Problem 23 - College Algebra

Question

Write an equation in slope-intercept form of a linear function \(f\) whose graph satisfies the given conditions. The graph of \(f\) is perpendicular to the line whose equation is \(3 x-2 y-4-0\) and has the same \(y\) -intercept as this line.

Step-by-Step Solution

Answer

The equation of the linear function \(f\) that is perpendicular to the given line and has the same y-intercept is \( y = -\frac{2}{3}x + 2 \).

Step 1: Get the Equation in Slope-Intercept Form

Firstly, take the given equation \(3x - 2y - 4 = 0\) and convert it into slope-intercept form \(y = mx + c\) to determine the slope and y-intercept of the original line. This can be done by rearranging the equation into \(y =\frac{3}{2}x + 2.\) Thus, the slope \(m\) of this line is \(\frac{3}{2}\) and the y-intercept \(c\) is \(2.\)

Step 2: Determine the Perpendicular Slope

Next, since the slope of the required line must be perpendicular to the slope of the given line, it's the negative reciprocal of \( \frac{3}{2} \), which is \(-\frac{2}{3}. This is the slope of the line we are looking for.

Step 3: Write down the equation of the line

Finally, form the equation of the line \(f\) in slope-intercept form using the slope derived from step 2 and the y-intercept derived from step 1. This gives the equation \(y = -\frac{2}{3}x + 2\).

Key Concepts

Linear Equations

Linear equations represent straight lines on a graph and are foundational in understanding algebra and higher mathematics. They come in various forms, but one of the most convenient for graphing is the slope-intercept form, expressed as \( y = mx + c \), where \( m \) is the slope of the line and \( c \) is the y-intercept. The slope determines the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis.

To convert any linear equation to this form, you'll need to manipulate it so y is isolated on one side of the equation. For example, if you have \( 3x - 2y = 4 \), you can rearrange it to find \( y = \frac{3}{2}x - 2 \), thus expressing it in slope-intercept form.

• Isolate \( y \) by moving all other terms to the opposite side of the equation.
• Ensure that the x-term coefficient represents the slope \( m \).
• Ensure that the constant term represents the y-intercept \( c \).

Remember, studying linear equations involves understanding how to graph them, find their slope, and interpret their intercepts, which are pivotal in solving many real-world problems.

Perpendicular Lines

Perpendicular lines are lines that intersect to form a right angle (90 degrees). A crucial property of perpendicular lines in Cartesian coordinates is that the product of their slopes is -1. This means if one line has a slope of \( m \), the line perpendicular to it will have a slope of \( -\frac{1}{m} \), also known as the negative reciprocal.

For instance, if a line has a slope of \( \frac{3}{2} \), as in our example, a line perpendicular to it will have a slope of \( -\frac{2}{3} \). To find the perpendicular slope:

• Take the reciprocal of the original slope. If the slope is \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
• Negate this value. If the reciprocal is \( \frac{b}{a} \), the perpendicular slope is \( -\frac{b}{a} \).

Recognizing this relationship helps in graphing perpendicular lines, solving geometric problems, and understanding the concepts of orthogonality in higher mathematics.

Y-Intercept

The y-intercept is where a line crosses the y-axis of a coordinate system. In the slope-intercept form of a linear equation \( y = mx + c \), the y-intercept is represented by \( c \). It's an essential feature as it gives a point \( (0, c) \) through which the line passes. This point is particularly helpful because it's one of the easiest points to identify and use when graphing a line.

For example, if you have a line with an equation of \( y = -\frac{2}{3}x + 2 \), this tells us that the graph of the line crosses the y-axis at the point (0,2). Knowing the y-intercept can also be useful in various applications, including finding common solutions in systems of equations and analyzing data in fields such as economics and science.

• Remember that every non-vertical line will have a unique y-intercept.
• The y-intercept is always at the point where x=0.

Understanding y-intercepts is fundamental for students as it aids in constructing graphs and enhances comprehension of linear relationships in quantitative data.