Problem 12 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. Graph the equation. $$ x-y=1 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is 1 and the y-intercept of the equation is -1.

Step 1: Find the x-intercept

In order to find the x-intercept, y is set to be 0 in the equation. The equation becomes: \(x - 0 = 1\). Solving for x gives that the x-intercept is 1.

Step 2: Find the y-intercept

Similarly, to find the y-intercept, x is set to be 0 in the equation. The equation becomes: \(0 - y = 1\). Solving for y gives that the y-intercept is -1.

Step 3: Graph the equation

Plot the x-intercept at (1,0) and the y-intercept at (0,-1) on the graph. The line passing through these points is the graph of the equation. This line will slope up from left to right because the line's equation is \(y = x - 1\), which is a line with a positive slope.

Key Concepts

Understanding the X-Intercept

To find the x-intercept of a linear equation, set the y-value to zero. This is because the x-intercept represents the point where the line crosses the x-axis, and along the x-axis, the y-value is always zero.

For the equation given, \(x - y = 1\), substitute zero for \(y\) to find the x-intercept:

• Substitute \(y = 0\) into the equation: \(x - 0 = 1\)
• Solve for \(x\):

Therefore, \(x = 1\).

This shows that the x-intercept is at the point (1, 0) on the graph. This means that the graph will cross the x-axis at this point, indicating an output of zero when the input is 1.

Discovering the Y-Intercept

The y-intercept is found by setting the x-value to zero. This point shows where the line crosses the y-axis, meaning that at this point, the x-value is zero.

Using the given equation, \(x - y = 1\), we find the y-intercept by substituting \(x = 0\):

• Substitute \(x = 0\) into the equation: \(0 - y = 1\)
• Solve for \(y\):

Thus, \(y = -1\).

This indicates that the y-intercept is at the point (0, -1) on the graph. At this spot, the graph crosses the y-axis. Understanding the location of the y-intercept helps in drawing the graph of the entire equation accurately.

Graphing Linear Equations Effectively

Graphing linear equations involves plotting intercepts and understanding the line's slope. For the equation \(x - y = 1\), we've already identified:

• The x-intercept is (1, 0)
• The y-intercept is (0, -1)

To graph this equation, plot these points on a coordinate grid. Then, draw a straight line through them. This line represents the equation \(y = x - 1\).

Understanding the slope is also crucial. Here, the slope is 1, indicating that for every unit increase in \(x\), \(y\) also increases by one. This positive slope means the graph will slant upwards, left to right. Such knowledge allows you to visualize and sketch the path of the line accurately, fostering a clear interpretation of the linear relationship.