Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$5 x-2 y=10$$

Step-by-Step Solution

Answer

The slope of the equation \(5x - 2y = 10\) is 5/2 and the y-intercept is -5.

Step 1: Solve the equation for y

The given equation is \(5x - 2y = 10\). To solve this for y, start by moving the term involving x to the other side of the equation. This is achieved by subtracting 5x from both sides of the equation yielding \(-2y = -5x + 10\).

Step 2: Convert the rearranged equation to the slope-intercept form

After solving for y, make it positive by dividing every term by -2.This changes the equation to \(y = 5/2x - 5\). This is the slope-intercept form of the line.

Step 3: Identify the slope and y-intercept

In this form \(y = mx + b\), m represents the slope and b is the y-intercept. From \(y = 5/2x - 5\), it is evident that the slope m = 5/2 and the y-intercept b = -5.

Key Concepts

Solving Linear Equations

Linear equations are fundamental in algebra and represent a straight line when graphed on a coordinate plane. Solving a linear equation means finding the value of the variable that makes the equation true. When solving for a specific variable, such as y, the goal is to get y by itself on one side of the equation.

For example, in the equation \(5x - 2y = 10\), we aim to solve for y. The process involves reversing operations that have been applied to y. In this case, we begin by subtracting 5x from both sides to isolate terms containing y. The resulting equation \( -2y = -5x + 10 \) is just one step away from expressing y solely in terms of x. Dividing each term by -2 gives us the y on one side: \(y = \frac{5}{2}x - 5\). This clear separation of y is important for analysis and graphing, and leads us directly to the slope-intercept form, which is key for understanding the graphical representation of the line.

Slope of a Line

The slope of a line is a measure of its steepness or incline and is denoted by the letter m. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Algebraically, it's the coefficient of x when a linear equation is in the slope-intercept form \(y = mx + b\).

In our exercise, once we've solved the equation \(5x - 2y = 10\) for y, we obtain the slope-intercept form \(y = \frac{5}{2}x - 5\), where \(\frac{5}{2}\) is the slope. This means for every 2 units we move horizontally (to the right or left), the line moves 5 units vertically (up or down). A positive slope, as in \(\frac{5}{2}\), indicates that the line rises as it moves from left to right, giving us valuable information about the direction and steepness of the line.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is represented as the variable b in the slope-intercept equation of a line, \(y = mx + b\). Essentially, it tells us where the line would be when x equals zero.

For our original equation, after manipulation, we identified the y-intercept as the constant term -5 in the slope-intercept form \(y = \frac{5}{2}x - 5\). This means that the line crosses the y-axis at the point (0, -5), giving us a precise starting point for drawing the line on a graph. The y-intercept is not just a spot on the grid; it can provide insights into real-world contexts, like the starting value of a quantity before any changes occur.