Problem 78 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ 4 y+12=0 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -3\). The graph is a horizontal line passing through the point (0, -3) on the y-axis.

Step 1: Rewrite the Equation In Slope Intercept Form

To express the function in slope-intercept form, we need to isolate \(y\). First, rewrite the function equation, which is initially \(4y + 12 = 0\). Subtract 12 from both sides to isolate \(4y\), to get \(4y = -12\). Afterwards, divide both sides by 4 to isolate \(y\), yielding \(y = -3\).

Step 2: Identify Slope and Intercept

Given the equation \(y = -3\), observe that \(m = 0\) (since there is no \(x\) term, implying the slope of the equation is zero), and \(b = -3\) which means the y-intercept is -3.

Step 3: Graph the Equation

Plot the y-intercept on the graph at -3 on the y-axis, and since the slope is 0, draw a straight line horizontally through the y-intercept point. It's a horizontal line because the slope is zero.

Key Concepts

Graphing Linear Equations

Understanding how to translate a linear equation into a visual graph is an essential skill in algebra. Graphing a linear equation involves plotting points on a coordinate plane and joining these points to form a straight line. The most common form of a linear equation is the slope-intercept form, which is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis.

To graph the equation from our exercise \( y = -3 \), we notice it's already in slope-intercept form with \( m = 0 \) and \( b = -3 \). Here's how to graph it:

• Begin by plotting the y-intercept (0, -3) on the graph.
• Since the slope \( m \) is 0, the line will be horizontal. This means all points on the line have the same y-coordinate (-3).
• Draw a straight horizontal line through the plotted point across the graph.

Remember, each point on the line is a solution to the equation, and the graph represents all the solutions for the linear equation presented.

Solving for y

Solving a linear equation for \( y \), to put it in slope-intercept form, requires isolating \( y \) on one side of the equation. Here's a simple set of steps:

• First, look for terms that include \( y \) and move them on one side of the equation by adding, subtracting, multiplying, or dividing both sides as needed.
• If \( y \) is multiplied by a coefficient, divide through by that coefficient to get \( y \) alone.
• Any terms not involving \( y \) will be part of the y-intercept \( b \) when you isolate \( y \).

Using the exercise example, we start with \( 4y + 12 = 0 \) and aim to get \( y \) on its own:
Subtract 12 from both sides: \( 4y = -12 \).
Now, divide both sides by 4: \( y = -3 \).
This final equation is now solved for \( y \), and it shows us that no matter what value of \( x \) we choose, \( y \) will always be -3.

Identifying Slope and Y-Intercept

The slope and y-intercept are critical to understanding the behavior of a line on a graph. The slope, represented by \( m \), indicates the steepness and direction of the line. When \( m > 0 \), the line rises from left to right; when \( m < 0 \), it falls. A slope of zero, as in our exercise \( y = -3 \), means the line is horizontal.

The y-intercept, represented by \( b \), is the point where the line crosses the y-axis. To find it, look for the term that remains after you've isolated \( y \). From our example, after solving for \( y \), the equation becomes \( y = -3 \), suggesting that the y-intercept \( b \) is -3. Therefore, the line crosses the y-axis at the point (0, -3). This y-intercept is a pinpoint location we use as the starting point for graphing the linear equation.