Problem 58 - Introductory Algebra for College Students

Question

Graph: \(4 x+6 y=12 .\)

Step-by-Step Solution

Answer

The graph of the equation \(4x + 6y = 12\) starts at the y-intercept of (0, 2) and has a slope of -2/3. Thus, for every three units moved to the right on the x-axis, the line moves down by two units on the y-axis.

Step 1 - Rewrite in slope-intercept form

The slope-intercept form of the linear equation is \(y = mx + c\). Let's rewrite the given equation in this form. To do this, subtract \(4x\) from both sides of \(4x + 6y = 12\) to get \(6y = -4x + 12\). Then, divide all terms by 6 to isolate y. This gives the equation in slope-intercept form: \(y = -\frac{2}{3}x + 2\).

Step 2 - Identify the slope and y-intercept

With the equation in slope-intercept form, one can easily identify the slope and the y-intercept. In \(y = -\frac{2}{3}x + 2\), the slope \(m = -\frac{2}{3}\) and the y-intercept \(c = 2\). The slope is the rate of change and signifies the incline of the line. The y-intercept is where the line crosses the y-axis.

Step 3 - Plot the graph

Start by drawing the y-intercept at point (0, 2) on the graph. Then, keep the slope in mind (-2/3) and from the y-intercept move down 2 units (since slope is negative) and to the right by 3 units (denominator of the slope fraction) to get the next point. Draw a line through these points to graph the equation

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to express a linear equation. It is given by the formula \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) represents the y-intercept. Writing an equation in this form helps to easily identify these two key components of a line.
To convert the given equation \( 4x + 6y = 12 \) into the slope-intercept form, we rearrange the equation so that \( y \) is isolated on one side. Subtract \( 4x \) from both sides to get \( 6y = -4x + 12 \).
Next, divide each term by 6, as we need to solve for \( y \). This simplifies to \( y = -\frac{2}{3}x + 2 \). Now, the equation is clearly in slope-intercept form, which greatly aids the process of graphing a line.

Slope and Y-Intercept

Knowing how to find the slope and y-intercept from a linear equation is crucial. Once an equation is expressed in the slope-intercept form \( y = mx + c \), identifying the slope and y-intercept becomes straightforward. In our equation, \( y = -\frac{2}{3}x + 2 \), the coefficient of \( x \) \(-\frac{2}{3}\) is the slope \( m \), and 2 is the y-intercept \( c \).
The slope \( m \) of \(-\frac{2}{3}\) tells us that the line declines by 2 units vertically for every 3 units it moves horizontally. It signifies the steepness and direction of the line.
The y-intercept is the point where the line crosses the y-axis, which in this case is \( (0, 2) \). This point is vital for starting your plot as it provides a fixed reference to draw your line.

Plotting Graphs

Plotting a graph using the slope-intercept form involves a few clear steps. Start by plotting the y-intercept, which is a pivotal point on the graph. For our equation, the y-intercept is at \( (0, 2) \). Mark this point clearly on the y-axis.
Next, utilize the slope \( m = -\frac{2}{3} \). The negative sign indicates that the line will slant downwards. From the y-intercept \( (0, 2) \), move down 2 units (a reflection of the numerator) and then 3 units to the right (the denominator's influence). This brings you to the next point at \( (3, 0) \).
With these two points, you already have a basic guide for the graph. Simply draw a line that passes through the y-intercept and this new point. Extending this line will give the full representation of the linear equation \( 4x + 6y = 12 \). Check your graph by verifying it intersects at the y-intercept and follows the correct slope.