Question

The lines \(y=2 x-1\) and \(y=a x+2\) are perpendicular if \(a=\) _______.

Short answer

a = -1/2.

Step-by-step solution

Understand Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. For line equations in the slope-intercept form, y=mx+b, 'm' represents the slope.

Identify the Slopes

The first line's equation is y=2x-1, so its slope (m1) is 2. The second line's equation is y=ax+2, so its slope (m2) is 'a'.

Set Up the Perpendicular Condition

Since the lines are perpendicular, the product of their slopes should be -1. Therefore, we set up the equation m1 * m2 = -1. Substitute the known slopes: 2 * a = -1.

Solve for 'a'

To find 'a', solve the equation 2a = -1. Divide both sides by 2: a = -1/2.

Key concepts

slope-intercept form

The slope-intercept form is a way of writing the equation of a line, making it easy to understand the line's slope and y-intercept. These are the two key characteristics that tell you how steep the line is and where it crosses the y-axis.

In its general form, the equation is written as:
y = mx + b
In this formula:

• y is the dependent variable (sometimes called the output).
• x is the independent variable (sometimes called the input).
• m represents the slope of the line.
• b is the y-intercept, the point where the line crosses the y-axis.

When solving problems, having an equation in slope-intercept form allows you to quickly identify these crucial features, aiding in understanding and graphing the line.

slopes of lines

Understanding the slopes of lines is fundamental in algebra and geometry, particularly when dealing with parallel and perpendicular lines.

The slope of a line tells you how steep the line is. It is a ratio that compares the vertical change to the horizontal change between two points on the line, often referred to as 'rise over run'.

The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)

• (x1, y1) and (x2, y2) are two distinct points on the line.
• The difference in the y-values (y2 - y1) represents the vertical change or rise.
• The difference in the x-values (x2 - x1) represents the horizontal change or run.

When dealing with slopes, it's important to recognize two special cases for perpendicular lines:

If two lines are perpendicular, the product of their slopes is -1. For example, if one line's slope is 2, the other must be -1/2.

Vertical lines have an undefined slope because they do not run horizontally, while horizontal lines have a slope of 0 as they do not rise vertically.

Understanding these slope concepts is crucial for solving problems involving perpendicular and parallel lines.

algebraic equations

Algebraic equations are mathematical statements that show the equality between two expressions. These equations can be simple or complex and are essential tools in algebra for modeling real-world situations, solving problems, and understanding mathematical relationships.

In the context of slope-intercept form and slopes of lines, understanding how to manipulate algebraic equations is key to finding solutions.

For example, consider we have the equation for two lines being perpendicular. If their slopes are multiplied and result in -1, you can set up an equation to solve for a missing variable.

• Use the rule that the product of the slopes of two perpendicular lines is -1:
  
  m1 * m2 = -1
  
  Consider two lines with equations in the form y = m1x + b and y = ax + c.
  
  
• Identify that m1 is the slope of the first line, and a (or m2) is the slope of the second line.
• Substitute these slopes into the equation: m1 * a = -1
• Solve for a: a = -1/m1

With algebraic equations, always remember to perform the same operation on both sides to maintain equality. Understanding how to manipulate these equations is vital in solving problems involving slopes and lines.