form \( Ax + By = C \).
02
Calculate the Slope (m)
The slope \( m \) of the line can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the given points \( (0,0) \) and \( \left(-\frac{1}{2}, \frac{1}{3}\right) \) into the formula:\[ m = \frac{\frac{1}{3} - 0}{-\frac{1}{2} - 0} = \frac{\frac{1}{3}}{-\frac{1}{2}} = -\frac{2}{3} \]
03
Use the Point-Slope Form to Find the Equation
With the slope determined, we can use the point-slope form \( y - y_1 = m(x - x_1) \). Using the point \( (0,0) \):\[ y - 0 = -\frac{2}{3}(x - 0) \] which simplifies to \( y = -\frac{2}{3}x \).
04
Convert to Standard Form \( Ax + By = C \)
To convert \( y = -\frac{2}{3}x \) to the standard form, multiply every term by 3 to eliminate the fraction:\[ 3y = -2x \]Rearrange to get:\[ 2x + 3y = 0 \].
05
Verify the Equation with Both Points
Verify that both points \( (0,0) \) and \( \left(-\frac{1}{2}, \frac{1}{3}\right) \) satisfy the equation \( 2x + 3y = 0 \). Both points satisfy the equation, confirming it is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a line is one of the most popular ways to express the equation of a line. It is represented by the formula \( y = mx + c \), where:
For instance, in the equation \( y = -\frac{2}{3}x \), the slope is \(-\frac{2}{3}\), and the line passes through the origin (0,0), indicating \( c = 0 \).
This is crucial for graphing a line easily as you know where to start (the y-intercept) and how to move across the plane based on the slope (rise over run).
- \( m \) is the slope of the line
- \( c \) is the y-intercept, or the point where the line crosses the y-axis
For instance, in the equation \( y = -\frac{2}{3}x \), the slope is \(-\frac{2}{3}\), and the line passes through the origin (0,0), indicating \( c = 0 \).
This is crucial for graphing a line easily as you know where to start (the y-intercept) and how to move across the plane based on the slope (rise over run).
Standard Form of a Line
The standard form of a line is another common way to express linear equations. It is written as \( Ax + By = C \). This form is particularly useful for certain types of algebraic manipulations and can make checking solutions at specific points more straightforward.
The coefficients \( A \), \( B \), and \( C \) can be integers, which can help eliminate fractions and simplify the equation. To convert from slope-intercept form to standard form, follow these steps:
The coefficients \( A \), \( B \), and \( C \) can be integers, which can help eliminate fractions and simplify the equation. To convert from slope-intercept form to standard form, follow these steps:
- Multiply all terms by the least common multiple to eliminate fractions.
- Rearrange terms to get the required format \( Ax + By = C \).
Point-Slope Form
Point-slope form is particularly valuable when you know the slope and a point on the line. The formula is \( y - y_1 = m(x - x_1) \), where:
This simplification comes from plugging in (0,0) into the original point-slope equation. It offers a direct path to finding a line equation without the need to calculate the y-intercept separately if you don't already know it.
- \( (x_1, y_1) \) is a known point on the line.
- \( m \) is the slope of the line.
This simplification comes from plugging in (0,0) into the original point-slope equation. It offers a direct path to finding a line equation without the need to calculate the y-intercept separately if you don't already know it.
Slope Calculation
Calculating the slope of a line is essential for understanding how the line behaves. The slope is a measure of how steep a line is and in which direction it slants. You can calculate the slope \( m \) using the formula:
For example, with points \((0,0)\) and \((-\frac{1}{2}, \frac{1}{3})\), the slope is:
Knowing the slope allows you to write the equation in various forms and helps with graphing and understanding the line's behavior.
- \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
For example, with points \((0,0)\) and \((-\frac{1}{2}, \frac{1}{3})\), the slope is:
- \( m = \frac{\frac{1}{3} - 0}{-\frac{1}{2} - 0} = -\frac{2}{3} \)
Knowing the slope allows you to write the equation in various forms and helps with graphing and understanding the line's behavior.
