Problem 59 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and \(y\) -intercept. c. Use the slope and y-intercept to graph the linear function. \(3 x+y-5-0\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -3x + 5\) with a slope of -3 and a y-intercept of 5.

Step 1: Rewrite To Slope-Intercept Form

First, simplify the given equation \(3x + y - 5 = 0\) to put it into slope-intercept form. This can be done by isolating \(y\). Firstly we need to subtract \(3x\) from both sides, yielding \(y = -3x + 5\).

Step 2: Identify The Slope And Y-Intercept

From the slope-intercept form (\(y = mx + c\)), it is clear that the slope (\(m\)) of the line is -3 and the y-intercept (\(c\)) is 5.

Step 3: Graph The Function

Let's start with the y-intercept at (0,5) on the graph. Then, from this point, use the slope to find the next point. Since the slope is -3, if we move 1 step to the right, we have to move down 3 steps (negative indicates downward direction). Connect these points to make the line.

Key Concepts

Linear Function Graphing

Graphing a linear function is a foundational skill in algebra that allows students to represent equations visually. The most convenient way of doing this is by using the slope-intercept form of a linear equation, which is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Once you have an equation in slope-intercept form, graphing it becomes straightforward. Start by plotting the y-intercept on the Cartesian plane. This point is where the line will cross the y-axis. Then, use the slope to determine the steepness and direction of the line. The slope, often read as 'rise over run', tells you how many units to go up or down for every one unit you go right. To graph the linear function from the exercise, \( y = -3x + 5 \), you start at the point (0, 5) and then follow the slope by moving 3 units down for every 1 unit to the right because the slope is -3.

Determining Slope

What is Slope?

The slope of a line is a measure of its steepness and direction. To find it, look for the coefficient of \( x \) in the slope-intercept form equation, \( y = mx + b \).

In our exercise, the equation \( y = -3x + 5 \) has a slope \( m \) of -3. This means that for every increment of 1 unit across the x-axis, the line descends 3 units on the y-axis. A negative slope indicates a line that goes downwards from left to right, while a positive slope means the line is upward sloping. Always remember that slope is a ratio: it can be expressed as a fraction (such as -3/1 in this case), but whether given as a fraction or an integer, it indicates the same rate of change for y in relation to x.

Pro Tip:

If the slope is a fractional value, that can help you get more accurate points when plotting your graph. For example, a slope of -3/2 would mean you move down 3 units for every 2 units you move to the right.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis and is essential in graphing linear functions. It is represented by the \( b \) value in the slope-intercept equation, \( y = mx + b \).

In the given exercise, the y-intercept is 5, which we can determine from the equation once it has been rearranged to slope-intercept form: \( y = -3x + 5 \). This tells you that when \( x \) is zero, the y-value of the graph will be 5. You would mark this point on the y-axis as your starting place for graphing the linear function.

Remember:

The y-intercept can sometimes be 'hidden' if the equation is in a different form or if \( b \) is zero. However, in the slope-intercept form, its identification is clear-cut, and plotting it is the first step in graphing a linear equation.