Problem 13
Question
Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((0,3)\) and \((2,1)\)
Step-by-Step Solution
Verified Answer
The standard form of the equation is \(x + y = 3\).
1Step 1: Find the slope (m) of the line
The slope of a line passing through two points is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in the points \((x_1, y_1) = (0, 3)\) and \((x_2, y_2) = (2, 1)\), we find:\[ m = \frac{1 - 3}{2 - 0} = \frac{-2}{2} = -1 \]
2Step 2: Find the y-intercept (b)
Since one of the points provided is the y-intercept itself, \((0, 3)\), we can directly use it to find the intercept:When \(x = 0\), \(y = 3\). Hence, \(b = 3\).
3Step 3: Write the equation in slope-intercept form
Using the slope \((m = -1)\) and y-intercept \((b = 3)\), write the equation of the line in slope-intercept form:\[ y = mx + b \]Substitute \(m\) and \(b\) to get:\[ y = -x + 3 \]
4Step 4: Convert the equation to standard form
The standard form of a line is \(Ax + By = C\), where \(A, B,\) and \(C\) are integers and \(A\) should be non-negative. Start with the equation from Step 3:\[ y = -x + 3 \]To transform it, add \(x\) to both sides:\[ x + y = 3 \]
Key Concepts
Slope-Intercept FormStandard FormY-Intercept
Slope-Intercept Form
The slope-intercept form of a linear equation is a direct way to express the equation of a line. It's written as \( y = mx + b \). In this formula:
For example, if the slope \( m \) is positive, the line rises as it moves from left to right. If \( m \) is negative, the line falls.
The number \( b \) shows exactly where the line hits the y-axis. In our exercise, we found that the slope was \( -1 \) and the y-intercept \( b \) was \( 3 \). This gave us the slope-intercept form: \( y = -x + 3 \).
Simply put, using the slope-intercept form makes understanding the relationship between \( x \) and \( y \) immediate and visible.
- \( m \) represents the slope of the line.
- \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
For example, if the slope \( m \) is positive, the line rises as it moves from left to right. If \( m \) is negative, the line falls.
The number \( b \) shows exactly where the line hits the y-axis. In our exercise, we found that the slope was \( -1 \) and the y-intercept \( b \) was \( 3 \). This gave us the slope-intercept form: \( y = -x + 3 \).
Simply put, using the slope-intercept form makes understanding the relationship between \( x \) and \( y \) immediate and visible.
Standard Form
The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A, B, \) and \( C \) are constants. A key characteristic of the standard form is that \( A \) should be a non-negative integer, and all constants are usually written as integers.
Add \( x \) to both sides to achieve \( x + y = 3 \).
Now, \( A = 1, B = 1, \) and \( C = 3 \) are the integers representing the equation in its standard form.
This transformation is simple but requires attention to ensure all coefficients are integers and \( A \) is non-negative.
- This form is prevalent because it can easily describe both vertical and horizontal lines.
- It's useful for solving systems of equations because it aligns well with matrix operations.
Add \( x \) to both sides to achieve \( x + y = 3 \).
Now, \( A = 1, B = 1, \) and \( C = 3 \) are the integers representing the equation in its standard form.
This transformation is simple but requires attention to ensure all coefficients are integers and \( A \) is non-negative.
Y-Intercept
The y-intercept is a vital component in a linear equation. It is the point where the line crosses the y-axis, noted as \( (0, b) \).
In our worked problem, the point \( (0, 3) \) was given as one of the line's points, indicating it directly as the y-intercept with \( b = 3 \).
This information simplifies finding the equation because it directly plugs into the slope-intercept form, \( y = mx + b \).
Having the y-intercept in hand means you have a solid anchor for sketching or analyzing the line, making it an essential piece of information in linear equations.
- This means the x-coordinate is always \( 0 \) at the y-intercept.
- Knowing the y-intercept allows you to quickly place a starting point on a graph.
In our worked problem, the point \( (0, 3) \) was given as one of the line's points, indicating it directly as the y-intercept with \( b = 3 \).
This information simplifies finding the equation because it directly plugs into the slope-intercept form, \( y = mx + b \).
Having the y-intercept in hand means you have a solid anchor for sketching or analyzing the line, making it an essential piece of information in linear equations.
Other exercises in this chapter
Problem 13
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