Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Use slopes to solve Exercises \(39-40\). The line passing through \((-1, y)\) and \((1,0)\) is perpendicular to the line joining \((2,3)\) and \((-2,1) .\) Find \(y\).

Step-by-Step Solution

Answer

The value for \(y\) is 4.

Step 1: Determine the slope of the line joining points (2,3) and (-2,1)

We will calculate the slope (\(m1\)) of this line using slope formula \((y2-y1)/(x2-x1)\). So, \(m1 = (1-3)/(-2-2) = -2/-4 = 0.5\)

Step 2: Determine the slope of the line that's perpendicular

Now, since the slope of a line perpendicular to another is the negative reciprocal of that line's slope, the slope (\(m2\)) of the line passing through (-1, y) and (1, 0) is -1/m1, which equals -1/0.5 = -2.

Step 3: Use the slope to find y

Since we know that the line passes through points (-1, y) and (1, 0) and that it has slope -2, we can use its slope to find the point y. Using the slope formula again \((y2-y1)/(x2-x1) = m2\), we substitute the points and the slope: \((0-y)/(1-(-1)) = -2\). Solving this equation will find us the y-value.

Step 4: Solving for y

With equation (0-y)/(1-(-1)) = -2, which simplifies to -y/2 = -2, we can solve for y by multiplying both sides by 2. This gives us y = 4.

Key Concepts

Slope Calculations

Slope calculations are a fundamental part of understanding relationships between lines in coordinate geometry. The slope, often represented as 'm', measures the steepness of a line and is calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula finds the change in the y-values divided by the change in the x-values between two points For the exercise, we first calculated the slope of the line joining the points (2,3) and (-2,1). This step involved subtracting the y-coordinates and x-coordinates of these points:

• \( m_1 = \frac{1 - 3}{-2 - 2} = \frac{-2}{-4} = 0.5 \)

After finding this slope, we turned our attention to finding the slope of the line perpendicular to it.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, plays a crucial role in finding the slope of lines and understanding their relations on a plane. It involves assigning coordinates to points on a plane and using those coordinates to understand geometric principles like slopes, perpendicularity, and parallelism. In our exercise, we took points provided in a two-dimensional plane:

• (2,3) and (-2,1) for the first line
• (-1,y) and (1,0) for the line we want to solve for.

By understanding how coordinates map out lines, we use these points to calculate slopes using the slope formula and find relationships like perpendicularity.

Algebraic Equations

Algebraic equations are expressions of equality between two algebraic expressions. They are vital in solving problems involving unknowns, such as 'y' in our exercise. With the slope of the second line known, we used the slope formula to create an equation.Given the second line passes through the coordinates (-1, y) and (1, 0) and we know its slope is -2, we set up the equation:

• \( \frac{0-y}{1 - (-1)} = -2 \)

This sets up a linear equation with the unknown y, simplifying to:

• \( -\frac{y}{2} = -2 \)

Solving this equation by multiplying both sides by 2 results in y being equal to 4. Thus, algebraic equations allow us to solve for unknowns by manipulating equality expressions.