Problem 15 - 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. $$x+y=6$$

Step-by-Step Solution

Answer

The slope of the line is -1 and the y-intercept of the line is 6.

Step 1: Solve for y

In the given equation \(x+y=6\), let's isolate y by subtracting \(x\) from both sides. This yields the equation \(y=-x+6\).

Step 2: Identify the slope and y-intercept

From the equation \(y=-x+6\), it is determined that the slope \(m\) is -1 (the coefficient of \(x\)) and the y-intercept \(b\) is 6 (the constant term).

Key Concepts

Linear Equations

Understanding linear equations is fundamental to mastering algebra. These equations can be recognized by their standard form, which is Ax + By = C. However, for many applications, especially graphing, it's beneficial to manipulate these into the slope-intercept form, given by y = mx + b. Here, m represents the slope and b represents the y-intercept.

Setting a linear equation to this form involves solving for y in such a way that y stands alone on one side of the equation, and everything else is on the other side. This often involves moving terms across the equals sign by adding, subtracting, multiplying, or dividing both sides of the equation by the same numbers.

For example, starting with the equation
\[\begin{equation}x + y = 6,\end{equation}\]
we subtract x from both sides to isolate y, resulting in
\[\begin{equation}y = -x + 6.\end{equation}\]
This simple transformation enables us to immediately identify the slope and y-intercept without further need for calculation. The ability to rewrite linear equations into slope-intercept form thus provides a powerful tool for quickly analyzing and graphing linear relationships.

Slope of a Line

The slope of a line is a measure of its steepness or incline and is represented as m in the slope-intercept form. Mathematically, slope is defined as the ratio of the rise (the vertical change) to the run (the horizontal change) between two points on the line.

In the equation y = mx + b, the slope is the coefficient of x. If the slope is positive, the line rises from left to right; conversely, a negative slope indicates that the line falls from left to right. A zero slope means the line is horizontal, and an undefined slope (when the denominator of the rise over run ratio is zero) signifies a vertical line.
For the example at hand,
\[\begin{equation}y = -x + 6,\end{equation}\]
the slope m is -1. This tells us that for every one unit the line moves horizontally to the right, it moves down by one unit. This consistent rate allows linear equations to predict values and model relationships with remarkable clarity.

Y-Intercept

The concept of a y-intercept is pretty straightforward yet very important in graphing and understanding the implications of linear equations. The y-intercept is the point where the line crosses the y-axis on a graph. In the slope-intercept form of a linear equation, y = mx + b, it is represented by b. This value tells you the point at which x is zero, meaning it provides a starting point for the line on the graph.

In the equation
\[\begin{equation}y = -x + 6,\end{equation}\]
the y-intercept b is 6. This means that when x is zero, y is 6. To plot the y-intercept on a coordinate grid, you simply find 6 on the y-axis and place a point. It's from this point that you use the slope to determine the direction and steepness of the line as you plot additional points to create the graphical representation of the equation. Identifying the y-intercept is essential for both understanding the equation's graph and for writing the equation from the graph.