Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Use slopes to solve Exercises \(39-40\). The line passing through \((5, y)\) and \((1,0)\) is parallel to the line joining \((2,3)\) and \((-2,1) .\) Find \(y\)

Step-by-Step Solution

Answer

The value of \(y\) is 2.

Step 1: Calculating the slope of the first line

We are given two points on this line, \((2,3)\) and \((-2,1)\), using these, we can calculate the slope using the formula \[slope = \frac{y2 - y1}{x2 - x1}\] Plugging in the values of the points, we get \( slope = \frac{1 - 3}{ -2 - 2} = -0.5 \)

Step 2: Equating the slopes for parallel lines

Since the lines are parallel, their slopes are equal. Again, using the formula for the slope but this time for the line passing through \( (5, y) \) and \( (1,0) \), we equate \(-0.5\) to \( \frac{0- y}{1 - 5}\).

Step 3: Solve for y

Solving for \(y\) in the equation, we get \(y = 2\).

Key Concepts

Linear Equations

In algebra, a linear equation is an equation that can be written in the form \( ax + by = c \) where \(a\text{, }b\text{, and }c\)) are constants and \(x\text{ and }y\)) are variables. These equations graph as straight lines when plotted on a coordinate system.

The beauty of linear equations lies in their simplicity and the fact that they model many real-world situations; everything from budgeting personal finances to measuring the speed of a car can be described using linear equations. Their predictability allows us to make forecasts and calculate various scenarios.

To solve a problem involving two linear equations, like finding a point that lies on both lines, we often use methods like substitution or elimination. But when we know that lines are parallel, as in the given exercise, we leverage their properties to find solutions quickly and efficiently.

Slope Formula

Understanding the slope formula is essential in algebra, especially when studying lines. The slope is the measure of the steepness and direction of a line, and is usually represented by the letter \(m\text{.}\)) The formula to calculate the slope between two points \( (x_1, y_1) \text{and}\) \( (x_2, y_2) \text{)\)) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \text{.}\]

A slope can be positive, negative, zero, or undefined. Positive slopes rise to the right, while negative slopes fall to the right. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line.

When working with the slope formula, it's important to consistently order the points to avoid sign errors. Differential steepness in lines is what sets them apart; and just like fingerprints, no two non-parallel lines share the same slope. In our exercise, we used the slope formula to determine the unknown \(y\) value by first finding the slope of a line parallel to the one through the points given.

Parallel Lines Algebra

In algebra, parallel lines are lines in a plane that never intersect. A key property of parallel lines is that they have the same slope. This characteristic is extremely useful when solving problems involving parallel lines, as it allows us to set the slopes equal to each other to find unknown coordinates.

When asked to find a point on a line parallel to another, like in our exercise, we calculate the slope of the known line and apply it to the point-slope form equation of the line we are analyzing. This is based on the premise that if two lines are parallel, the equation \( y - y_1 = m(x - x_1) \) will hold true for both lines if they have the same slope \(m\)

Moreover, understanding parallel lines in algebra helps to grasp concepts like angles formed by transversals, and it lays the foundation for more complex geometry. In our exercise, the concept of parallel lines allowed us to set up an equation that led to finding the correct value of \(y\) by ensuring both lines shared the same slope.