Problem 42 - Elementary Algebra

Question

Graph the line that passes through the given point and has the given slope. (Objective 3 ) $$(-1,0), m=\frac{3}{4}$$

Step-by-Step Solution

Answer

Graph the line using the slope-intercept form: start at \((-1,0)\), then move up 3 and right 4 to plot another point and draw the line.

Step 1: Identify the point-slope form

To graph the line, we use the point-slope form of a linear equation, which is given by \( y - y_1 = m(x - x_1) \). Here, \( m = \frac{3}{4} \), \( x_1 = -1 \), and \( y_1 = 0 \).

Step 2: Substitute values into the formula

Substitute the known values into the point-slope equation: \( y - 0 = \frac{3}{4}(x - (-1)) \), simplifying it to \( y = \frac{3}{4}(x + 1) \).

Step 3: Simplify the equation

Expand the equation to make it easier to graph. Distribute \( \frac{3}{4} \) through the parenthesis: \( y = \frac{3}{4}x + \frac{3}{4} \).

Step 4: Find the y-intercept

In the equation \( y = \frac{3}{4}x + \frac{3}{4} \), identify the y-intercept, which is the constant term \( \frac{3}{4} \). This means the line crosses the y-axis at \( y = \frac{3}{4} \).

Step 5: Plot the point and draw the slope

Start by plotting the given point \((-1, 0)\) on the coordinate plane. Use the slope \( m = \frac{3}{4} \) to determine another point. From \((-1, 0)\), move 3 units up and 4 units to the right, landing at \((3, 3)\). Plot this second point.

Step 6: Draw the line

Connect the points \((-1, 0)\) and \((3, 3)\) with a straight line. This line represents the equation \( y = \frac{3}{4}x + \frac{3}{4} \), with the given slope and passing through the specified point.

Key Concepts

Point-Slope Form

The point-slope form is a way to write the equation of a line. It is useful when you know the slope of a line and one point on the line. The form is written as:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) represents the slope of the line, and \((x_1, y_1)\) is a given point on the line.
To use this form

• Replace \(x_1\) and \(y_1\) with the coordinates of the point.
• Replace \(m\) with the slope.

This gives you the equation specific to the line passing through the known point.
In our exercise, the point given is \((-1,0)\) and the slope is \(\frac{3}{4}\). So, we substitute these values into the formula like so:

• \(y - 0 = \frac{3}{4}(x - (-1))\)

This form can then be simplified to help plot or understand the line's behavior.

Y-intercept

The y-intercept is where the line crosses the y-axis on a graph. It is an important feature of a linear equation written in slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept.
In a graph, the y-intercept tells us at what point the line meets the vertical axis. It provides a starting point to draw the rest of the line when you have the slope.
In the equation from our exercise \(y = \frac{3}{4}x + \frac{3}{4}\):

• The term \(\frac{3}{4}\) without the \(x\) is the y-intercept.

This means the line will cross the y-axis at \(y = \frac{3}{4}\). Knowing the y-intercept helps in quickly plotting the line on a graph because you immediately know a point on that line.

Slope

The slope of a line describes its steepness and direction, calculated as "rise over run." It is represented by \(m\) in the equation of a line. Slope is a crucial aspect because it tells you how the line rises or falls as you move along it.

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope means the line is horizontal.
• An undefined slope means the line is vertical.

The slope formula is \(m = \frac{\text{change in } y}{\text{change in } x}\).
For our exercise, the slope \(m\) is \(\frac{3}{4}\). This indicates that for every 4 units you move right along the x-axis, the line moves up 3 units. Using these small movements, you can plot the line by starting from one point and using the slope to find another point.