Problem 111 - Precalculus Essentials

Question

The graph of the linear function \(5 x+6 y-30=0\) is a line passing through the point (6,0) with slope \(-\frac{5}{6}\).

Step-by-Step Solution

Answer

The slope of the given linear function is \(-\frac{5}{6}\), which matches the slope given in the problem and the point (6,0) lies on the line, confirming the function correctly represents the problem.

Step 1: Identify the Given Linear Function

The given linear function is \(5x + 6y - 30 = 0\). We can re-write this function in the \(y=mx+b\) format, which is easier to interpret graphically. When we isolate y, we get \(y = (-\frac{5}{6})x + 5\).

Step 2: Finding the Slope from the Linear Function

In the \(y=mx+b\) format, m is the slope of the line. From our rewritten function \(y = (-\frac{5}{6})x + 5\), it's clear that the slope of the line is \(-\frac{5}{6}\). This matches the slope given in the problem statement, which confirms the accuracy of our function transformation.

Step 3: Checking the Given Point

We need to confirm that the point (6,0) lies on the line represented by our function. We can do this by substituting x=6 and y=0 into our function and checking that it holds true. Substituting these values into our original equation \(5x + 6y - 30 = 0\) we get \(5*6 + 6*0 - 30 = 0\). Simplifying, it results in 0, confirming that the given point does indeed lie on the line.

Key Concepts

Slope of a Line

Understanding the slope of a line is fundamental in algebra and geometry. The slope is a measure of the steepness or incline of a line. To find the slope, we can use the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\), where \(m\) represents the slope and \( (x_1, y_1)\) and \( (x_2, y_2)\) are any two points on the line.

When we look at the linear function \(5x + 6y - 30 = 0\), and rewrite it as \(y = (-\frac{5}{6})x + 5\), we can immediately see that the number in front of \(x\), which is \( -\frac{5}{6}\), is the slope of our line. This value tells us that for every 6 units we move horizontally, the line moves down 5 units vertically, as the negative sign indicates a downward slope. It's essential to match the sign and value of the slope with the graph's incline to understand the direction and steepness of the line.

Graphing Linear Equations

Graphing linear equations involves plotting a line on a coordinate plane based on the equation of the line. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In graphing, we first plot the y-intercept, and then use the slope to determine the direction and steepness of the line. For instance, with the equation \(y = (-\frac{5}{6})x + 5\), you would start at the point \( (0, 5)\) on the y-axis and then apply the slope \( -\frac{5}{6}\). If we create a table of values by selecting various x-coordinates and solving for y, we can plot more points to draw a more accurate line. Checking that the point (6,0) aligns with our line provides an extra layer of confirmation that our graph is correct.

Point-Slope Form

The point-slope form is another format to represent the equation of a line and is particularly useful when we know a point on the line and its slope. The equation in point-slope form is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \( (x_1, y_1)\) is the known point.

For our linear function \(5x + 6y - 30 = 0\), we isolated the slope as \( -\frac{5}{6}\) and confirmed the line passes through (6,0). Using point-slope form, we would write this as \(y - 0 = (-\frac{5}{6})(x - 6)\). This form is particularly useful for quickly sketching the graph of a line when starting with a point other than the y-intercept and when the slope is already known.