Problem 8 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(y=-3 x-5\)

Step-by-Step Solution

Answer

The y-intercept is (0, -5) and the x-intercept is \((-\frac{5}{3}, 0)\).

Step 1: Finding the y-intercept

The y-intercept occurs when the value of \(x\) is 0. Substitute \(x = 0\) into the equation \(y = -3x - 5\) and solve for \(y\). This gives us \(y = -3(0) - 5 = -5\). Thus, the y-intercept is the point \((0, -5)\).

Step 2: Finding the x-intercept

The x-intercept occurs when the value of \(y\) is 0. Substitute \(y = 0\) into the equation \(y = -3x - 5\) and solve for \(x\). This gives us \(0 = -3x - 5\). Add 5 to both sides: \(5 = -3x\), then divide by -3 to solve for \(x\): \(x = -\frac{5}{3}\). The x-intercept is the point \((-\frac{5}{3}, 0)\).

Step 3: Drawing the graph

To graph the equation \(y = -3x - 5\), start by plotting the y-intercept \((0, -5)\) on the graph. Then plot the x-intercept \((-\frac{5}{3}, 0)\). Draw a straight line through these two points, extending it in both directions, to represent the graph of the equation.

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is the point where the graph crosses the x-axis. At this point, the y-value is always zero. To find the x-intercept, we need to set the equation to zero for the y-variable and solve for x. For example, in the equation \( y = -3x - 5 \), set \( y = 0 \) to find the x-intercept:

• Substitute \( y = 0 \) into the equation: \( 0 = -3x - 5 \).
• Add 5 to both sides to get \( 5 = -3x \).
• Divide each side by -3 to isolate \( x \): \( x = -\frac{5}{3} \).

So, the x-intercept is \(-\frac{5}{3}\) and the intersecting point is \( \left(-\frac{5}{3}, 0 \right) \). Understanding where the graph crosses the x-axis is crucial because it helps in visualizing and analyzing the behavior of the line.

Identifying the y-intercept

The y-intercept in a linear equation is where the line crosses the y-axis. This occurs when the value of x is zero. Finding the y-intercept is straightforward: substitute \( x = 0 \) into the equation and solve for y.Here's how it works with our equation, \( y = -3x - 5 \):

• Set \( x = 0 \) in the equation: \( y = -3(0) - 5 \).
• This simplifies to \( y = -5 \).

Thus, the y-intercept is the point \((0, -5)\). Understanding the y-intercept is important as it provides a starting point for graphing a line and understanding its alignment on the y-axis.

Plotting and interpreting linear graphs

Graphing linear equations involves plotting intercepts and drawing a line through them. This line represents all solutions of the equation. To start graphing, follow these simple steps:To graph the equation \( y = -3x - 5 \):

• Plot the y-intercept \( (0, -5) \) on the y-axis.
• Then, plot the x-intercept \( \left(-\frac{5}{3}, 0\right) \) on the x-axis.
• Draw a straight line through these points, extending it across the graph.

This graph visually represents the linear relationship between x and y as defined by the equation. Graphs help in understanding trends, direction, and the solution set of a linear function. Once you have the graph, you can visually interpret additional features such as slope and direction, facilitating deeper insights into the equation's behavior.