Problem 43 - Algebra 2

Question

Find the slope and \(y\) -intercept of the graph of each function. $$ -x-2 y=12 $$

Step-by-Step Solution

Answer

The slope of the graph is \(-\frac{1}{2}\) and the y-intercept is -6.

Step 1: Convert to Slope-Intercept Form

The first step is to convert the given linear equation to slope-intercept form, which is: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, solve the given equation for \(y\). Starting with the equation \(-x - 2y = 12\), add \(x\) to both sides to get \(-2y = x + 12\). Then, divide both sides by -2 to isolate \(y\), yielding \(y = -\frac{1}{2}x - 6\).

Step 2: Identify the Slope

In the slope-intercept form \(y = mx + b\), the coefficient of \(x\) is the slope \(m\). Hence, from the equation \(y = -\frac{1}{2}x - 6\), the slope \(m\) is \(-\frac{1}{2}\).

Step 3: Identify the Y-Intercept

In the slope-intercept form \(y = mx + b\), the constant term \(b\) represents the y-intercept. From the equation \(y = -\frac{1}{2}x - 6\), the y-intercept \(b\) is -6. This is the point on the graph where the line crosses the y-axis, which is \((0, -6)\).

Key Concepts

Finding Slope

Understanding how to find the slope of a linear equation is crucial in graphing and analyzing the behavior of the line it represents. The slope indicates how steep the line is and the direction it slants. To find it, remember the slope-intercept form of a linear equation, which is
\( y = mx + b \),
where \( m \) is the slope. The slope can be positive or negative, and this sign will dictate if the line inclines upwards or downwards respectively as you move from left to right on a graph.

For the given equation \(-x - 2y = 12\), after rearranging it to the slope-intercept form, we identify the slope as the coefficient of \( x \), which is \(-\frac{1}{2}\). This means that for every single unit the x-coordinate increases, the y-coordinate decreases by half a unit, indicative of a downward slope. Simplifying this concept:

• If the slope is negative, the line goes down hill as it moves left to right.
• If the slope is positive, the line goes up hill as it moves left to right.
• If the slope is zero, the line is perfectly horizontal.
• If the slope is undefined, the line is perfectly vertical.

Identifying Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. To identify the y-intercept, look at the constant term in the slope-intercept form of the equation (\( y = mx + b \)). Here, \( b \) is the y-intercept and it represents not just a value, but a specific point on the y-axis, specifically at \((0, b)\).

In our exercise, after converting the equation \( -x - 2y = 12 \) to slope-intercept form, we obtained the y-intercept as \( -6 \). Therefore, the line intersects the y-axis at the point \( (0, -6) \). It's an important feature because it provides a starting point from which to draw the graph of the equation. Often, it's the first point students plot when graphing a linear equation.

Linear Equations

Linear equations form the foundation of algebra and are the simplest type of equations to understand and work with. They describe straight lines when graphed on a Cartesian plane and are expressed in various forms, including slope-intercept form, point-slope form, and standard form. Among these, the slope-intercept form, denoted as
\( y = mx + b \),
is especially valuable because it directly shows both the slope and the y-intercept, enabling you to quickly sketch the graph of the line without much calculation.

In practice, to graph a linear equation, start at the y-intercept and use the slope to determine the direction and steepness of the line. The slope's numerator tells you the vertical change, while the denominator indicates the horizontal change. Always remember that linear equations create straight lines, so you only need two points to draw the entire line, but having the slope and y-intercept makes plotting even simpler.