Question

Which of the following lines is perpendicular to \(y=-2 x+3\) and has the same \(y\) -intercept as \(y=2 x-3 ?\) $$\begin{array}{l} y=-\frac{1}{2} x+3 \\ y=\frac{1}{2} x-3 \\ y=\frac{1}{2} x+3 \\ y=2 x+3 \\ y=2 x-3 \end{array}$$

Short answer

The line that is perpendicular to \(y = -2x + 3\) and has the same y-intercept as \(y = 2x - 3\) is \(y = \frac{1}{2}x - 3\).

Step-by-step solution

Identify the Slope of the Given Line

The slope-intercept form of a line is given by the equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the given line \(y = -2x + 3\), the slope \(m\) is -2.

Determine the Slope of the Perpendicular Line

Two lines are perpendicular if the product of their slopes is -1. Therefore, the slope of the line that is perpendicular to \(y = -2x + 3\) is given by: \[ m_1 \times m_2 = -1 \ (-2) \times m_2 = -1 \ m_2 = \frac{1}{2} \] So, the slope of the perpendicular line is \(\frac{1}{2}\).

Identify the y-Intercept of the Given Line

The y-intercept of the second given line \(y = 2x - 3\) is -3.

Write the Equation of the Desired Line

The desired line has a slope of \(\frac{1}{2}\) and a y-intercept of -3. Therefore, the equation of the line is: \(y = \frac{1}{2}x - 3\).

Find the Matching Equation in the List

Compare the equation of the desired line \(y = \frac{1}{2}x - 3\) to the given options: \(y = -\frac{1}{2}x + 3\) \(y = \frac{1}{2}x - 3\) \(y = \frac{1}{2}x + 3\) \(y = 2x + 3\) \(y = 2x - 3\) The correct match is \(y = \frac{1}{2}x - 3\).

Key concepts

Perpendicular Lines

To solve the problem, it's essential to understand the concept of perpendicular lines. Perpendicular lines intersect at a 90-degree angle. This special property also influences their slopes. If two lines are perpendicular, the product of their slopes equals -1.

For instance, if one line has a slope of 'm,' the other line, which is perpendicular to it, will have a slope of '-1/m.' This relationship is crucial when deriving the equation of a line that's perpendicular to another line.

Slope-Intercept Form

In mathematics, the slope-intercept form is a way of writing down a straight-line equation: \(y = mx + b\).

Here, 'm' represents the slope of the line and 'b' represents the y-intercept, which is where the line crosses the y-axis. This form makes it easy to identify both the slope and the y-intercept at a glance.
In our problem, recognizing the slope and y-intercept of given lines allows us to determine the equation of a perpendicular line that fits the required criteria.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the equation \(y = mx + b\), it is denoted by 'b.'

Understanding this concept helps you quickly find where the line starts on the y-axis, making it easier to graph the line and solve related problems. In our given problem, one line has a y-intercept of 3 and the other has a y-intercept of -3. Identifying these values helps us find the correct line among the given options.

Linear Equations

Linear equations describe straight lines and are often written in the slope-intercept form \(y = mx + b\). These equations are fundamental in algebra and coordinate geometry.

They help us understand relationships between variables, and their graphs always yield straight lines. Linear equations are easy to manipulate and help solve a wide variety of problems, making them a foundational skill in mathematics.

Product of Slopes

When two lines are perpendicular, the result of multiplying their slopes gives -1. This fact comes from the geometric property that perpendicular lines intersect at a 90-degree angle.

In algebraic terms, if \(m_1\) is the slope of the first line and \(m_2\) is the slope of the second line, then the relationship can be written as: \[ m_1 \times m_2 = -1 \] Knowing this, we can easily find the slope of a line perpendicular to a given line, which is crucial in finding the equation of the perpendicular line in our problem.