Problem 38 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$-6 x+7 y=-14$$

Step-by-Step Solution

Answer

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

Step 1: Write the Equation in Slope-Intercept Form

The slope-intercept form of a line's equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. First, solve the given equation \(-6x + 7y = -14\) for \( y \). Add \(6x\) to both sides to get \(7y = 6x - 14\).

Step 2: Solve for y

Divide every term by 7 to isolate \( y \). The equation becomes \( y = \frac{6}{7}x - 2 \). Now the equation is in the slope-intercept form.

Step 3: Identify the Slope and y-Intercept

From the equation \( y = \frac{6}{7}x - 2 \), the slope \( m \) is \( \frac{6}{7} \), and the y-intercept \( b \) is \(-2\).

Step 4: Graph the Line

Start by plotting the y-intercept, which is the point (0, -2). From this point, use the slope \( \frac{6}{7} \), which means "rise 6, run 7," to plot another point. Moving up 6 units and over 7 units to the right from \( (0, -2) \), plot the point \( (7, 4) \). Draw a line through these points to complete the graph of the line.

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. In the slope-intercept form of a linear equation, represented by \( y = mx + b \), \( m \) is the slope. This value indicates how much the line rises or falls as we move along the x-axis. For the given equation, after converting it into slope-intercept form, we find that the slope \( m \) is \( \frac{6}{7} \). This tells us that for every 7 units we move horizontally—meaning along the x-axis—the line goes up by 6 units vertically. Slope has a few key characteristics:

• Positive slope: The line rises from left to right.
• Negative slope: The line falls from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

In our case, a slope of \( \frac{6}{7} \) indicates a mild upward trajectory as we move from left to right.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept. It is the value of \( y \) when \( x = 0 \). This is because, at the y-intercept, the line meets the y-axis, where all points have an x-coordinate of 0.For the equation \( y = \frac{6}{7}x - 2 \), the y-intercept \( b \) is -2. This means that the line crosses the y-axis at the point (0, -2).

If you were to graph this, you'd start by plotting the point on the y-axis directly 2 units below the origin. This is an essential step in the process of graphing, providing a reliable starting point for plotting the rest of the line.

Graphing Linear Equations

Graphing linear equations involves translating an equation in slope-intercept form, \( y = mx + b \), onto the coordinate plane. This process visually represents the line described by the equation.

To graph the line from the equation \( y = \frac{6}{7}x - 2 \), begin by plotting the y-intercept at (0, -2). This gives the initial point on the graph.

Next, use the slope, which is \( \frac{6}{7} \), to determine the direction and steepness of the line. From the y-intercept, move 6 units up ("rise") and 7 units to the right ("run"), putting you at the point (7, 4). Plotting this second point enables you to draw a straight line through both points, extending in both directions.

Remembering these key steps:

• Identify and plot the y-intercept \( (0, b) \).
• Use the slope for direction and spacing \( (rise, run) \).
• Draw a line through the points.

This method ensures a precise representation of the linear equation on the graph.