Problem 47 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts and the y-intercepts of the line. Graph the equation. Label the points where the line crosses the axes. $$ -9 x+y=36 $$

Step-by-Step Solution

Answer

The x-intercept of the line is at point (-4,0) and the y-intercept is at point (0,36).

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the given equation and solve for x. So, \(-9x + 0 = 36\) reduces to \(x = -36 / 9 = -4\). The x-intercept is at the point (-4, 0).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and then solve for y. So, \(-9 * 0 + y = 36\) reduces to \(y = 36\). Hence, the y-intercept is at the point (0, 36).

Step 3: Graphing and Labeling

Plot the x-intercept and y-intercept on a graph, draw a straight line through these two points. The point at which the line crosses the X-axis is the x-intercept, and where it crosses the Y-axis is the y-intercept. So, label the point (-4,0) as the x-intercept and (0, 36) as the y-intercept.

Key Concepts

Understanding the X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis of a graph. Typically, this is the spot where the value of y is zero.
Think of it as the line's way of saying hello to the x-axis! In mathematical terms, to determine the x-intercept from an equation, we set \( y = 0 \) and solve for \( x \).
In our exercise, the equation \( -9x + y = 36 \) becomes \( -9x + 0 = 36 \) when \( y \) is zero.
By solving \( -9x = 36 \), we find that \( x = -4 \). Thus, the x-intercept is \((-4, 0)\).

• To find the x-intercept, set \( y = 0 \) in the equation.
• Solve for \( x \) to locate the intercept point on the x-axis.

Finding the x-intercept helps in understanding where the line will "rest" on the x-axis, making it a crucial step in graphing linear equations effectively.

Determining the Y-Intercept

The y-intercept of a line represents the point where the line meets the y-axis. This intersection occurs when the value of \( x \) is zero.
Imagine it as the starting point of your line on the y-axis.
To find the y-intercept in an equation, set \( x = 0 \) and solve for \( y \). In our example \( -9x + y = 36 \), letting \( x \) be zero simplifies it to \( 0 + y = 36 \).
The answer is \( y = 36 \), giving us a y-intercept at \((0, 36)\).

• Set \( x = 0 \) to find the y-intercept.
• Calculate \( y \) for the intercept position on the y-axis.

The y-intercept is essential for graphing because it acts as one of the key anchors that guides us on how the entire line should be drawn.

Graphing Linear Equations

Graphing linear equations involves plotting points where the line interacts with the axes, creating a visual representation of the equation.
First, plotting the x-intercept and the y-intercept provides two exact points through which the line will pass.
In our case, these points are \((-4, 0)\) and \((0, 36)\). Place these points on graph paper or graphing software.
Next, connect them with a straight line, ensuring it extends across the graph.

• Plot the x-intercept \((-4, 0)\) on the graph.
• Plot the y-intercept \( (0, 36)\) on the graph.
• Draw a straight line through these points to represent the equation.

The graph gives us a clear look at the behavior of the line described by the equation \( -9x + y = 36 \) and helps in understanding its slope and intercepts. This visual approach enhances comprehension and makes it easier to predict the line's behavior in extension.