Problem 90 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$y=x+8$$

Step-by-Step Solution

Answer

The y-intercept is 8 and the x-intercept is -8.

Step 1: Identify y-intercept

To identify the y-intercept, set \(x = 0\) in the given equation \(y = x + 8\). Replacement of \(x\) yields, \(y = 0 + 8 = 8\). Thus, the y-intercept is 8.

Step 2: Identify x-intercept

To identify the x-intercept, set \(y = 0\) in the given equation \(y = x+8\). Replacement of \(y\) gives \(0 = x + 8\). Solving this equation, we get, \(x= -8\). Thus the x-intercept is -8.

Key Concepts

Understanding the x-intercept

When we talk about the x-intercept of a linear equation, we are referring to the point where the graph of the equation crosses the x-axis. This is the point where the value of the y-coordinate is zero because any point on the x-axis has a y-value of 0.
To find the x-intercept, you follow a straightforward method:

• Set the y value of the equation to 0.
• Solve for x, the remaining variable.

In the context of our example equation, which is given as \( y = x + 8 \), we set \(y = 0\). This gives us the equation \(0 = x + 8\). Solving for \(x\), we subtract 8 from both sides to get \(x = -8\). Therefore, the x-intercept is at the point (-8, 0). This tells us exactly where the graph crosses the x-axis.

Exploring the y-intercept

A y-intercept of a linear equation is where the graph of the line crosses the y-axis. At this point, the value of x is zero because all points on the y-axis have x-values of 0.
To determine the y-intercept, the procedure is similarly direct:

• Set the x value in the equation to 0.
• Solve for y.

For the equation \(y = x + 8\), we substitute \(x = 0\). This results in \(y = 0 + 8\), which simplifies to \(y = 8\). Therefore, the y-intercept is at the point (0, 8).
This point indicates where the line will cross the y-axis, providing a key point for graphing the equation.

Basic principles of solving linear equations

Solving linear equations involves finding the values of the variables that satisfy the equation. A linear equation typically looks like \( y = mx + b \), where \(m\) and \(b\) are constants.
When solving, the goals are:

• Isolate the variable.
• Use basic arithmetic operations, like addition or subtraction, to simplify the equation.

For the example equation \(y = x + 8\), we solve for the intercepts by isolating either x or y. Solving the equation by setting \(y = 0\) gives the x-intercept, and setting \(x = 0\) gives the y-intercept.
Understanding the intercepts is fundamental, as they provide critical points to sketch the line on a graph, helping to visualize the relationship between x and y.