Problem 38 - Elementary Algebra

Question

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ -6 x+7 y=-14 $$

Step-by-Step Solution

Answer

Slope: \(\frac{6}{7}\); y-intercept: -2.

Step 1: Rearrange the Equation

First, we need to rearrange the given equation into the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Starting with the equation: \(-6x + 7y = -14\), we first add \(6x\) to both sides to isolate the \(y\) term: \(7y = 6x - 14\).

Step 2: Solve for y

Now, to solve for \(y\), divide every term by 7: \( y = \frac{6}{7}x - 2 \). Now, the equation is in slope-intercept form, with \(m = \frac{6}{7}\) and \(b = -2\).

Step 3: Identify the Slope and y-Intercept

From the slope-intercept form \( y = \frac{6}{7}x - 2 \), we can identify the slope \( m \) as \( \frac{6}{7} \) and the y-intercept \( b \) as \(-2\).

Step 4: Graph the Line

To graph this line, start by plotting the y-intercept \((0, -2)\) on the coordinate plane. Then use the slope \(\frac{6}{7}\), which means "rise over run". From \( (0, -2) \), move up 6 units and to the right 7 units to plot the next point \((7, 4)\). Draw a straight line through these points to extend the graph of the line.

Key Concepts

Graphing Linear Equations

Graphing linear equations allows us to visualize the relationship defined by an equation on a coordinate plane. A linear equation like the one given can be represented as a straight line. To graph a linear equation, it's essential first to rearrange it into the slope-intercept form, which is \(y = mx + b\). This format makes it easy to identify key components of the equation that assist in drawing the graph.

• The slope \(m\) indicates the tilt or steepness of the line.
• The \(y\)-intercept \(b\) shows where the line crosses the y-axis.

You start by plotting the \(y\)-intercept, which provides a precise starting point. From there, use the slope to determine the direction and incline of the line. With just these two elements, the entire line can be drawn. Remember, a straight edge or ruler can help ensure the line is drawn accurately across the plane.

Identifying Slope

In the slope-intercept form \(y = mx + b\), the slope \(m\) is crucial. It tells us how steep the line is and the direction it goes, either upward or downward. The slope is often referred to as "rise over run," helping us understand how much the line goes up or down for a given horizontal distance. For example, the slope \(\frac{6}{7}\) from the equation \(y = \frac{6}{7}x - 2\) means:

• The line rises 6 units vertically for every 7 units it runs horizontally.
• A positive slope like \(\frac{6}{7}\) makes the line tilt upwards from left to right.

The greater the absolute value of the slope, the steeper the line. Negative slopes tilt downward, making this an essential point to differentiate when graphing.

Y-Intercept

The \(y\)-intercept is another critical component in graphing linear equations. Represented by \(b\) in the slope-intercept form \(y = mx + b\), it tells you exactly where the line will cross the y-axis. This is the starting point when plotting the line on a graph.For an equation like \(y = \frac{6}{7}x - 2\), the \(y\)-intercept is \(-2\). This means:

• The line crosses the y-axis at the point \((0, -2)\).
• This point is found by setting \(x = 0\), and solving for \(y\).

Once you have your \(y\)-intercept plotted, it serves as the foundation for applying the slope, giving a clear visual cue of where the line will begin before following the slope's guidance across the plane.