Problem 69 - Algebra 1: Concepts and Skills

Question

What is the slope of a line parallel to the graph of the equation 16x -32y = 160? $$A.2$$ $$B.\frac{1}{2}$$ $$C.5$$ $$D.-5$$

Step-by-Step Solution

Answer

The slope of the line parallel to the graph of the equation 16x -32y = 160 is 0.5 (Option B).

Step 1: Convert given equation to slope-intercept form.

In order to find the equation's slope, convert the equation 16x -32y = 160 to the y = mx + b form. Begin by moving the term involving x to the other side of the equation:\n -32y = -16x + 160. Next divide every term by -32 to isolate y: y = 0.5x -5. Now, the equation is in the slope-intercept form.

Step 2: Identify the slope.

After converting to the slope-intercept form, it can be easily observed that the coefficient of x is the slope of the line. From the slope-intercept form of the equation y = 0.5x -5, the slope (m) is 0.5.

Step 3: State the slope of the parallel line.

Parallel lines have the same slope. Therefore, the slope of a line parallel to the graph of the given equation is the same as the slope of the given equation which is 0.5.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form, denoted as \(y = mx + b\), is fundamental when dealing with linear equations. In this representation, \(m\) stands for the slope of the line, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. To find the slope of a parallel line, we need the original line in slope-intercept form.

Converting to this form is done by isolating \(y\) on one side of the equation. For example, the equation \(16x - 32y = 160\) can be rearranged by subtracting \(16x\) from both sides and then dividing by \(-32\), yielding \(y = 0.5x - 5\). This reveals that the slope is \(0.5\), and the y-intercept is \(-5\).

Parallel Lines

When we discuss parallel lines in the coordinate plane, we refer to lines that never intersect and remain the same distance apart, no matter how extended they are. A key property of parallel lines is that they always have the same slope. This means if you are given an equation of a line and need to find the slope of a line parallel to it, you simply take the slope of the given line.

Using the example \(y = 0.5x - 5\), any line with a slope of \(0.5\) is guaranteed to be parallel to this line. This is a critical concept for understanding how lines relate to each other in geometry.

Coefficient of x

In any linear equation in slope-intercept form, the coefficient of \(x\) holds significant meaning – it is the slope of the line. This coefficient dictates the steepness and direction (upwards or downwards as x increases) of the line. If it’s positive, the line slopes upwards; if negative, the line slopes downwards.

So, when you are presented with an equation like \(y = 0.5x - 5\), the coefficient of \(x\) is \(0.5\). It tells us how to rise and run: for every 1 unit we run (move to the right), we rise \(0.5\) units (move up).

Slope Calculation

Calculating the slope is all about understanding the rate of change between two points on a line. It's a measurement of how steep a line is. The slope can be determined using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\), where \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.

However, if we start with the slope-intercept form, the slope calculation is straightforward because it’s directly given by the coefficient of \(x\). In the example provided, you don’t need additional points or calculations; the slope \(m\) is already determined to be \(0.5\), as extracted directly from the equation \(y = 0.5x - 5\).