Problem 10 - 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. $$2 x+6 y=30$$

Step-by-Step Solution

Answer

The x-intercept of the given equation is \( 15 \) and the y-intercept is \( 5 \).

Step 1: Find the x-intercept

To find the x-intercept, substitute \( y = 0 \) into the equation. We have \( 2x + 6 * 0 = 30 \), which simplifies to \( 2x = 30 \). Solving for \( x \), gives \( x = 15 \). So, the x-intercept is \( 15 \).

Step 2: Find the y-intercept

To find the y-intercept, substitute \( x = 0 \) into the equation. We have \( 2 * 0 + 6y = 30 \), which simplifies to \( 6y = 30 \). Solving for \( y \), gives \( y = 5 \). So, the y-intercept is \( 5 \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe straight lines when graphed on a coordinate plane. A standard form of a linear equation in two variables, like this one, is expressed as \( Ax + By = C \). Here, \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables. In the given equation, \( 2x + 6y = 30 \), \( A = 2 \), \( B = 6 \), and \( C = 30 \). Linear equations always represent straight lines due to their constant rate of change.
To break it down further:

• The term with \( x \) (\( 2x \)) determines the slope in the horizontal direction.
• The term with \( y \) (\( 6y \)) affects the slope in the vertical direction.
• The constant term (\( 30 \)) offsets the line along the coordinate axes.

By understanding each of these parts, solving and interpreting linear equations becomes easier.

Coordinate Geometry

Coordinate Geometry, often called analytic geometry, allows for the exploration of geometry using algebra. It helps to precisely place geometric shapes on a grid, known as the coordinate plane. The plane consists of two axes:

• The horizontal axis (x-axis)
• The vertical axis (y-axis)

The intersections of lines with these axes are known as intercepts.
To find the intercepts:

• **X-intercept:** This occurs where the graph crosses the x-axis. At this point, the y-coordinate is zero. For instance, substituting \( y = 0 \) in the equation \( 2x + 6y = 30 \), we solve for \( x \).
• **Y-intercept:** This occurs where the line crosses the y-axis. Here, the x-coordinate is zero. For the y-intercept, substituting \( x = 0 \) into the same equation gives us the value of \( y \).

Understanding where a line intersects the axes can give insights into the line's direction and position within the plane.

Mathematical Solutions

Solving linear equations involves operations to find the values of variables that make the equation true. In terms of intercepts, this means isolating one variable to find specific points.**Steps to find intercepts:**

• Set one variable to zero and solve for the other.
• For the x-intercept in \( 2x + 6y = 30 \), set \( y = 0 \):
  - The equation simplifies to \( 2x = 30 \).
  - Dividing both sides by 2 gives \( x = 15 \).
• For the y-intercept, set \( x = 0 \):
  - The equation becomes \( 6y = 30 \).
  - Dividing by 6, we find \( y = 5 \).

These solutions show that the intercepts are the specific points where the line defined by the equation crosses the axes. Intercepts are useful not just for graphing, but also for understanding relationships in linear equations.