Problem 32 - Introductory Algebra for College Students

Question

Use intercepts and a checkpoint to graph equation. \(10 y=60-40 x\)

Step-by-Step Solution

Answer

The x-intercept is (1.5,0), the y-intercept is (0,6) and the check point is (1,2). These points can be used to draw a line representing the equation.

Step 1: Find the x-intercept

Set y=0 and solve for x in the equation \(10 y=60-40 x\). So putting y=0 gives \(0=60-40x\). Solving for x gives x=\(\frac{60}{40}=1.5\). So the x-intercept is (1.5, 0).

Step 2: Find the y-intercept

Set x=0 and solve for y in the equation \(10 y=60-40 x\). So setting x=0 gives \(10y=60\). Solving this equation gives y=\(\frac{60}{10}=6\). So, the y-intercept is (0,6).

Step 3: Identify the Check Point

The check point can be any point that is not on the x or y-axis. An easy check point to calculate would be x=1. So, substituting x=1 in to the equation gives \(10y=60-40*1=20\). Solving for y gives y=\( \frac{20}{10}=2\). So, the check point is (1,2).

Step 4: Draw the graph

Plot the x and y intercepts and the check point on the graph. Draw a line through these points.

Key Concepts

Understanding Intercepts

When graphing linear equations, intercepts are crucial tools. They are points where the line crosses the axes.

There are two types of intercepts:

• X-intercept: This is where the line crosses the x-axis. To find it, set the y-value to zero in the equation and solve for x. In our example, by substituting 0 for y in the equation \(10y = 60 - 40x\), we find the x-intercept at (1.5, 0).
• Y-intercept: This is where the line crosses the y-axis. To determine it, set the x-value to zero and solve for y. Substituting 0 for x in the equation gives us the y-intercept at (0, 6).

Finding these intercepts helps form the skeleton of your line, making it easier to graph.

They are simple yet effective in giving us valuable points that outline the general direction and position of the line on the graph.

Navigating the Coordinate System

The coordinate system, or Cartesian plane, is like a map for plotting points of equations. Each point is defined by a pair of numbers, called coordinates, usually in the form of \((x, y)\).

Here's a quick guide on how it works:

• The horizontal axis is the x-axis, and the vertical axis is the y-axis.
• Where they intersect is the origin, marked as (0, 0).
• A point on the graph is defined by its x and y coordinates, indicating its distance from the origin.

This system allows us to visualize the equation we are dealing with.
In this exercise, the x-intercept (1.5, 0) and the y-intercept (0, 6) sum up to form a path on the graph, explaining the relationship between x and y in the equation \(10y = 60 - 40x\).

By plotting these intercepts along with a checkpoint, we can trace a line that exactly represents the equation's solution.

Building a Graph: A Step-by-Step Solution

When constructing a graph, following a clear process ensures accuracy and understanding. Here’s how we do it for a linear equation using intercepts and a checkpoint:

• Find the X-Intercept: Set y to 0 and solve for x. For our equation, this results in the point (1.5, 0).
• Find the Y-Intercept: Set x to 0 and solve for y, leading to the point (0, 6).
• Choose a Checkpoint: Select a simple value for x or y that’s not at the axes – like x = 1. This point, calculated as (1, 2), confirms the line’s direction.
• Draw the Line: Use these points to plot on the coordinate plane. Connect them with a straight line to visualize the equation \(10y = 60 - 40x\).

Each step helps verify the accuracy of the graph.
This process ensures your visualization matches the mathematical relationship defined in the equation. It’s a reliable approach to understanding and solving linear equations systematically.