Problem 82 - Calculus Volume 1

Question

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. \(6 x-5 y+15=0\)

Step-by-Step Solution

Answer

Slope \(m = \frac{6}{5}\); \(y\)-intercept \(b = 3\); Graph: line through (0,3) and (5,9).

Step 1: Rearrange the Equation

Start by rearranging the given equation into the slope-intercept form, which is \(y = mx + b\). The given equation is \(6x - 5y + 15 = 0\). Move all terms involving \(y\) to one side: \(-5y = -6x - 15\).

Step 2: Solve for y

Divide every term by -5 to solve for \(y\): \[y = \frac{6}{5}x + 3\]. This puts the equation in the form \(y = mx + b\).

Step 3: Identify the Slope and Y-intercept

The coefficient of \(x\) in the equation \(y = \frac{6}{5}x + 3\) represents the slope \(m\). So, \(m = \frac{6}{5}\). The constant term is the \(y\)-intercept \(b = 3\).

Step 4: Graph the Line

Start by plotting the \(y\)-intercept \((0,3)\) on the graph. Use the slope \(\frac{6}{5}\) which means 'rise 6 units and run 5 units' to locate another point. From \((0,3)\), move up 6 units and right 5 units to reach the next point \((5,9)\). Draw a straight line through these points to complete the graph.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a method that simplifies the equation of a straight line. It is expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) stands for the \(y\)-intercept. This form is extremely useful because it directly gives you the two crucial components needed to graph a line: the slope and the \(y\)-intercept. By rearranging an equation into this form, you can quickly identify how steep the line is and where it crosses the \(y\)-axis.

• Slope \(m\): Indicates the steepness of the line. A larger absolute value means a steeper slope.
• \(y\)-Intercept \(b\): Shows where the line crosses the \(y\)-axis.

By rearranging equations into this form, the process becomes much more manageable, as seen in the original exercise where we transform \(6x - 5y + 15 = 0\) into \(y = \frac{6}{5}x + 3\). This makes it clear that the slope \(m\) is \(\frac{6}{5}\) and the \(y\)-intercept \(b\) is 3.

Graphing Linear Equations

Once a linear equation is in slope-intercept form, graphing becomes straightforward. The first step is to plot the \(y\)-intercept, the point where the line crosses the \(y\)-axis.
For example, with the equation \(y = \frac{6}{5}x + 3\), the \(y\)-intercept is 3, so you plot the point \((0, 3)\) on the graph.
After plotting the \(y\)-intercept, use the slope \(\frac{6}{5}\) to find another point on the line. The slope represents the vertical change ("rise") over the horizontal change ("run"). For \(\frac{6}{5}\), rise 6 units upwards and run 5 units to the right to find another point \((5, 9)\).

• Plot \(y\)-Intercept: Start at the \(y\)-intercept \((0, b)\).
• Use Slope: From the \(y\)-intercept, follow the slope to plot other points.
• Draw Line: Connect the points with a straight line, extending as needed.

This technique ensures accurate and clean graphing of linear equations.

Slope and Y-Intercept

Understanding the slope and \(y\)-intercept of a line is fundamental to mastering linear equations. The slope \(m\) demonstrates how much the line inclines or declines as you move along the \(x\)-axis. If the slope is positive, the line goes upwards; if negative, it goes downwards.

The \(y\)-intercept \(b\) is the starting point of the line on the \(y\)-axis. It tells you where the graph intersects when \(x = 0\). This is critical as it grounds your graphing framework. For instance, the expression \(y = \frac{6}{5}x + 3\) clearly specifies that the starting point is at \(y = 3\), with a gradually increasing slope.

• Positive Slope: Line rises as \(x\) increases.
• Negative Slope: Line falls as \(x\) increases.
• Zero Slope: Horizontal line, with no rise or fall.
• \(y\)-Intercept: Indicates where the line starts on the \(y\)-axis.

Mastering these components allows for accurate prediction and graphing of lines, ensuring a solid understanding of linear equations.