Understanding the Problem: Finding the Y-intercept of a Perpendicular Line

The question asks us to find the y-intercept of a specific line. We are given two key pieces of information about this line:

• It passes through a known point, (2, 2).
• It is perpendicular to another given line, 3x + y = 3.

To find the y-intercept, we first need to determine the equation of the required line. We can do this by finding its slope and using the given point.

Step-by-Step Solution

Step 1: Find the Slope of the Given Line

The given line has the equation \(3x + y = 3\). To find its slope, we can rewrite this equation in the slope-intercept form, which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Rearranging the equation \(3x + y = 3\):

Subtract \(3x\) from both sides:

\(y = -3x + 3\)

Comparing this to \(y = mx + c\), we can see that the slope of the given line is \(m_1 = -3\).

Step 2: Find the Slope of the Required Line

We are told that the required line is perpendicular to the line \(3x + y = 3\). For two non-vertical lines to be perpendicular, the product of their slopes must be -1.

Let \(m_2\) be the slope of the required line. Then, we have:

\(m_1 \times m_2 = -1\)

Substituting the value of \(m_1\):

\(-3 \times m_2 = -1\)

To find \(m_2\), divide both sides by -3:

\(m_2 = \frac{-1}{-3} = \frac{1}{3}\)

So, the slope of the required line is \(\frac{1}{3}\).

Step 3: Find the Equation of the Required Line

We know the required line passes through the point \((x_1, y_1) = (2, 2)\) and has a slope \(m = \frac{1}{3}\). We can use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\).

Substitute the known values into the point-slope form:

\(y - 2 = \frac{1}{3}(x - 2)\)

Now, we can simplify this equation to get it into slope-intercept form \(y = mx + c\):

\(y - 2 = \frac{1}{3}x - \frac{1}{3}(2)\)

\(y - 2 = \frac{1}{3}x - \frac{2}{3}\)

Add 2 to both sides of the equation:

\(y = \frac{1}{3}x - \frac{2}{3} + 2\)

To add \(- \frac{2}{3}\) and 2, we need a common denominator:

\(y = \frac{1}{3}x - \frac{2}{3} + \frac{6}{3}\)

\(y = \frac{1}{3}x + \frac{4}{3}\)

This is the equation of the required line in the form \(y = mx + c\).

Step 4: Determine the Y-intercept

In the slope-intercept form of a linear equation, \(y = mx + c\), the constant term \(c\) represents the y-intercept. The y-intercept is the point where the line crosses the y-axis, which occurs when \(x = 0\).

From the equation we found, \(y = \frac{1}{3}x + \frac{4}{3}\), the constant term is \(\frac{4}{3}\).

Alternatively, set \(x = 0\) in the equation:

\(y = \frac{1}{3}(0) + \frac{4}{3}\)

\(y = 0 + \frac{4}{3}\)

\(y = \frac{4}{3}\)

The y-intercept is \(\frac{4}{3}\).

Conclusion: Finding the Y-intercept

The y-intercept of the line that passes through (2, 2) and is perpendicular to the line \(3x + y = 3\) is \(\frac{4}{3}\). This corresponds to one of the given options.

Revision Table: Key Concepts for Finding Y-intercept

Additional Information: Coordinate Geometry Basics

Coordinate geometry is a branch of mathematics that uses coordinates to study geometric shapes and figures. Understanding the relationships between lines, points, and slopes is fundamental.

• Every point in a 2D plane can be represented by an ordered pair \((x, y)\).
• A line is defined by an equation that the coordinates of all points on the line satisfy.
• Parallel lines have the same slope.
• Vertical lines have undefined slopes and equations of the form \(x = a\). Their y-intercept is undefined unless the line is the y-axis itself (\(x=0\)).
• Horizontal lines have a slope of 0 and equations of the form \(y = b\). Their y-intercept is \(b\).

Finding the equation of a line is a common task in coordinate geometry. The method used depends on the information given (e.g., two points, a point and a slope, slope and y-intercept).