Problem 31 - Introductory Algebra for College Students

Question

Use intercepts and a checkpoint to graph equation. \(25 y=100-50 x\)

Step-by-Step Solution

Answer

The eqaution \(y=4-2x\) has x-intercept at (2,0), y-intercept at (0,4), and a checkpoint at (1,2). Join these points to create a straight line graph which represents the equation.

Step 1: Convert to slope-intercept form

Divide all terms by 25 to isolate y on the left side of the equation. This results in the equation \(y=4-2x\)

Step 2: Find the x and y intercepts

The y-intercept is the value of y when x=0. Substituting 0 for x in the equation gives \(y=4\), so the y intercept is 4. The x-intercept is the value of x when y=0. Substituting 0 for y in the equation gives \(x=2\), so the x intercept is 2.

Step 3: Find a checkpoint

A checkpoint is a point on the line that is not the x or y intercept. A good choice is when \(x=1\). Substituting 1 for x in the equation gives \(y=2\), so (1,2) is a checkpoint.

Step 4: Draw the graph

Using graph paper, plot the x and y intercepts and the checkpoint. Draw a straight line through these points. This line is the graph of the equation. Remember that the line should extend beyond these points in both directions, as it goes infinitely in both the positive and negative direction.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an essential tool in graphing. It is written as \(y = mx + b\). Here, \(m\) represents the slope, which tells us how steep the line is and whether it rises or falls as we move from left to right.
The \(b\) is the y-intercept, indicating the point where the line crosses the y-axis. In the given exercise, the equation \(y = 4 - 2x\) is already in slope-intercept form.

• \(m = -2\): This negative slope means the line falls as we move to the right.
• \(b = 4\): The line crosses the y-axis at \(y = 4\).

Understanding this form simplifies graphing because it provides both the slope and the y-intercept directly.

X-Intercept

The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis.
To find it, we set \(y = 0\) in the equation and solve for \(x\).
This tells us where the line meets the x-axis, which is crucial for drawing an accurate graph.
In this exercise, setting \(y = 0\) in the equation \(y = 4 - 2x\) results in \(0 = 4 - 2x\).
Solving this, we find \(x = 2\).
So, the x-intercept of the line is at the point (2, 0).
By identifying this intercept, we have a precise marker to plot on the x-axis when sketching the line.

Y-Intercept

The y-intercept is the point where the graph of a linear equation crosses the y-axis.
It can be found by setting \(x = 0\) in the equation, leaving us with the value of \(y\).
In our equation \(y = 4 - 2x\), substituting \(x = 0\) gives us \(y = 4\).
This means the line crosses the y-axis at the point (0, 4).
The y-intercept is a starting point for graphing and helps in determining the direction of the line alongside the slope.
It's always a fixed point on the y-axis, making it easy to locate and plot.

Coordinate Plane

Graphing a linear equation takes place on the coordinate plane, which consists of two axes: the horizontal x-axis and the vertical y-axis.
Each point on this plane is represented by a pair of coordinates \((x, y)\).
To graph the equation \(y = 4 - 2x\), we use the coordinate plane to plot points like the x-intercept (2, 0) and the y-intercept (0, 4).
A checkpoint, like (1, 2), helps confirm that the line is accurately drawn.

• Plot each intercept: (2, 0) and (0, 4).
• Mark the checkpoint: (1, 2).
• Draw a line through these points, extending infinitely in both directions.

The coordinate plane is a visual aid that allows us to see the relationship between the equation's components and how the line stretches across both axes.