Problem 9 - Introductory Algebra for College Students

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 is (10, 0) and the y-intercept is (0, 4).

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 in the equation. This gives \(2x + 5*0 = 20\). Solving for x gives \(x = 20/2 = 10\). So the x-intercept is (10, 0).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 in the equation. This gives \(2*0 + 5y = 20\). Solving for y gives \(y = 20/5 = 4\). So the y-intercept is (0, 4).

Key Concepts

Understanding the x-intercept

In algebra, the x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of y is always zero.
This makes sense because, on the x-axis, the y-coordinate is always zero. To find the x-intercept from a linear equation such as \( 2x + 5y = 20 \), you simply set \( y = 0 \) and solve for \( x \).

• Replace \( y \) in the equation: \( 2x + 5(0) = 20 \).
• Solve for \( x \): \( 2x = 20 \).
• Divide both sides by 2: \( x = 10 \).

The x-intercept is represented as a point on the graph and is written as \( (10, 0) \). This means the graph will touch or cross the x-axis at 10.

Discovering the y-intercept

The y-intercept is another important aspect of a linear equation. It is the point where the graph crosses the y-axis. At this specific point, the value of x is always zero.
This is because the y-axis represents when the x-coordinate is zero. Finding the y-intercept involves a slightly different approach because you set \( x = 0 \) in the equation.

• Replace \( x \) in the equation: \( 2(0) + 5y = 20 \).
• Simplify and solve for \( y \): \( 5y = 20 \).
• Divide both sides by 5 to find \( y \): \( y = 4 \).

The resulting point, the y-intercept, is denoted as \( (0, 4) \). This indicates that the graph will intersect the y-axis at 4.

Linear equations explained

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They are usually in the form \( Ax + By = C \).
In a linear equation, both x and y variables can be represented graphically with a straight line on a coordinate plane.
The line defined by a linear equation flows smoothly without any curves or angles.Let's break down some important traits of linear equations:

• Two variables: Often involves x and y.
• Simplicity: No exponents or variables multiplied together.
• Straight line: The graph of the equation forms a straight line.

Finding intercepts by setting x or y to zero gives valuable insights into the relationships represented by this line. Each intercept provides a starting point for graphing linear equations, making it easier to visualize and predict relationships in data.