Problem 40 - College Algebra with Corequisite Support

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

x-intercept: (-1, 0); y-intercept: (0, 5). Draw a line through these points.

Step 1: Identify the Linear Equation

The given equation is \( y - 5 = 5x \). This can be rewritten in slope-intercept form: \( y = 5x + 5 \). This equation is a straight line with slope \( m = 5 \) and y-intercept \( b = 5 \).

Step 2: Find the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. To find it, set \( x = 0 \) in the equation \( y = 5x + 5 \). This gives \( y = 5(0) + 5 = 5 \). Hence, the y-intercept is \((0, 5)\).

Step 3: Find the X-Intercept

The x-intercept is the point where the line crosses the x-axis. To find it, set \( y = 0 \) in the equation \( y = 5x + 5 \). This gives \( 0 = 5x + 5 \). Solving for \( x \), we subtract 5 from both sides: \(-5 = 5x\). Dividing both sides by 5, we get \( x = -1 \). Hence, the x-intercept is \((-1, 0)\).

Step 4: Plot the Intercepts on a Graph

To graph the line, plot the x-intercept \((-1, 0)\) and y-intercept \((0, 5)\) on the coordinate plane.

Step 5: Draw the Straight Line

With both intercepts plotted, draw a straight line through these points. This line represents the equation \( y = 5x + 5 \).

Key Concepts

Understanding X-Intercepts

The x-intercept of a line is the point where the graph crosses the x-axis. At this point, the value of the y-coordinate is zero. To find the x-intercept from an equation like \( y = 5x + 5 \), set \( y = 0 \). This way, you solve for \( x \) in the equation.

• Set \( y = 0 \): \( 0 = 5x + 5 \)
• Solve for \( x \) by subtracting 5: \( -5 = 5x \)
• Divide by 5 to isolate \( x \): \( x = -1 \)

Thus, the x-intercept is \((-1, 0)\). It indicates where the line touches the x-axis of a graph.
Understanding the x-intercept can help determine how a line behaves in relation to the x-axis.

Understanding Y-Intercepts

The y-intercept is where a graph crosses the y-axis. At this special point, the x-coordinate is zero. Finding it involves substituting \( x = 0 \) into any linear equation, like \( y = 5x + 5 \), and solving for \( y \).

• Set \( x = 0 \): \( y = 5(0) + 5 \)
• This simplifies to \( y = 5 \)

Therefore, the y-intercept is \((0, 5)\). This point gives a starting location on the graph. Knowing the y-intercept helps visualize where the line begins in relation to the y-axis.

Graphing Linear Equations

Graphing a linear equation means representing it visually on a coordinate plane. To begin, identify the x- and y-intercepts, which serve as reference points. These intercepts are crucial for plotting the line accurately.

• Plot the x-intercept: example \((-1, 0)\)
• Plot the y-intercept: example \((0, 5)\)
• Draw a straight line through these points

By connecting these intercepts with a straight line, you've graphed the equation. This line visually represents all solutions to the equation. Graphing is simple with intercepts because it eliminates choosing random points.

Understanding Slope-Intercept Form

The equation \( y - 5 = 5x \) can be rewritten in what's known as the slope-intercept form, \( y = mx + b \). This form provides quick insights about the line's characteristics.

• \( m \) represents the slope: change in \( y \) for a unit change in \( x \)
• \( b \) is the y-intercept, the starting point on the y-axis

For the equation \( y = 5x + 5 \), the slope \( m \) is 5, indicating how steeply the line rises. The y-intercept \( b \) is 5, showing where the line crosses the y-axis. Using this form makes it easy to graph and understand the line without needing to manually calculate intercepts each time.
Thinking in terms of slope and intercept simplifies understanding linear relationships in everyday algebra.