Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ 7 x+2 y=56 $$

Step-by-Step Solution

Answer

The x-intercept is (8, 0) and the y-intercept is (0, 28).

Step 1: Find the x-intercept

The x-intercept is the point where the graph of the equation crosses the x-axis. At this point, the value of \( y \) is zero. Substitute \( y = 0 \) in the equation \( 7x + 2y = 56 \):\[7x + 2(0) = 56 \7x = 56 \x = \frac{56}{7} \x = 8\]The x-intercept is \( (8, 0) \).

Step 2: Find the y-intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. At this point, the value of \( x \) is zero. Substitute \( x = 0 \) in the equation \( 7x + 2y = 56 \):\[7(0) + 2y = 56 \2y = 56 \y = \frac{56}{2} \y = 28\]The y-intercept is \( (0, 28) \).

Key Concepts

Understanding Linear Equations

Linear equations are mathematical statements that show equality between two expressions, involving one or more variables. They usually form a straight line when graphed, hence the name "linear." A typical linear equation in two variables, such as the one given, can be expressed in the standard form: \(Ax + By = C\). In our example, \(7x + 2y = 56\), \(A = 7\), \(B = 2\), and \(C = 56\).Here's why linear equations are important:

• They describe straight-line relationships between variables.
• They are frequently used in economics, science, and engineering to model relationships.
• They help us find important features of graphs such as intercepts.

To solve for \(x\) and \(y\) using this equation, we find specific points which make understanding movements along a graph easier.

The Art of Graphing

Graphing is a visual way to represent mathematical equations and relationships. It involves plotting points on the Cartesian plane (coordinate system) to show how variables relate. For linear equations like \(7x + 2y = 56\), the goal is to create a straight line.To graph this equation, you'll need:

• The x-intercept, where the line crosses the x-axis, is found by setting \(y = 0\) and solving for \(x\). In this case, it results in the point \( (8, 0) \).
• The y-intercept, where the line crosses the y-axis, is found by setting \(x = 0\) and solving for \(y\). Here, it results in the point \( (0, 28) \).

By plotting these intercepts on the coordinate plane and connecting the dots, you can draw the line characterized by the equation. This line illustrates the direct relationship between \(x\) and \(y\).

Navigating the Coordinate System

The coordinate system, or Cartesian plane, is composed of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It helps in plotting points and displaying graphs of equations.Each point on this plane is identified by a pair of coordinates \( (x, y) \), which shows its position.In our linear equation example, the coordinates for intercepts are:

• x-intercept: \((8, 0)\) indicates that the graph crosses the x-axis at \(x = 8\).
• y-intercept: \((0, 28)\) shows that the graph crosses the y-axis at \(y = 28\).

Understanding this grid system is crucial because:

• It allows for visual interpretation of algebraic solutions.
• It's a key tool for solving equations graphically.
• This system lays the foundation for more complex topics like inequalities and calculus.

Using the coordinate system, you can visually analyze and comprehend the behavior of equations and their solutions.