Problem 92 - College Algebra

Question

Then use the TRACE feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=2 x+4$$

Step-by-Step Solution

Answer

The slope of the line \(y = 2x + 4\) as given by its coefficient and calculated using two points from the line, is \(2\).

Step 1: Identify the Slope from the Equation

The given equation of the line is \(y = 2x + 4\). The slope-coefficient from this equation is \(2\).

Step 2: Choose two Points on the Line

Let's pick two points on the line. For simplicity, choose the points where \(x=0\) and \(x=1\). When plugged into the equation, this gives the points \((0, 4)\) and \((1, 6)\) respectively.

Step 3: Compute the Slope Using the Two Points

To calculate the slope using the two points obtained from the previous step, apply the formula \((y2 - y1) / (x2 - x1)\), which gives \((6 - 4) / (1 - 0) = 2\).

Step 4: Verify the Results

The result obtained from Steps 1 and 3 are the same, which verifies that the slope of the line is indeed \(2\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is foundational in algebra and coordinate geometry. The standard format for this form is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) stands for the y-intercept, the point where the line crosses the y-axis.

When you're given an equation like \(y = 2x + 4\), it's easy to identify the slope directly. Here, the coefficient of \(x\) is \(2\), indicating that the slope of the line is \(2\). This number tells us that for every one unit increase in \(x\), \(y\) increases by two units.

The y-intercept \(b\) in this case is \(4\), meaning the line will cross the y-axis at \((0,4)\). This intercept provides a starting point for graphing the line or understanding its position in relation to the origin.

Linear Equations

Linear equations form straight lines when graphed on a coordinate plane and are an important part of understanding algebraic relationships. These equations often appear in the form \(ax + by = c\) or in the slope-intercept form mentioned earlier.

In our example, \(y = 2x + 4\) is a linear equation. It's linear because \(y\) depends on the first power of \(x\), which means the graph of this equation will be a straight line.

To solve a linear equation, you can isolate the variable \(y\) (or \(x\) if needed), which allows us to graph the line or use it for other algebraic calculations. Solving linear equations can include operations like addition, subtraction, multiplication, and division, along with respect to the properties of equality.

Graphing Lines

Graphing lines is a way to visualize the relationship between two variables described by a linear equation. A coordinate plane is used, with a horizontal x-axis and a vertical y-axis.

To graph a line like \(y = 2x + 4\), start by plotting the y-intercept \((0, 4)\). Use the slope, or rise over run, to find another point. With a slope of \(2\), move up two units and right one unit from the y-intercept to reach the next point on the line, at \((1, 6)\).

By drawing a line through these points and extending it in both directions, you create the graph of the linear equation. It's crucial to understand how to graph lines accurately, as this skill is widely used in various fields, including mathematics, engineering, and the physical sciences.