Problem 40 - College Algebra

Question

For each of the following exercises, find and plot the \(x\) -and \(y\) -intercepts, and graph the straight line based on those two points. $$y-5=5 x$$

Step-by-Step Solution

Answer

The x-intercept is (-1, 0) and the y-intercept is (0, 5).

Step 1: Rearrange the Equation into Slope-Intercept Form

The equation given is: \(y - 5 = 5x\). First, we want to solve for \(y\) to express it in slope-intercept form, \(y = mx + b\). The expression becomes: \[y = 5x + 5\]

Step 2: Find the x-intercept

To find the \(x\)-intercept, we set \(y = 0\) and solve for \(x\).Starting with the equation \(0 = 5x + 5\), subtract 5 from both sides to get:\[-5 = 5x\]Now, divide by 5:\[x = -1\]So, the \(x\)-intercept is at \((-1, 0)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, we set \(x = 0\) and solve for \(y\).Substitute 0 for \(x\) in the equation:\(y = 5(0) + 5\)\(y = 5\)So, the \(y\)-intercept is at \((0, 5)\).

Step 4: Plot the Points and Graph the Line

Plot the intercepts \((-1, 0)\) and \((0, 5)\) on a coordinate plane. Draw a straight line passing through these two points. This line represents the graph of the equation \(y = 5x + 5\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing and analyzing lines. This form is written as: \(y = mx + b\), where

• \(m\) stands for the slope of the line.
• \(b\) represents the y-intercept, the point where the line crosses the y-axis.

Slope tells you how steep the line is, and the direction it goes.
If \(m\) is positive, the line rises as you move from left to right. If \(m\) is negative, it falls.
The y-intercept \(b\) is where the line meets the y-axis. This is a fixed starting point on the graph.

Graphing Linear Equations

Graphing linear equations is a fundamental skill that allows us to visually see how different values of \(x\) and \(y\) relate to each other.
The process involves using the equation in its slope-intercept form, \(y = mx + b\), to identify two main features: the slope \(m\) and the y-intercept \(b\).
By plotting the y-intercept on the graph as a starting point:

• From this point, use the slope \(m\), rise over run, to determine the next point.
• This means moving vertically (rise) and horizontally (run) from the y-intercept.

Once you have at least two points, draw a line through them. This line shows all possible solutions to the equation.

Finding Intercepts

Finding intercepts is a crucial step in graphing a line because these points provide easy reference locations for sketching the graph.
There are two types of intercepts in the context of linear equations:

• The x-intercept is where the graph crosses the x-axis. To find it, set \(y = 0\) and solve for \(x\).
• The y-intercept is where the line crosses the y-axis. This is found by setting \(x = 0\) and solving for \(y\).

In the example, once we have these intercepts, they can serve as guideposts for plotting the entire line on a coordinate plane.
Make sure to double-check these points against the slope and y-intercept to ensure their accuracy.