Problem 41 - Prealgebra

Question

What equation represents the table of values? $$\begin{array}{|c|c|c|c|c|}\hline x & -4 & -8 & -12 & -16 \\\\\hline y & 6 & 8 & 10 & 12 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The equation representing the table of values is \(y = -\frac{1}{2}x + 4\).

Step 1: Determine the type of relationship

Examine the table to see if the relationship between the values of \(x\) and \(y\) appears linear, exponential, or another type. Since the table presents a consistent change in \(y\) as \(x\) changes by a fixed amount, it suggests a linear relationship.

Step 2: Calculate the change in y-values

Calculate the difference between consecutive \(y\) values: \[ 8 - 6 = 2, \quad 10 - 8 = 2, \quad 12 - 10 = 2. \]The changes are consistent, confirming a linear relationship with a constant slope of 2.

Step 3: Calculate the change in x-values

Calculate the difference between consecutive \(x\) values:\[ -8 - (-4) = -4, \quad -12 - (-8) = -4, \quad -16 - (-12) = -4. \]The changes are consistent and each step decreases by 4.

Step 4: Find the slope of the line

The slope \(m\) of the line is given by the ratio of the change in \(y\) to the change in \(x\):\[ m = \frac{2}{-4} = -\frac{1}{2}. \]

Step 5: Use point-slope form to find equation

Use the point-slope form of the line equation, which is \(y = mx + b\), where \(b\) is the y-intercept. Using the point (-4, 6):\[ 6 = -\frac{1}{2}(-4) + b \ 6 = 2 + b \ b = 4. \]

Step 6: Write the final equation

Substitute the values of \(m\) and \(b\) back into the equation:\[ y = -\frac{1}{2}x + 4. \]

Key Concepts

Slope

In the realm of linear equations, understanding the concept of slope is crucial. The slope measures how steep a line is and is symbolized by the letter \( m \). It is calculated by comparing the change in the vertical direction (\( y \)-values) to the change in the horizontal direction (\( x \)-values). This is often stated as "rise over run." In simple terms:

• The "rise" refers to how much the line moves in the vertical direction between two points (difference in \( y \)-values).
• The "run" refers to how much the line moves horizontally between those same two points (difference in \( x \)-values).

For the exercise given, we found the slope by observing that each step results in an increase of 2 in the \( y \)-direction while decreasing 4 in the \( x \)-direction. This results in a slope \( m \) of \(-\frac{1}{2}\). A negative slope indicates the line is decreasing—it's slanting downwards as one moves along the line from left to right.

Point-Slope Form

The point-slope form is a powerful tool for writing the equation of a line when you're given a point and a slope. This form is useful because it allows you to plug in known values and easily find the complete equation. The formula for point-slope form is: \[ y - y_1 = m(x - x_1) \] Where:

• \( m \) is the slope.
• \((x_1, y_1)\) is a specific point on the line.

Once you've chosen a point from the data table, say \((-4, 6)\), you can substitute it into the point-slope formula along with the slope \(-\frac{1}{2}\) to construct an equation. Working through the algebra, we obtain: \[ 6 = -\frac{1}{2}(-4) + b \] Solving this allows us to find the value of \( b \), which helps us to ultimately express the line equation in slope-intercept form.

Y-Intercept

The y-intercept, represented as \( b \) in a line equation, is the point where the line crosses the y-axis. It's an important component of the slope-intercept form \( y = mx + b \). At this point, the \( x \)-coordinate is zero. Thus, the y-intercept is the value of \( y \) when \( x = 0\).
Finding the y-intercept involves substituting a known point into the linear equation and solving for \( b \). In our exercise, once we knew the slope and selected the point \((-4, 6)\), we could isolate \( b \) in the equation. Placing \( -\frac{1}{2} \) for \( m \), our equation came out to be: \[ 6 = -\frac{1}{2}(-4) + b \] Simplifying this, we determined \( b \) to be 4. With this value, the final equation becomes \( y = -\frac{1}{2}x + 4 \), clearly showing that the line crosses the y-axis at \( y = 4 \). This intercept guides us to understand how graph lines start on the y-axis and ensures we can accurately sketch or interpret the line on a graph.