Problem 18 - Precalculus with Limits

Question

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( y = x - 10 \)

Step-by-Step Solution

Answer

The slope of the line is 1 and the y-intercept is \(y = -10\). The line starts at \(y = -10\) on the y-axis and rises to the right at a 45-degree angle.

Step 1: Identify the slope

The slope \(m\) of the line is the coefficient of \(x\) in the equation. Here, the coefficient of \(x\) is 1, so the slope of the line is 1.

Step 2: Identify the y-intercept

The y-intercept \(c\) of the line is the constant term in the equation. Here, the constant term is -10, so the y-intercept of the line is \(y = -10\).

Step 3: Sketch the line

With a slope of 1, the line increases by 1 for every 1 unit increase in \(x\). Starting from the y-intercept at \(y = -10\), draw a line that rises to the right at a 45-degree angle, since a slope of 1 corresponds to a 45-degree angle with the x-axis.

Key Concepts

Slope

The slope of a line represents how steep the line is. It's basically the rate at which the line ascends or descends as you move along it. The slope is crucial for understanding the behavior of linear equations.

In the equation of a line, which is often written in the form \(y = mx + c\), the slope is denoted by the letter \(m\). It tells us how much the \(y\)-value (or output) changes for a given change in the \(x\)-value (input). For instance, with a slope of 1, for every unit of change in \(x\), the change in \(y\) will match it.

• A positive slope means the line goes up as you move right.
• A negative slope makes the line go down as you move right.
• A zero slope means the line is horizontal.
• An undefined slope (division by zero) occurs in vertical lines.

Y-intercept

The \(y\)-intercept is where the line crosses the \(y\)-axis. It's the point that tells us the value of \(y\) when \(x\) is zero. This intercept is another important aspect of a linear equation because it gives us a starting point for graphing the line.

From the equation \(y = mx + c\), the \(y\)-intercept is represented by \(c\). It gives you the position on the \(y\)-axis where the line hits. So, in the equation \(y = x - 10\), the \(y\)-intercept is -10. This means that when \(x\) is 0, \(y\) is -10, and our line crosses the \(y\)-axis at point (0, -10).

Understanding the \(y\)-intercept helps in quickly drawing the beginning of the line on a graph.

Equation of a Line

The equation of a line in its simplest form is written as \(y = mx + c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. This form is known as the slope-intercept form, and it's widely used because it gives immediate insights into the behavior of the line.

By identifying \(m\) and \(c\) in this equation, you can quickly determine the steepness of the line and where it begins on the \(y\)-axis. This form is particularly useful in graphing and understanding linear relationships in different contexts.

Knowing how to interpret and manipulate the equation of a line is fundamental in solving various mathematical problems as well as real-life applications, such as economics, physics, and everyday situations. Whether you're calculating cost, speed, or growth trends, linear equations provide the basic framework.

Graphing Lines

Graphing a line is about visualizing a linear equation on the coordinate plane. To draw the graph of a line, you'll need both the slope and the \(y\)-intercept.

• Step 1: Plot the \(y\)-intercept on the graph. This point is where your line will start.
• Step 2: Use the slope to find another point on the line. Starting from the \(y\)-intercept, apply the slope \(m\) to determine the next point. For a slope of 1, you go up 1 unit and right 1 unit.
• Step 3: Connect these points with a straight line, extending it across the graph.

Graphing not only helps in understanding the solution better but also provides an intuitive grasp of the line's behavior. A visual representation makes it easier to interpret the relationship between variables and predict their outcomes.