Problem 62 - Algebra 1

Question

Find the x-intercept and the y-intercept of the graph of the equation. $$y=x-5$$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is 5 and the y-intercept is -5. In terms of points, the x-intercept is at the point (5,0) and the y-intercept is at the point (0,-5).

Step 1: Find the x-intercept

To find the x-intercept, set \(y=0\) and solve the equation for \(x\). In the given equation, when substituting \(y=0\), we get \(0=x-5\). Solving for \(x\) gives \(x=5\). So, the x-intercept is 5.

Step 2: Find the y-intercept

To find the y-intercept, set \(x=0\) and solve the equation for \(y\). In the given equation, when substituting \(x=0\), we get \(y=0-5\). Solving for \(y\) gives \(y=-5\). So, the y-intercept is -5.

Step 3: State the Intercept Points

From step 1 and step 2, \(x\)-intercept is at the point (5,0) and \(y\)-intercept is at the point (0,-5).

Key Concepts

Intercepts

Intercepts are key points where a line or graph meets the axes on a coordinate plane. Understanding how to find these can enhance your grasp of graphing linear equations.
To find the **x-intercept**, determine where the graph crosses the x-axis. This happens when the value of y is zero. Substitute **0** for **y** in the equation and solve for **x**. In our example, using the equation \(y = x - 5\):

• Set \(y = 0\): \(0 = x - 5\)
• Solve for \(x\): \(x = 5\)

Therefore, the x-intercept is at the point (5, 0).
The **y-intercept** is the point where the graph crosses the y-axis. This is found by setting **x** to zero and solving for **y**. Using the same equation:

• Set \(x = 0\): \(y = 0 - 5\)
• Solve for \(y\): \(y = -5\)

Thus, the y-intercept is at the point (0, -5).Identifying intercepts allows for easier graph plotting and gives a quick view of where the line sits on a graph.

Solving Equations

Solving equations is a fundamental part of algebra and essential for finding intercepts. A linear equation is typically expressed as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Let's break down the process of finding values within an equation like \(y = x - 5\).To solve for **x-intercepts**, follow these steps:

• Set \(y\) to zero, because the x-intercept occurs on the x-axis, where y is always zero.
• Substitute into the equation: \(0 = x - 5\).
• Add 5 to both sides to solve for \(x\): \(x = 5\).

Now, for **y-intercepts**:

• Set \(x\) to zero since the y-intercept is where the line touches the y-axis.
• Substitute this into the equation: \(y = 0 - 5\).
• Solve for \(y\), yielding \(y = -5\).

These solved equations reveal where the line intersects with each axis, crucial for accurate graphing.

Coordinate Plane

The coordinate plane is a two-dimensional space where we graph linear equations. It consists of a horizontal axis, called the x-axis, and a vertical axis, known as the y-axis. These axes intersect at the origin point (0,0).
When you graph a linear equation on the coordinate plane, every solution to the equation is represented as a point with coordinates (x, y). Knowing the intercepts can quickly help you sketch a graph by providing two important points on the line. To understand how to graph **y = x - 5**:

• Begin by plotting the intercepts on the plane: the x-intercept at (5, 0) and the y-intercept at (0, -5).
• Draw a straight line through these points, extending in both directions.
• Ensure the line continues indefinitely, orienting as the graph suggests.

The coordinate plane allows you to visualize the relationship between variables in a linear equation, making it easier to interpret and analyze data. Recognizing how lines interact with the x-axis and y-axis via intercepts deepens your comprehension of graph behavior.