Problem 77 - Algebra A Combined Function

Question

Write an equation in standard form of the line that contains the point (-1,2) and is perpendicular to the line \(y=3 x-1\)

Step-by-Step Solution

Answer

The standard form is \(x + 3y = 5\).

Step 1: Understand the Problem

We need to find the equation of a line that is perpendicular to another line given by \(y = 3x - 1\) and passes through the point \((-1, 2)\). The equation must be in standard form.

Step 2: Identify Slope of Given Line

The given line has the equation \(y = 3x - 1\), which is in slope-intercept form \(y = mx + b\). The slope \(m\) is 3.

Step 3: Determine Perpendicular Slope

Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of the given line is 3, the slope of the perpendicular line will be \(-\frac{1}{3}\).

Step 4: Use Point-Slope Form

To find the equation of the line, use the point-slope form: \(y - y_1 = m(x - x_1)\). Here \((x_1, y_1)\) is the point \((-1, 2)\) and \(m = -\frac{1}{3}\). So the equation is \(y - 2 = -\frac{1}{3}(x + 1)\).

Step 5: Simplify the Equation

Simplify the point-slope form equation:\[y - 2 = -\frac{1}{3}(x + 1)\]\[y - 2 = -\frac{1}{3}x - \frac{1}{3}\]Add 2 to both sides:\[y = -\frac{1}{3}x - \frac{1}{3} + 2\]\[y = -\frac{1}{3}x + \frac{5}{3}\].

Step 6: Convert to Standard Form

Rearrange the equation \(y = -\frac{1}{3}x + \frac{5}{3}\) to standard form \(Ax + By = C\). Multiply through by 3 to clear fractions:\[3y = -x + 5\]Then rearrange:\[x + 3y = 5\].

Key Concepts

Standard Form

The standard form of a linear equation is an incredibly useful representation when dealing with lines. In mathematics, it is generally expressed as \(Ax + By = C\). This form allows you to see both the x and y components clearly on one side of the equation.
The coefficients \(A\), \(B\), and \(C\) are integers, and \(A\) should be a positive integer if possible. When converting other forms of linear equations into standard form, it can be helpful to eliminate any fractions by multiplying through by the least common multiple.

• Standard form: \(Ax + By = C\)
• \(A\), \(B\), \(C\) are integers
• \(A\) is positive

This form is particularly useful because it allows both variables to remain easily visible. It's also the go-to form when dealing with certain mathematical operations, like determining intersections by solving systems of linear equations.

Slope-Intercept Form

The slope-intercept form is a very intuitive way to express the equation of a line, making it easy to determine the slope and y-intercept right off the bat. It is typically written as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. The slope \(m\) describes how steep the line is, while \(b\) shows where the line crosses the y-axis.
Understanding the components:

• Equation form: \(y = mx + b\)
• \(m\): the slope of the line
• \(b\): y-intercept (where the line crosses the y-axis)

This format is very user-friendly when you want to quickly draw a graph of a line or ascertain its rise-over-run. It's especially useful when you need to determine or compare slopes of multiple lines or find parallel or perpendicular lines.

Point-Slope Form

The point-slope form is particularly handy when you already have a specific point on the line and the slope, and you want to find the line's equation. This form is written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.
Using this form is straightforward:

• Equation form: \(y - y_1 = m(x - x_1)\)
• \((x_1, y_1)\) is a known point on the line
• \(m\) is the slope of the line

The point-slope form is perfect when you're given a point and a directional slope, making problem-solving quick and efficient. You can also convert it to either slope-intercept form or standard form with some algebraic manipulation. This flexibility makes it a powerful tool for understanding the geometry of lines and curves.