Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Do not graph the equation. $$2 x+5 y=20$$

Step-by-Step Solution

Answer

The \(x\)-intercept of the graph is 10 and the \(y\)-intercept is 4.

Step 1: Find the \(x\)-intercept

Set \(y\) to be 0 in the equation, then solve for \(x\). In the given equation: \(2x + 5(0) = 20\), which simplifies to \(2x = 20\). Solving for \(x\) we get \(x = 20/2 = 10\). So the \(x\)-intercept is 10.

Step 2: Find the \(y\)-intercept

Set \(x\) to be 0 in the equation, then solve for \(y\). In the given equation: \(2(0) + 5y = 20\), which simplifies to \(5y = 20\). Solving for \(y\) we get \(y = 20/5 = 4\). So the \(y\)-intercept is 4.

Key Concepts

X-intercept Calculation

Understanding how to find the x-intercept of a linear equation is a crucial skill in algebra. The x-intercept is the point where the line crosses the x-axis on a graph. This occurs when the value of y is zero. To calculate the x-intercept, we follow a simple process where we set the value of y to zero and solve for x.

For the equation 2x + 5y = 20, following this method:

• First, we substitute y with 0, making the equation 2x = 20.
• Then, we solve for x by dividing both sides by 2, resulting in x = 10.

Remember that the x-intercept can be written as a point, in this case, (10, 0), indicating that the line crosses the x-axis at the point where x is 10.

Y-intercept Calculation

In contrast to the x-intercept, the y-intercept is the point where the line crosses the y-axis. In order to find this point, we need to figure out the value of y when x equals zero. The y-intercept gives an insight into the starting value of the equation when no x-values are influencing the outcome.

Using our same linear equation 2x + 5y = 20:

• Insert 0 for x, which transforms our equation to 5y = 20.
• Then, we find y by dividing both sides by 5, which simplifies to y = 4.

So, the y-intercept is at (0, 4), meaning that the line intersects the y-axis at the point where y is 4.

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 the foundation for understanding how different variables interact with each other. They are represented by the form ax + by = c, where a, b, and c are constants, and x and y are variables.

The graph of a linear equation is always a straight line, and this line is fully described by its slope and intercepts. As seen in the given equation, 2x + 5y = 20, we can identify the behavior of the line without graphing just by finding its x- and y-intercepts. Linear equations are key to not only algebra but also to understanding a vast array of real-world relationships.

Algebraic Problem-Solving

When tackling algebraic problem-solving, we need to develop systematic strategies. These strategies typically involve identifying what we're solving for, arranging the equation accordingly, and executing arithmetic operations. Patience and practice are essential to become efficient at problem-solving in algebra.

To illustrate, with our equation 2x + 5y = 20, the problem-solving process involves two steps:

• Identifying the required intercepts (x-intercept and y-intercept).
• Isolating and solving for the desired variable while treating the other variable as zero.

By carefully and methodically working through these steps, we're able to reach a solution without the need to graph the equation, showcasing the power of algebraic manipulation and problem-solving techniques.