Problem 61 - Algebra 1

Question

Find the x-intercept and the y-intercept of the graph of the equation. $$x+5 y=10$$

Step-by-Step Solution

Answer

The x-intercept is \(x = 10\) and the y-intercept is \(y = 2\).

Step 1: Find the x-intercept

Set \(y = 0\) in the equation, and then solve for \(x\). This gives: \[x + 5(0) = 10\]which simplifies to \(x = 10\). Therefore, the x-intercept of the graph of the equation is \(x = 10\).

Step 2: Find the y-intercept

Now set \(x = 0\) in the equation, and then solve for \(y\). This gives: \[0 + 5y = 10\]which simplifies to \(y = 2\). Therefore, the y-intercept of the graph of the equation is \(y = 2\).

Key Concepts

X-Intercept

Understanding the concept of an x-intercept is fundamental when studying linear equations. The x-intercept is the point where the graph of an equation crosses the x-axis. In other words, it is the value of x when y is zero.

To find the x-intercept from a linear equation, you simply set the y variable to zero and solve for x. This process essentially tells us where our line will touch the x-axis if extended. For instance, in the equation \(x + 5y = 10\), setting \(y = 0\) results in \(x = 10\). Hence, the x-intercept is the point \((10, 0)\), which indicates that the line crosses the x-axis 10 units to the right of the origin.

Y-Intercept

Conversely, the y-intercept is the point at which the graph of an equation intersects with the y-axis. To find this point, set the x variable to zero and solve for y. The y-intercept tells us the value of y when x is zero, meaning it is where the line crosses the y-axis.

In our example equation \(x + 5y = 10\), placing x to zero, we get \(5y = 10\), and by dividing both sides by 5, we find \(y = 2\). This reveals that the y-intercept is located at point \((0, 2)\), showing that the line crosses the y-axis 2 units above the origin.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are of the form \(ax + by = c\), where a, b, and c are constants. The solutions to linear equations are a straight line when graphed on a two-dimensional plane.

These equations are convenient because they're simple to manipulate algebraically, and they're an essential component of a variety of applied and theoretical disciplines. Linear equations are also the starting point for understanding more complex functions and mathematical models.

Graphing Linear Equations

When it comes to graphing linear equations, we're essentially plotting the solutions to the equation on a coordinate plane. By finding the x-intercept and the y-intercept, we can draw a line through these two points, which will be the graph of the equation. You can also use additional points that satisfy the equation to confirm the accuracy of the graph.

Graphing is a powerful tool as it provides a visual representation of the solutions and helps understand the relationship between the variables. It is important to practice plotting points and drawing lines correctly to master graphing linear equations. Remember, the slope of the line indicates how steep the line is, and it is calculated as the change in y divided by the change in x between two distinct points on the line.