Problem 41 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ x+3 y-3=0 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -(1/3)x + 1\). The slope is \(-1/3\) and the y-intercept is 1. A graph of the equation shows a slope of \(-1/3\) starting from the point (0,1).

Step 1: Rewrite Equation in slope-intercept form

The first step will be rearranging the given equation, \(x + 3y - 3 = 0\), into the slope-intercept form. This could be done by isolating \(y\) on one side of the equation. So the equation becomes \(3y = -x + 3 \). Finally, divide through by 3 to isolate \(y\), obtaining \(y = -(1/3)x + 1\).

Step 2: Define Slope and y-intercept

From the equation obtained in step 1, we can therefore determine that the slope \(m\) of the line is \(-1/3\) and the y-intercept \(b\) is 1.

Step 3: Graph the Equation

To graph the equation, plot the y-intercept first, which is the point (0, 1). Starting from this point, use the slope to find additional points. The slope \(-1/3\) means for every 1 unit decrease in \(x\), \(y\) increases by 3. Alternatively, for every 3 units increase in \(x\), \(y\) decreases by 1. Draw a line through these points to complete the graph.

Key Concepts

Linear Equations

A linear equation is an equation that can represent a straight line on a graph. It is usually written in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations involve only the operations of addition, subtraction, and multiplication by a constant. They don't contain any products of variables or higher powers like squares or cubes, which keeps them 'linear.'

Some common characteristics of linear equations include:

• They form a straight line when graphed.
• The solution of a linear equation is typically a line that extends infinitely in two directions on a graph.
• They have a constant rate of change, which is seen as the slope of the line.

Linear equations can describe many real-world relationships, such as determining costs, predicting trends, or calculating distances.

Graphing Linear Equations

Graphing linear equations involves visually representing a linear relationship on a graph. The process helps in understanding the behavior of variables in the equation. The most common method for graphing linear equations is by converting them into the slope-intercept form, \(y = mx + b\), which makes it simpler to identify both the slope and y-intercept. This standard form helps in clearly laying out the graphical interpretation of the equation both in terms of direction and starting point on the y-axis.

To graph a linear equation:

• Start by identifying the y-intercept on the graph, which is the value of \(b\) when \(x = 0\).
• Use the slope \(m\), which denotes the rise over run, to determine additional points. For example, a slope of \(-1/3\) implies moving down 1 unit in y for every 3 units right in x.
• Connect these points with a straight line extending in both directions to complete the graph.

Graphing provides a visual understanding and helps in predicting how changes in one variable affect another in practical situations.

Slope and Y-intercept

The slope and y-intercept are crucial components of a linear equation in the slope-intercept form \(y = mx + b\). These components directly influence the line's direction and position on a graph.

The **slope** \(m\) represents the rate of change. It's the steepness and direction of the line, calculated as the change in y divided by the change in x ("rise over run"). A positive slope means the line is ascending, while a negative slope points downward. In our example, the slope of \(-1/3\) suggests the line falls as you move from left to right.

The **y-intercept** \(b\) is where the line intersects the y-axis. It reveals the value of \(y\) when \(x\) is zero, often serving as a starting point when plotting the graph. For an equation like \(y = -(1/3)x + 1\), the y-intercept is 1, indicating the line crosses the y-axis at \((0,1)\).

Understanding these concepts not only aids in plotting graphs but also in interpreting and predicting the effects of any changes to the equation variables.